Hot electron spectroscopy and...

INSTITUTE OF PHYSICS PUBLISHING REPORTS ON PROGRESS IN PHYSICS

Rep. Prog. Phys. 67 (2004) 1863–1914 PII: S0034-4885(04)72190-9

Hot electron spectroscopy and microscopy

J Smoliner, D Rakoczy and M Kast

Institut fur Festkorperelektronik,TU-Wien, Floragasse 7, A-1040 Wien, Austria

E-mail: [email protected]

Received 28 June 2004Published 27 August 2004Online at stacks.iop.org/RoPP/67/1863doi:10.1088/0034-4885/67/10/R04

Abstract

Semiconductor heterostructures, such as double-barrier resonant tunnelling diodes andsuperlattices, are nowadays used for many applications. One very versatile and powerfulmethod to study electronic transport in heterostructures is hot electron spectroscopy. Hotelectron spectroscopy can be carried out in two complementary versions: device-basedtechniques usually employ so-called hot electron transistors (HETs), while ballistic electronemission microscopy (BEEM) uses a scanning tunnelling microscope (STM) as the source ofballistic electrons.

In this review, spectroscopic results obtained by these two methods are compared anddiscussed. It is shown that BEEM results are strongly influenced by electron refraction effects,while the behaviour of HET devices is dominated by inelastic scattering effects in the baseand drift region of the device. Thus, STM-based BEEM/S and HET-based spectroscopy aregenuinely complementary methods, which yield supplementary results.

(Some figures in this article are in colour only in the electronic version)

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1. Motivation

Hot electron spectroscopy is a valuable tool for studying transport in heterostructures. It can becarried out in two different ways: first, as device-based hot electron spectroscopy and second, asa scanning tunnelling microscope (STM)-based method usually referred to as ballistic electronemission microscopy (BEEM). As the activities in both fields are extremely broad, this articlecannot and does not intend to give a complete survey of the field. An excellent review on hotelectron spectroscopy was written by Heiblum [1], and more general topics on hot electrontransport can be found in the book by Balkan [2]. There is a recent review on BEEM byNarayanamurti [3] and two somewhat older review articles by Prietsch [4] and Bell [5] which,together, give an excellent overview about the developements in BEEM.

However, STM-based BEEM/S and HET-based spectroscopy are usually performed bycompletely different research groups, with only a little communication between the twocommunities. Therefore, till now the available literature still lacks a critical comparison ofboth techniques in order to identify their specific strengths and weaknesses. To make such acomparison, we can take advantage of the unique situation that at our institute experimentalactivities in device-based hot electron spectroscopy as well as BEEM have been carried out formany years. During this time, many complementary device-based hot electron spectroscopyand BEEM/S measurements were performed on the same types of heterostructure samples,which has lead to the rare opportunity of bringing together the expertise of researchers in thesetwo fields.

In this review, therefore, we want to discuss and compare the basic features of BEEM/Sand device-based hot electron spectroscopy, especially the properties of the respective hotelectron injectors. To show the spectroscopic strengths and weaknesses of both methodswe will discuss selected results on superlattice transport obtained by both techniques.To illustrate the possibilities of spatially resolved measurements, some BEEM results onsingle impurities embedded in AlAs barriers will be reviewed in the last section of thisarticle.

2. Introduction: state of the art in hot electron spectroscopy on heterostructures

2.1. Semiconductor heterostructures

In modern semiconductor physics, heterostructures play a key role in the development of newdevice concepts and in fundamental scientific research. In heterostructures two semiconductorswith different material properties, i.e. different band gaps, are grown on each other. Dependingon the alignment of valence and conduction bands, potential steps occur at the interface.Various parameters, such as the composition of the compound semiconductors involved, thelayer thicknesses or the doping concentrations in principle allow the engineering of any desiredpotential profiles and band structure properties of a heterostructure sample [6].

With the invention of growth techniques such as molecular beam epitaxy (MBE) [7, 8]and metallorganic chemical vapour deposition (MOCVD) [9] in the 1970s, it became possible togrow semiconductor layers on an atomic scale. This was the starting point for the developmentof heterostructure-based quantum structures. Standard quantum mechanics textbook exampleslike potential barriers or quantum wells could now be fabricated simply by growing multilayersof alternating semiconductor materials.

The first experimental demonstrations of quantum physics in heterostructures werepublished in the early 1970s. A clear manifestation of the quantum-size effect in the opticalspectra of GaAs–AlGaAs quantum wells was demonstrated by Dingle et al [10]. The group of

Hot electron spectroscopy and microscopy 1865

Esaki studied the tunnelling current in biased GaAs/AlGaAs double-barrier heterostructures[11] and the observed maxima in the current were associated with resonant tunnellingprocesses.

In 1970, Esaki and Tsu [12] proposed the possibility of achieving a negative differentialconductivity (NDC) in one-dimensional, man-made crystals fabricated by growing periodicstructures with layers of alternating semiconductor materials. In such superlattices a parabolicband would break into so-called minibands, separated by small forbidden gaps and havingBrillouin zones determined by the superperiod. This proposal opened up a new and quicklyemerging research field; since up to then solid state physics was limited to conventional crystalsin which the fundamental parameters like lattice constants, band gaps, effective masses of thecharge carriers as well as their mobilities, etc are predefined by nature. The large numberof design parameters now made it possible to fabricate superlattices with superperiods in thenanometer range featuring narrow minibands. In the last three decades a lot of effort hasbeen put into the exploitation of semiconductor superlattices in order to study both opticaland electron transport properties in great detail. Probably, the most sophisticated devicewhich emerged from this research activity is the superlattice-based quantum cascade laser(QCL). Based on the idea of generating stimulated emission in superlattices, the group ofCapasso developed a unipolar cascade laser in a superlattice structure, which operates in themid-infrared regime [13].

Another interesting feature in heterostructure device technology is the possibility ofreducing the dimensionality of quantum confined systems. Two-dimensional electron gases(2DEGs), quantum wires (1D-systems) or quantum dots (0D-systems) can be achieved by(i) special growth techniques like cleaved edge overgrowth, modulation doping and self-assembly of dots, by (ii) standard etch processes or by (iii) using gate electrodes which locallydeplete a two-dimensional electron gas to achieve 1D and 0D structures. Based on suchlow-dimensional electron systems, fundamental physical effects like the quantum hall effect(QHE) [14, 15] have been discovered which have had a great impact on modern solid statephysics.

2.2. Ballistic electron transport in heterostructures

Since device dimensions have reached the size of typical mean-free-paths (mfps) of electronsin crystals, ballistic electron transport effects have gained importance in semiconductor devicephysics. The development of heterostructure-based hot electron transistors (HET) as perfectlydesignable playgrounds for ballistic electrons facilitated the investigation of nonequilibriumelectron transport in semiconductor bulk and heterostructure materials. In 1981, Heiblum [16]proposed a family of novel three-terminal devices, which are based on the injection of a quasi-monoenergetic hot electron beam used to study electron transport in the transit region of thedevices. This proposal was supported by Hesto et al [17], who presented numerical simulationsof a n+nn+ device for ballistic and quasi-ballistic electron spectroscopy.

Somewhat later, Hayes et al [18] studied hot electron transport through heavily dopedGaAs layers utilizing a HET based on Shannon’s camel transistor [19]. Supported byMonte Carlo calculations, the mfp of hot electrons in the base layer was determined to bearound 400 Å. This led to the conclusion that in such devices electron–electron scattering iscomparable in strength to the very efficient longitudinal optical phonon scattering process.In a subsequent work of this group, Levi et al [20,21] provided evidence of quasi-ballistic hotelectron transport through heavily doped GaAs layers.

As a substantial step forward in hot electron spectroscopy, Heiblum et al [22] introducedthe tunnelling hot electron transfer amplifier (THETA), which allows a systematic study of


hot electron effects in GaAs/AlGaAs-based heterostructure devices [23]. The energetic widthof the ballistic electron beam was measured to be about 60 meV for hot electrons with excessenergies of about 300 meV above the thermal electrons. In the following, hot electron transferfrom the � valley into the L valleys of GaAs in the presence of hydrostatic pressure was alsoreported [24].

In 1987, the THETA device was used to study electron interference effects [25] in the widequantum well formed by the emitter barrier and the collector barrier. The strong modulation inthe emitter current provides information about the nonparabolicity coefficient of bulk GaAs.In the same year, Levi et al [26] presented results of electron transport dynamics in wide GaAsquantum wells in two-terminal structures. They found that the lifetime of � electron states indouble-barrier resonant tunnelling structures depends on the well width, longitudinal optical(LO) phonon scattering and the transfer into the subsidiary X and L minima.

The THETA device was also used to demonstrate single optical phonon emission in theGaAs transit region as well as in the AlGaAs collector barrier [27]. The LO phonon scatteringtimes were estimated to be 200 fs for 85 meV electrons in n+-type GaAs and 550 fs for 40 meVelectrons in undoped AlGaAs. In the same year, Bending et al [28,29] published a quantitativestudy of the behaviour of an injected electron distribution in a magnetic field. They showedthat the dependence of the distribution on a perpendicular magnetic field can be describedquasi-classically, but with an effective mass 2–3 times larger than one would expect. Thediscrepancy was supposed to be due to a large degree of momentum randomization in theelectron distribution incident at the collector barrier of the THETA device.

Brill and co-workers used the THETA device to study electron heating effects in highlydoped GaAs. The injected hot electrons heat the cold electrons which are confined in thethin, doped GaAs layer of the device. This manifests itself in a collector current which islarger than the input current [30, 31]. Later, they also reported on ballistic electron transportstudies in high-purity GaAs at low temperatures [32, 33]. The mfp was determined to beseveral microns long for electrons with energies just below the LO phonon emission thresholdin GaAs. The dominant scattering mechanism in this energy regime was suggested to be due toimpact ionization of neutral impurities. It was also found that the mfp scales roughly inverselywith the impurity concentration.

In 1990, Choi et al [34] presented a study of quantum transport and phonon emission ofnonequilibrium hot electrons utilizing a HET. The energy distribution of the injected electronbeam was measured using both single and double-barrier analysers. The experimental resultsprovided evidence of single-particle interference in current transport and the existence ofphonon replicas in the hot electron distribution.

Ballistic hole transport was also demonstrated utilizing a hot-hole transistor [35]. Theenergetic width of the ballistic hole beam was determined to be 35 meV and the mfp forballistic holes was found to be 14 nm. The resonances in the injected current were used tosupport the light nature of the holes. In a subsequent paper the mfp of light holes in slightlydoped GaAs was measured to lie in the range of 300–360 nm, provided that the energy wasbelow the LO phonon threshold. The dominant scattering mechanism in this energy regimewas stated to be interband elastic scattering via ionized impurities [36].

Another application for hot electron transport is the investigation of higher valleys inheterostructures. For AlAs single barrier structures it was shown by hot electron spectroscopy,that the transferred electrons undergo strong inelastic scattering within the AlAs barrier andrelax down through the ladder of X point subbands before being re-emitted into the baselayer [37]. Note that lateral hot electron transport and hot electron noise measurements aswell as hot electron electroluminescence spectroscopy can also be used to study intervalleyscattering rates [38, 39].


Finally, Lyon et al [40] introduced ballistic electron luminescence spectroscopy (BELS),which involves the injection of ballistic electrons into a p-type semiconductor at lowtemperature together with the measurement of the luminescence produced when the electronsrecombine with neutral acceptors. This technique was successfully utilized for ballistic electrontransport in GaAs, in order to observe multi-phonon scattering of high energy electrons, and todetermine band offsets of heterojunctions. Later, similar experiments were also realized usinga STM as an emitter for ballistic electrons [41, 42].

2.3. Hot electron transistors

Besides the aspect of utilizing hot carrier transistors to study hot carrier effects in their baseand transit regions, the technique of hot electron spectroscopy gained much interest in the late1980s to exploit the knowledge about quasi-monoenergetic, ballistic electron transport in orderto study tunnelling structures like resonant tunnelling diodes and superlattices.

The first attempt to incorporate a resonant tunnelling structure into a HET as an analyserwas reported by Capasso et al [43] in 1987. In this work, a double-barrier heterostructureis implemented in a p-i-n heterojunction where the emitter consists of a wide-gap p+ layer.This device concept allows information on the hot electron energy distribution to be obtaineddirectly from the measured resonant tunnelling collector current, without requiring the use ofderivative techniques as requested in conventional hot electron spectroscopy.

Later, Choi et al [44] realized this device concept on the basis of the unipolar THETAdevice. The differential conductance G = dI/dVE showed features which were attributedto coherent tunnelling through the double-barrier structure. A second device, incorporatinga single barrier, was used to measure the shape of the hot electron energy distribution. In asubsequent paper the authors presented a model which allows one to calculate the energydistribution of the hot electron beam, taking optical phonon emission and plasmon emissioninto account [45]. They found that the relative contribution of these two mechanisms dependson the base doping density ND .

England et al [46,47] presented the first HET incorporating a GaAs/AlGaAs-superlatticebetween the electron injection barrier and the collector barrier. The superlattice was heavilydoped in order to be used as the base contact. Using this device the authors observed weakfeatures in the injector tunnelling characteristic which were attributed to hot electron transportthrough superlattice minibands.

In 1990, Vengurlekar et al [48] presented a study of hot electron transport through short-period InP/InGaAs-superlattices using an n-p-n bipolar transistor. Using the same device,Beltram et al [49] reported scattering induced tunnelling of hot electrons through biasedsuperlattices in 1990, too. At a fixed injection energy of the hot electron beam, currentresonances appear for all alignments of Wannier–Stark states with the electron injectionenergy.

In 1992, Kuan et al [50] demonstrated that the hot electron distribution of an electron beamcan be used as a probing tool to determine the band structure of a semiconductor superlattice.The results fit quite accurately to miniband positions calculated using a transfer matrix method.Inspired by this work, Rauch et al [51] introduced a three-terminal device, based on the THETAdevice of Heiblum et al [27], which performed hot electron spectroscopy with an energeticresolution of 25 meV. Using this device, several experiments have been carried out successfully,like mapping superlattice miniband positions and widths [51], quenching of miniband transportdue to external electric fields [52, 53], the observation of coherent and incoherent electrontransport in undoped GaAs/AlGaAs-superlattices [54, 55], and ballistic transport in magneticfields [56].


In a complementary experiment, Petrov et al [57] used BELS [58] to map the energypositions of minibands in short-period superlattices. The luminescence results provide precisemeasurements of the ballistic electron energies, while the electrical transfer ratio is sensitive tothe peaks and valleys in the transmission through the miniband. The combination of optical andhot electron spectroscopy eliminates the uncertainty in translating applied voltages to electronenergies, which occurs in semiconductor tunnelling experiments.

HETs are also interesting for practical applications. On InP, Miyamoto et al [59] presentedHETs with a buried metal gate for high speed applications. The highest estimated cutofffrequency of their devices is approximately 1 THz.

Magnetic applications can also be realized with HET devices. The unique properties ofmagnetic tunnel junction (MTJ) devices have led to the development of an advanced, highperformance, non-volatile magnet random access memory (RAM) with a memory cell densityapproaching that of dynamic RAM and read–write speeds comparable to static RAM. Both,giant magnetoresistance (GMR) and MTJ devices are examples of spintronic materials, inwhich the flow of spin-polarized electrons is manipulated by controlling the orientation ofmagnetic moments in magnetic thin film systems via external magnetic fields. Latest resultson three-terminal hot electron magnetic tunnel transistors suggest that there are also otherpromising applications of spintronic materials [60–62].

Finally, infrared detectors can also be realized on the basis of HETs. For instance,Yao and co-workers [63] reported an infrared HET (IHET) with extremely high detectivity,D∗ > 7.5 × 1012 cm2 √

Hz/W at 4.2 K .

2.4. STM-based hot electron microscopy/spectroscopy of heterostructures

Possibly motivated by the promising first results of device-based hot electron spectroscopy,it was also tried to combine the benefits of hot electron spectroscopy with the outstanding spatialresolution of a tunnelling microscope. As a result of these attempts, ballistic electron emissionmicroscopy/spectroscopy (BEEM/S) was introduced by Bell and Kaiser in 1988 [64, 65].BEEM/S is a three-terminal extension of STM, and utilizes an STM tip to inject hot electronsinto a semiconductor via a thin metallic base layer. In this way, an energetically tunableelectron injector is obtained, which in principle can be used on any semiconductor samplefor spectroscopic purposes. On specially designed samples incorporating an avalanche diode,even single electron sensitivity can be achieved [66].

The functionality of STM-based BEES is the same as in device-based hot electronspectroscopy, but with higher flexibility in the material systems used, and including imagingcapabilities. However, the advantage of spatial resolution has to be paid with a reducedenergetic resolution, as we will show later.

Initially, BEEM has been used to conduct Schottky barrier and band structure character-ization in various material systems, including technologically important semiconductors suchas Ga(Al)As [67–69], Ga(In)P [70–72], Ga(As)N [73–75], Si [76, 77] and SiC [78].

A recent example for a band structure investigation with BEEM was given by Reddyet al [79]. They investigated GaNxP1−x samples for various nitrogen concentrations betweenzero and 3.1%. For x �= 0, they observed a fine structure in the ballistic spectra, whichdepended on the nitrogen concentration. This phenomenon was interpreted as a splitting inthe degeneracy of the X valley due to the nitrogen induced, intense perturbation of the GaPlattice.

Another huge area of applications for BEEM was opened up by the investigation ofsemiconductor heterojunctions. For instance, the band offsets of various semiconductor–semiconductor interfaces have been measured [69, 80]. Additionally, the transport


behaviour through single barriers [81, 82], resonant tunnelling double barriers [83, 84] andsuperlattices [85, 86] have been studied extensively.

The first report of the direct detection of quantum confined states in buried AlGaAs–GaAs–AlGaAs double-barrier resonant tunnelling diodes (DBRTDs) was given by Sajoto andco-workers [83]. Quasi-bound states and band structure effects were described in dependenceof the temperature using cryogenic BEEM for temperatures in the range of 77–300 K. Themeasured BEEM thresholds were found to be in good agreement with the energetically favouredtransmission levels yielded by calculations. However, theoretical studies (e.g. [88]) as well astunnelling and BEES experiments on Al(Ga)As-based DBRTDs show, that a simple descriptionusing merely the conduction band edge of the heterostructure is not always sufficient to explainfully the transport properties in BEEM experiments. As a consequence, intensive theoreticalstudies were carried out. The latest ones can be found in [89–92].

Especially on GaAs-based samples it was found, that the influence of higher valleyscannot always be neglected. The first tunnelling experiments on this topic were carried outby Mendez and co-workers [93, 94], who observed resonant tunnelling via X point statesin AlAs–GaAs–AlAs and AlGaAs–GaAs–AlGaAs heterostructures. Later, a DBRTD in theX valley band profile of an AlAs–GaAs–AlAs–GaAs–AlAs structure was demonstrated byShieh and Lee [95]. With BEEM, the positions of higher conduction band minima had alreadybeen determined on bulk GaAs as well as on single AlAs barriers, before the general interestfocused on the ballistic transport behaviour of RTDs on the GaAs/AlGaAs [64,69,96–99], butalso on the InAs/AlSb material system [100].

With further progress in the fabrication of nano-structured devices also one- and zero-dimensional structures became available, i.e. quantum wires and quantum dots. For theinvestigation of such structures it is even more important to have a local probe techniquewith nanometer resolution.

For instance, Eder et al [101] and Rakoczy et al [102] studied quantum wires fabricatedby optical lithography. To demonstrate that BEEM is a powerful tool for the investigation oflaterally patterned structures, they used GaAs/AlGaAs heterostructures and superimposed alateral pattern of 200 nm wide wires with a pitch width of 800 nm. Finally, the semiconductorwas partially removed by wet-chemical etching to form the protruding wires. BEEM imagesdepicting the wires in good quality could be obtained in this way.

A different source for low-dimensional structures is provided by self-assembled quantumdots. A quite elegant way of producing such quantum dots is provided by high strain epitaxy,e.g. on InAs/GaAs [103,104] and on Ge/Si(001) [105]. Under certain growth conditions, a thinlayer of material deposited on a substrate with a high lattice mismatch will undergo a processof self-organization and form so-called self-assembled quantum dots (SAQDs). This methodis capable of delivering large arrays of quantum dots with a quite narrow size distribution anda reproducible (opto)electronic behaviour—an essential prerequisite for the development ofquantum dot lasers [106, 107]. To mention another example, (self-assembled) quantum dotscan be used to fabricate single electron transistors (see, e.g. [108]), which can, for instance, beexploited in high-density computer memories.

As quantum dots are currently a field of high common interest, they are also studied byBEEM. On the InAs system Rubin et al [109] took BEEM images of InAs dots buried under a75 Å thick GaAs cap layer and obtained a good contrast between the dots and ‘off-dot’ regions.Rakoczy et al [110,111] studied the band offset between InAs ‘on surface’ dots and the GaAsunderneath. For GaSb self-assembled quantum dots on GaAs [112–114], the type II bandlineup results in a barrier in the conduction band instead of a potential well as in InAs dots.As this results in spatially indirect transitions in optical experiments, reliable information onthe band offset cannot be obtained by optical methods. Traditional transport experiments can


also not be used. Therefore, BEEM is the only experiment providing local information on theconduction band profile of GaSb dots.

Other material systems of interest are phosphide related quantum dots. They are especiallyinteresting, since phosphide materials are widely used for optical applications in the visiblespectrum [115–117]. Again, BEEM is extremely useful to determine the band offsets on thedots as well as higher energy levels within the dots. Therefore, BEEM yields information onthe expected optical properties of these dots.

Quantum dots on Si were also studied by BEEM. As an example the work of Klemencet al [118] shall be mentioned here, which is concerned with BEEM on Ge self-assembled dotsembedded in Si. Klemenc and co-workers used MBE grown Ge dots on a Si(100) substrate,which were capped with a 10 nm thick epitaxial Si layer and an additional 3 nm thick layer ofCoSi2. They found that the inhomogeneous strain causes a change in the electronic surfacestructure, resulting in a lowering of the surface band-gap on top of the buried Ge dots. Detectionof these dots by BEEM was possible, although the STM images yielded no visible surfacecontrast for the two-fold buried dots.

Another field, where BEEM shows its strength in spatial resolution is the investigationof buried dislocations and defects. Here, several material systems were investigated, e.g.InGaAs/GaAs heterostructures [119, 115], which are interesting for pseudomorphic highelectron mobility transistors, but also silicide systems like NiSi2 [120, 121] and CoSi2[122–129].

BEEM was also applied successfully to metal–insulator–semiconductor (MIS) structures,to study the transport related insulator properties on a microscopic scale, for instance, in buriedCaF2/Si [130], SiO2/Si [131] and also in aluminium oxide junctions [132, 133]. BEEM wasfurther used to shed light on quantum interference effects and the distribution of trapped chargesin SiO2-based metal-oxide-semiconductor structures (MOS) [134–136].

Another interesting application of BEEM was presented recently by Kurnosikov et al[137], who performed ballistic transport studies of the barrier properties of tunnel junctionsgrown without any auxiliary Schottky barrier. They used Co–Al2O3–Ru tunnel junctions asBEEM samples and measured an effective barrier height of 1.7 V. Further they observed, thata continuous current injection into a single point of the junction increased the local barrierheight. As possible causes for this phenomenon charging effects and degradation of the barrierstructure were suggested. Additionally, they found first evidence of pinholes directly showingin a BEEM image by increased transport through these areas.

With the increasing interest in the field of spintronics, the magnetic properties of Co–Cuthin films [138] and magnetic nanostructures [139, 140] were studied by BEEM. In theliterature, this method is sometimes referred to as ballistic electron magnetic microscopy(BEMM). Under ultrahigh vacuum conditions it was demonstrated by Lu et al [142] that it ispossible to image magnetic domains with nanometer resolution In the following, other groupsused this technique to determine hot electron attenuation lengths in magnetic films [143].Magnetic multilayers [144] and embedded ferromagnetic films were studied by BEEM, too.

For a broader survey of the widespread activities in BEEM/BEES, the review byNarayanamurti and Kozhevnikov [3] is strongly recommended.

3. Basic principles of hot electron spectroscopy

In hot electron spectroscopy, electrons with kinetic energies high above the thermal energykBT (‘hot’ electrons) are used to probe the electronic states of heterostructure samples.If the electrons stay unscattered during this process, they are often referred to as ‘ballistic’


A

A

Ic

It

Vt

n-type collector

STM-tip

metal base

xz

e-

y

Figure 1. Sketch of a BEEM set-up with a Schottky diode as a sample (not to scale). The tunnelvoltage Vt is applied between the STM tip and the metal surface of the sample, the tunnel currentIt in this circuit is kept constant by the STM feedback loop. A second amperemeter measures theballistic current Ic between the metal base and the semiconductor collector.

electrons. All sources of ballistic electrons reported so far utilize a three-terminal transistor-like configuration, where electrons tunnel from an emitter electrode into a thin base electrode.Between the base and the subsequent collector, another single barrier or a more complex barrierstructure has to be placed in order to separate the base and the collector electrode. If the kineticenergy of the tunnelling electrons is high enough and the base is so thin that at least a certainpercentage of the electrons can traverse the base without being scattered inelastically, thoseelectrons are eligible to overcome the collector barrier and can be transferred ‘ballistically’ intothe collector region of the sample. Two configurations are typically used: one utilizes a STMto establish the injector structure (BEEM), while the other one uses MBE grown injectors,which are directly included within the sample under investigation (HET device).

As already mentioned at the very beginning, the aim of this review is a critical comparisonof both techniques in order to identify the strengths and weaknesses of hot electron spectroscopyusing HET devices on the one hand and the STM based BEEM/S technique on the other hand.For this purpose, we have to discuss the basic principles of both techniques first.

3.1. BEEM/S

The configuration using a STM as an emitter for ballistic electrons is frequently referredto as ballistic electron emission microscopy/spectroscopy (BEEM/BEES). STM with itsspectroscopic and imaging capabilities has been a well-established measurement techniquefor many years now. Since a lot of textbooks exist on this topic, its principle is not describedhere. One can find an introduction to STM, for example, in [145–147]. For BEEM/S, thesample has to have a conducting top (base) electrode. An additional contact on the samplebackside acts as collector electrode for ballistic electrons. A typical sample is sketched infigure 1. Most BEEM samples are semiconductor structures with a well-conducting metallayer at the top acting as a base. A frequent choice for the base material is Au. To facilitate themeasurement of the ballistic electrons, an n-type semiconductor has to be used as a collector.In this case, the natural band bending accelerates the ballistic electrons in the semiconductoraway from the metal–semiconductor interface and therefore prevents them from leaking backinto the base. To record a BEEM spectrum, the STM tip is kept at a fixed xy-position whilethe tunnel voltage is varied within a chosen interval and the ballistic current in dependence ofthe tunnel voltage, Ic(Vt ), is measured at this position in constant current mode.


Ef

Efcollector

STM-tip

e-

eVt

eVb

vacuum barrier

Ez

z

base

IcVb

Vt

Ic

Figure 2. Left part: schematic energy diagram of a BEEM experiment on a Schottky diode.Electrons tunnel from the STM tip into the base. If eVt > eVb , electrons will be able to surmountthe Schottky barrier and enter the semiconductor. Right part: ballistic current spectrum Ic(Vt )

corresponding to the energy diagram on the left (for varying Vt ). Defining zero by the Fermi levelof the base, the current sets in at Vt =Vb .

The conduction band profile of a simple Schottky diode sample is shown in figure 2,together with the corresponding ballistic current.

Due to the applied tunnel voltage, with the tip under negative voltage with respect tothe base, the potential for electrons lies much higher in the STM tip than in the base. Thus,electrons entering the base by tunnelling through the vacuum barrier have energies high abovethe Fermi level in the base metal, i.e. they are so-called ‘hot’ electrons. As long as theenergy provided by the tunnel voltage, eVt , is smaller than the Schottky barrier height, noelectrons will be transferred into the semiconductor, thus the ballistic current is zero. Graduallyincreasing Vt finally leads to a situation where some of the injected electrons can surmount theSchottky barrier and progress through the conduction band of the semiconductor. This causesa measurable ballistic current Ic at the collector electrode. The higher the tunnel voltage, themore electrons will contribute to Ic.

Keeping the position of the STM tip constant in xy with respect to the sample surfaceand varying the tunnel voltage in a certain interval while recording the corresponding ballisticcurrent represents the spectroscopic part of BEEM, often also explicitly denoted as BEES. Theplot of Ic(Vt ) is usually called BEEM spectrum. Note that in the graph for the ballistic currentdepicted in figure 2 the axes are swapped compared to the usual representation, to illustratemore clearly the connection with the energy diagram on the left.

3.2. Hot electron transistors

If the use of a STM as a hot electron emitter is not possible or desired, purely device-basedhot electron emitters can also be realized. Such HET devices are powerful tools to studynonequilibrium electron transport in semiconductor bulk and heterostructure material systems.In figure 3 the conduction band profile of a tunnelling hot electron transfer amplifier (THETA)is shown. The picture was adapted from [22] and represents the standard design of an all-semiconductor unipolar HET. Between the emitter and the base a thin layer of a wide-gapmaterial acts as a tunnelling barrier. By applying a negative emitter voltage electrons are


200 250 300 350 400 450

-0.1

0.0

0.1

0.2

0.3

0.4

Ef

AlxGa

1-xAs

InjectionBarrier

IB

IE I

C

AlxGa1-x AsCollectorBarrier

Emitter

Base Collector

Ene

rgy

(eV

)

Distance (nm)

Figure 3. Conduction band profile of a typical THETA device under forward bias. Quasi-monoenergetic electrons are injected into the base region of the device. Depending on the collector-base bias VCB and the emitter–base bias VEB, electrons either reach the collector contact or relaxdown to the base contact.

injected into the base region of the device to form a quasi-monoenergetic ballistic electronbeam. The barrier placed between the base and the collector is used as a ‘spectrometer’, whichalso prevents thermal electrons in the base from flowing into the collector.

While traversing the base region the injected electrons undergo different transportmechanisms. Electrons which have been scattered at least once and thus have suffered energylosses or direction changes modify the energy distribution of the hot electrons and can contributeas ‘quasi-ballistic’ electrons to the collector current. By analysing the hot electron energydistribution, it is, therefore, possible to study ballistic and quasi-ballistic electron transport inthe base region of the device.

It should be pointed out, that the situation is completely different in the BEEMconfiguration. In the BEEM configuration with its metallic base layer, there is an additionalcriterion that only electrons under a very small angle of incidence will be allowed to enter thesemiconductor (see section 4.1). Any scattering process of an electron within this ‘acceptancecone’ will change its angle of incidence in a way so that it can no longer enter the collectorelectrode, and is therefore lost. As a consequence, the observed energetic distribution ofelectrons entering the collector will be close to the theoretically predicted distribution forgenuinely ballistic electrons. However, the total amount of electrons contributing to Ic will bevery small.

Unlike in BEEM measurements, the collector current in HET devices measured as afunction of the emitter–base bias VEB is not very useful for data analysis in hot electronspectroscopy. The reason for this lies in the nature of the STM, which keeps the tunnellingcurrent constant, regardless of the applied bias. Thus, the emitter current stays always thesame. In HET devices this is not the case. Here, the tunnelling current through the emitterbarrier exponentially depends on the emitter bias. To get comparable results with BEEMdata, a transfer ratio α = Ic/IE has, therefore, to be defined. Further, it can be shown thatthe portion of electrons arriving at the collector with a vertical component of the kineticenergy (Ez) equal to the collector barrier height is given by the differential transfer ratioα = dIc/dIE . This technique, which appears in literature as hot electron spectroscopy orballistic electron spectroscopy, was mainly used to demonstrate quasi-ballistic [20] and ballisticelectron transport [23], as well as to probe scattering mechanisms [27], nonparabolicity [25]and intervalley transfer [24] in the base region of the device.


Au GaAs

vxyGaAs

vxyAu z

y

Au-

GaA

sin

terf

ace

vzGaAs

vzAu

e–

e–

Au GaAs

z

y

Au-

GaA

sin

terf

ace

e–

e–

(a) (b)

Figure 4. (a) Electron refraction: by crossing the interface from Au to GaAs, the electron will gainvelocity parallel to the interface and lose velocity perpendicular to it. (b) Total reflection: If theincident angle with respect to the z-axis is larger than the critical angle, the electron cannot enterthe GaAs and is reflected.

4. Ballistic currents and transfer ratios in BEEM and HETs

We now consider the ballistic collector currents in BEEM and HET devices in a morequantitative way. Most non-atomistic models used to calculate the ballistic current are basedon the assumption, that during the transition through an interface the momentum parallel tothe interface, kxy , is conserved. Already in the initial work of Bell and Kaiser [64, 65] theconservation of the momentum parallel to the interface was assumed in order to derive amodel which described the experimental data. Although the ‘parallel momentum conservationlaw’, as it is often called, was initially derived for an idealised quantum mechanical system, itsoon became clear, that this conservation law is in fact valid for the vast majority of actuallyconducted BEEM experiments [148–150] as well as for most experiments with HET devices.

4.1. Electron refraction effects in BEEM

The fact that the xy-momentum is conserved during the passage through the interface has far-reaching implications in BEEM. Typical BEEM samples exhibit a huge potential step betweenthe base and the collector (caused by the Schottky barrier) as well as strongly different effectivemasses in the involved areas. As a consequence, electron ‘refraction’ effects are observed,which are described in detail below. Note that for refraction effects the influence of the potentialstep is usually even more significant than the mass difference (see (5)). In contrast to BEEM,the effective masses are more or less constant in HET devices and also the large potential stepcaused by the metal–semiconductor interface is missing. Therefore, electron refraction effectsare not an issue in HET devices.

As a typical example for electron refraction effects, figure 4 illustrates the situation at theAu–GaAs interface. The effective electron mass in the gold is equal to the free electron mass,m0, while the effective electron mass in the semiconductor, m∗, is considerably smaller (forGaAs: m∗ = 0.067m0). To quantify the consequences of this situation, we first write downthe energy of the motion parallel to the interface:

Exy = E(kxy) = h2k2xy

2m(z)= h2

2m(z)

(k2x + k2

y

). (1)

Using the relation given in equation (1) for the energy associated with the motion in thexy-plane and the momentum conservation law for kxy , the energy component parallel to the


interface changes according to:

EGaAsxy = EAu

xy

m0

m∗ . (2)

Another limitation is the conservation of the total energy E, which can be written as:

E = h2k2

2m(z)+ Epot = h2

2m(z)(k2

x + k2y + k2

z ) + Epot = Exy + Ez + Epot. (3)

Therefore, the energy associated with the movement vertical to the interface is given by:

EGaAsz = E − EGaAs

xy − Eb = E0 + EAuz + EAu

xy − EAuxy

m0

m∗ − Eb, (4)

EGaAsz = EAu

z − EAuxy

(m0

m∗ − 1)

− eVb0, (5)

where eVb0 = Eb−E0 is the height of the potential step at the interface, with E0 the conductionband minimum in the metal and Eb the conduction band minimum in the semiconductor. Thefirst subtrahend in this expression represents the decrease in Ez due to the change of effectivemass, while the second one just originates from the potential step.

For the particular case of an Au–GaAs interface, where the effective mass in the GaAsis just 6.7% of the electron mass in the Au layer, EGaAs

xy is almost 15 times higher than theenergy parallel to the interface in the gold. Electrons crossing the Au–GaAs interface will gainExy according to equation (2) and lose Ez according to equation (4), Therefore, they will berefracted away from the z-axis, as depicted in figure 4(a). Only those electrons which haveexactly a perpendicular angle of incidence at the interface, i.e. kxy = 0, will not undergo anyrefraction.

A direct consequence of this refraction away from the z-axis is, of course, the possibilityof total reflection at the Au–GaAs interface. Considering the configuration of a BEEMexperiment, it is obvious, that only electrons with kGaAs

z � 0 can travel through thesemiconductor and finally be collected at the backside of the heterostructure, i.e. contribute tothe ballistic current. This automatically gives an upper limit for kxy and, therefore, a maximumangle of incidence (with respect to the z-axis) for an electron in the base.

To calculate this critical angle for total reflection, one can use equation (3) to isolateEz and take this in turn to find an expression for the momentum component vertical to theinterface, kz:

kAuz =

√2m0

h2 (E − E0) − k2xy, (6)

kGaAsz =

√2m∗

h2 (E − Eb) − k2xy. (7)

From kGaAsz � 0 and equation (7) it follows directly, that:

k2xy � 2m∗

h2 (E − Eb). (8)

The angle of incidence with respect to the z-axis can be described by

sin(θAu) = kxy

kwith k = |k| =

√2m0(E − E0)

h2 . (9)

By combining relation (8) and equation (9), the critical angle is determined by:

sin2(θcrit) = m∗

m0

E − Eb

E − E0= m∗

m0

E′ − eVb0

E′ , (10)


with E′ = E − E0, the total electron energy referring to the conduction band minimum in thegold base, and eVb0, as defined above, the height of the potential step. In a typical experimentalsituation where the material in front of the interface is represented by the base layer of theBEEM sample, it is more convenient to refer all energies to the Fermi level in the base layerrather than to the conduction band minimum. This can be achieved by the simple substitutionof eV = E

′ − Ef and eVb = eVb0 − Ef leading to a critical angle of:

sin2(θcrit) = m∗

m0

eV − eVb

eV + Ef

, (11)

where eVb, now, is the usual Schottky barrier height and Ef is the position of the Fermi levelwith respect to the conduction band minimum in the base layer. eV + Ef is the total energy ofthe incident electron, e.g. provided by a tunnel voltage.

Electrons are only able to cross the interface, if their angle of incidence, θ , fulfils therequirement:

sin(θ) � sin(θcrit). (12)

Electrons with a larger angle of incidence will be reflected at the interface, as depicted infigure 4(b). In a BEEM experiment, this condition defines the opening angle of the so-calledacceptance cone for electrons in the metal film, which selects a fraction of those electrons,which have overcome the tunnel barrier, for further transmission into the semiconductor.

Taking typical values for a BEEM experiment on an Au–GaAs interface, one can seenicely the influence of the effective mass change: assuming a Schottky barrier height of 1 eVand an injection energy of 1.1 eV, the critical angle without considering the effective masswould be about 18˚. In contrast, taking the change in effective mass into account, the criticalangle will be just 4.5˚.

These small critical angles have a crucial effect on the lateral resolution of BEEM. Onlyelectrons within the acceptance cone will determine the lateral resolution, because only thosewill be able to enter the semiconductor and therefore to contribute to the ballistic current.Assuming a point-like electron source at the top of the base layer, the minimal lateral resolutionis determined by:

�x = 2d tan(θcrit), (13)

where d is the thickness of the base layer. Taking the critical angle calculated above, thelateral resolution will be 16 Å for a 100 Å thick base layer and 11 Å for a 70 Å thick one. Forinjection energies closer to the Schottky barrier height, the critical angle and, therefore, thelateral resolution will be even smaller. However, these numbers are, of course, only valid forthe lateral resolution at the metal–semiconductor interface. For buried structures, the lateralresolution will decrease with the depth of the buried feature.

Note that the determination of the lateral resolution by the acceptance cone further has theconsequence that scattering in the base layer will usually not deteriorate the lateral resolution.Scattering in the base will just lead to a general decrease of the ballistic current, because it willremove electrons from the acceptance cone by changing their angle of incidence.

Finally, because it will be needed later, also the restriction on the initial energy componentparallel to the interface shall be written down here explicitly. From equation (8) andEAu

xy = h2k2xy/2m0 it immediately follows, that:

EAuxy � m∗

m0(E − Eb). (14)

Expressing this in terms of EAuz rather than in terms of the total energy, by using E =

EAuxy +EAu

z +E0, and, again further replacing Eb directly by the barrier height, Eb = eVb0 +E0,


one obtains:

EAuxy � m∗

m0(EAu

xy + EAuz + E0 − eVb0 − E0), (15)

EAuxy � m∗

m0 − m∗ (EAuz − eVb0). (16)

Or, again referring to the more common Schottky barrier height with respect to the Fermi levelin the Au base, by substituting eVb0 = eVb + Ef :

EAuxy � m∗

m0 − m∗ (EAuz − Ef − eVb). (17)

4.2. The Bell–Kaiser model

When introducing BEEM, Bell and Kaiser presented a formula for the modelling of the ballisticcurrent, which has been widely utilized in BEEM/BEES since then [64]. Their first step indeveloping their model was to use the well-known formalism for tunnelling between planarelectrodes as an approximation. For simplicity, the STM tip and the base layer are assumedto be identical metals. Further, heterostructures incorporated into the collector, are not treatedby the original Bell–Kaiser Model.

At T = 0, electrons tunnelling from the STM tip to the metal base occupy tip stateswithin a half-shell (because of the restriction kz > 0) of the Fermi sphere between E = Ef

and E = Ef − eVt . Within the framework of this model, the tunnel current can be writtenas [64, 151]:

It (Vt ) = 2eA

(2π)3

∫d3kTtb(Ez)

hkz

m0[f (E) − f (E + eVt )]. (18)

For convenience, here as well as in the text that follows, all energies are referring to the STMtip conduction band minimum. The BEEM sample is assumed to be energetically loweredwith respect to the tip by a positive tunnel voltage Vt applied to the base layer of the sample.A is the effective tunnel area, f the Fermi function and Ttb(Ez) the tunnelling probability.Note that the expression hkz/m0 is the velocity component parallel to the z-axis.

Provided that the barrier between tip and base at Vt = 0 is a square barrier of height �

and width s, and will be distorted into a trapezoidal shape by applying a tunnel voltage, thetunnelling probability given by the WKB model can be approximated by [152]:

Ttb(Ez) = e−αs√

Ef +�−(eVt /2)−Ez (19)

with α = √8m0/h = 1.024 eV−1/2 Å−1.

Substituting the integral over the wave vector by integrals over the energy componentsassociated with kxy and kz yields:

It (Vt ) = C

∫ ∞

0dEzTtb(Ez)

∫ ∞

0dExy[f (E) − f (E + eVt )] (20)

with the constant C = 4πAm0e/h3.A similar expression can be directly obtained for the collector current. However, due to the

conservation of the total energy and the xy-momentum at the metal–semiconductor interface,additional restrictions apply to those tip states which can contribute to the collector current.The first one originates from the requirement, that the electrons must have enough energyto surmount the Schottky barrier between the base layer and the semiconductor ballistically.Because here, as mentioned above, Ez refers to the tip conduction band minimum and the


conduction band minimum in the base layer lies at −eVt with respect to the one in the tip, theminimum Ez which facilitates a ballistic entry of the semiconductor is:

Eminz = Ef − eVt + eVb. (21)

The second limit has its origin in the refraction at the metal–semiconductor interface and isgiven by equation (17). Again, one has to bear in mind that the energies now refer to the tip,and therefore:

Emaxxy = m∗

m0 − m∗ (Ez − Ef + eVt − eVb). (22)

Using these two restrictions as integration limits, one can express the collector current as:

Ic(Vt ) = RC

∫ ∞

Eminz

dEzTtb(Ez)

∫ Emaxxy

0dExy[f (E) − f (E + eVt )]. (23)

R is a measure of the attenuation due to scattering in the base layer. The attenuation lengthfor ballistic electrons in metals is in very good approximation energy independent within theenergy ranges used in BEEM experiments for the measurement of Schottky barrier heights,and, therefore, R shall be treated here as a constant.

To eliminate the prefactor C, which is usually unknown in an experiment, Bell andKaiser [64] suggested normalizing Ic(s0, Vt ) by It (s0, Vt ) for each voltage Vt . The expressionfor Ic then takes the following form, commonly known as the Bell–Kaiser formula:

Ic(Vt ) = RIt0

∫ ∞Emin

zdEzTtb(Ez)

∫ Emaxxy

0 dExy[f (E) − f (E + eVt )]∫ ∞0 dEzTtb(Ez)

∫ ∞0 dExy[f (E) − f (E + eVt )]

, (24)

where It0 is the constant tunnel current at which the ballistic spectrum is measured. The Bell–Kaiser formula is well suited for fitting ballistic electron spectra on simple Schottky diodes ina tunnel voltage range of up to ≈200 mV above the onset, i.e. eVb. Many authors use the Bell–Kaiser model even up to tunnel voltages of ≈600 mV above the onset. However, for such largeranges usually a noticeable deviation has to be taken into account. At higher tunnel voltagesnumerous scattering processes occur and energy dependent influences become important.

4.3. Heterostructure extension for the Bell–Kaiser model

As first shown by Sajoto et al [83], BEEM can also be used to investigate buried hetero-structures. To calculate the ballistic current through such structures, Smith and Kogan [153]introduced an extension of the initial description of the ballistic current. Essentially, theirmodel is a modification of the original Bell–Kaiser model and results in the simple extensionof the Bell–Kaiser formula (equation (24)) by an additional transmission coefficient Ths whichdescribes the properties of the heterostructure [153]:

Ic(Vt ) = RIt0

∫ ∞Emin

zdEzTtb(Ez)Ths(Ez)

∫ Emaxxy

0 dExy[f (E) − f (E + eVt )]∫ ∞0 dEzTtb(Ez)

∫ ∞0 dExy[f (E) − f (E + eVt )]

. (25)

Besides Ths, all variables are defined just as in equation (24). The integration limitsare again given by equations (21) and (22). Note that Ths does not only describe thetransmission behaviour of the buried heterostructure, but must also include the quantummechanical reflections in the region between the metal–semiconductor interface and the buriedheterostructure. The total coefficient Ths is usually calculated by a transfer matrix method, asdescribed in section 4.4.

It must be pointed out that the above formula only accounts for the coupling of thetransverse and the longitudinal energy via the upper integration limit Emax

xy . While this is


not a problem for simple Schottky barriers, the proper choice of Emaxxy for samples with burried

tunnelling structures is not straightforward.In our opinion, a much more instructive way to include this coupling, is to calculate the

transmission through the BEEM sample, starting at the metal–semiconductor interface, independence of both, Exy and Ez:

Ic(Vt ) = RIt0

∫ ∞Emin

zdEzTtb(Ez)

∫ ∞0 dExyThs(Ez, Exy)[f (E) − f (E + eVt )]∫ ∞

0 dEzTtb(Ez)∫ ∞

0 dExy[f (E) − f (E + eVt )]. (26)

In this expression Emaxxy no longer has to be declared explicitly, because this limit is now

implicitly included via the dependence of Ths on Exy . Note that the transmission coefficient ofthe vacuum barrier and the transmission of the heterostructure should not be combined into anoverall transmission coefficient since this leads to quantum interferences between the vacuumbarrier and the collector barrier when calculating Ic explicitly. In principle, such interferencescould exist, but in practice they are not observed in BEEM, because of the scattering processesat the non-epitaxial Au–GaAs interface and the electron–electron scattering in the Au-baselayer. Further details on the calculation of Ths can be found in section 4.5.

It should be mentioned that the prefactor R, which is usually used as fitting parameter, canbe calculated quantitatively when metal–semiconductor interface induced scattering processesare taken into account (MSIS model) [98]. Including the anisotropy of the effective mass,the energy dependence of the electron mfp in the metal base, and finite temperatures, BEEMspectra were calculated quantitatively with surprising accuracy within this model and withoutneeding further fitting parameters [154].

4.4. Transfer ratio analysis for HET devices

Like in BEEM/S the basic idea in device-based hot electron spectroscopy is to utilize anelectron beam with a narrow vertical energy distribution as a probing tool to study bandstructure and transport properties of heterostructures. For the data analysis, however, slightlydifferent procedures are applied. Usually only the ratio between the collector and the emittercurrent is of interest (see section 2.3). This so-called static transfer ratio of a semiconductorheterostructure is given by the convolution [50]:

α(Ez) =∫ ∞

0F(E′

z − Ez)T (E′z) dE′

z, (27)

where F(E′z − Ez) denotes the vertical energy distribution of the injected hot electron beam

and T (Ez) the energy dependent transmission function of the heterostructure.Although this looks much simpler than in BEEM/S, this is not the case since inelastic

scattering processes complicate the analysis in device-based hot electron spectroscopy.In contrast to that, inelastic scattering by LO phonons is usually not an issue for typicalBEEM samples.

The main reason for this somewhat surprising behaviour can be found in the phononscattering time, which has been determined to be ≈185 fs [27] in undoped GaAs. Lookingat typical heterostructure samples investigated by BEEM, like buried double-barrier resonanttunnelling diodes and superlattices [86,155,156], one can see that the classical electron transfertime through the active region of the corresponding samples is usually only up to ≈100 fs andis, therefore, well below the phonon scattering time. Thus, LO phonon emission will not occurin the active region of the sample. Phonon absorption is suppressed by the low temperatures,at which the experiments are usually carried out.


200 300 400 500 600 700 800 900-0.1

0.0

0.1

0.2

0.3

0.4

α=Ic/I

E

IC

IE

Minibands

SuperlatticeGaAs/Al

0.3Ga

0.7As

Al0.3

Ga0.7

AsInjection barrier

AlAs Etchstop layer

Drift region

Emitter

Base Collector

Ene

rgy

(eV

)

Distance (nm)

Ef

Ef

Figure 5. Conduction band profile of a HET incorporating a semiconductor superlattice betweenthe base and the collector. The etch stop layer has just technological purposes and is necessary toestablish the base contact to the device without generating a shortcut to the collector layer.

In contrast to that, phonon scattering in device-based hot electron spectroscopy is almostinevitable. The reason for this is twofold: First, the electron kinetic energies used in device-based hot electron spectroscopy are normally much lower than in BEEM, since one usuallytries to stay near the edge of the conduction band and in HET devices the huge potential stepbetween the base and the collector is usually missing. Second, the so-called drift region betweenthe emitter barrier and the collector barrier is usually relatively long to avoid the formationof quantized states and quantum resonances between those barriers. Thus, the transfer timesthrough the active region of HET devices easily reach values comparable to the phonon emissiontimes. For instance, for the sample shown in figure 5, the electron transfer time is approximately500 fs.

To illustrate the strong influence of inelastic scattering in device-based hot electronspectroscopy, we briefly consider the following three-terminal device: figure 5 shows theconduction band diagram of a typical HET [51] incorporating a short-period superlattice. Thesuperlattice in the collector region has 5 periods consisting of 2.5 nm AlGaAs barriers and6.5 nm GaAs wells (results taken from [157]). The mechanism of electron injection equalsthat of the THETA device: a ballistic electron beam is generated at a tunnelling barrier andreaches the superlattice after traversing a thin, highly doped n-GaAs base layer and a slightlyn-doped drift region. By tuning the energy of the ballistic electron beam with the emitter biasVEB it is possible to probe the band structure and the transport properties of the superlatticeat a fixed collector–base bias VCB. The probability of an injected electron to be transmittedthrough the superlattice is given by the static transfer ratio α = Ic/IE .

The corresponding transfer ratio is shown in figure 6. The calculated position of thefirst miniband is indicated with dotted lines. For energies below the first miniband the ballisticelectrons are reflected at the superlattice and relax down to the base contact. These electrons donot contribute to the collector current Ic. The onset at VE = −45 mV indicates the beginning ofelectron tunnelling through the first miniband. Increasing the emitter bias leads to an increasein the transfer ratio up to a critical emitter bias where the maximum overlap between the hotelectron energy distribution and the miniband is achieved. At higher emitter bias the electronbeam is reflected at the minigap, which separates the first and the second miniband, and thus,the transfer ratio decreases.


-0.02 -0.04 -0.06 -0.08 -0.10 -0.12 -0.14 -0.16 -0.18

0.00

0.01

0.02

0.03

0.04

1st M

inib

and

Phonon Replica

Tra

nsfe

r R

atio

α=I c

/I E

VE (V)

Figure 6. Measured transfer ratio of a 5-period superlattice consisting of 2.5 nm AlGaAs barriersand 6.5 nm GaAs wells. The calculated position of the first miniband is indicated with dotted lines.The first peak is attributed to ballistic electron tunnelling through the first miniband. At higheremitter bias phonon replicas contribute as inelastic background.

The following peaks in the transfer ratio are due to electrons which have been scattered byLO phonons while traversing the drift region and have lost n · hωLO = n · 36 meV (n = 1, 2)during the scattering processes. The transfer ratio does not drop to zero in between these peaks,due to the overlap of the peaks.

This is due to the fact that in HETs longitudinal optical phonon emission is the mostefficient scattering process at liquid helium temperatures. At energies above the phononthreshold (36 meV in GaAs), electrons which are injected into the drift region emit LO phononsand therefore reduce their kinetic energies by the amount of the phonon energy. In the measuredtransfer ratio these quasi-ballistic electrons contribute as inelastic background.

Consequently, inelastic scattering plays a major role in HET devices. Therefore, thetransfer ratio has to be written as a superposition of a ballistic part and an inelastic backgroundformed by phonon replicas. In equation (27) this can be included by writing the hot electronenergy distribution as a sum consisting of ballistic electrons and electrons, which lose energyby sequential phonon emission. The index i denotes the index of the emitted phonon.

F = Fball +∑

i

F iph. (28)

In hot electron spectroscopy Fball is utilized as a probing tool to study electron transportproperties in heterostructure-based resonant tunnelling structures like superlattices. Thephonon replicas contribute as unwanted inelastic background. Therefore, it is important tohave detailed knowledge about the shape and the width of the hot electron energy distribution.

4.5. The transfer matrix method

In the description of BEEM spectra as well as of HET transfer ratios the quantum mechanicaltransmission factor of the considered heterostructures has to be calculated. The method that isnormally used for this purpose is the so-called transfer matrix method (TMM) [158, 159]. Inthe following we outline the principles of this method including the modifications necessaryto account for the electron refraction effects mentioned earlier.

The transfer matrix method can be used to calculate the transmission of an arbitrarypotential shape by relatively simple means. Because it is based on the approximation of an


Vj

zj-1z0 zj zn-1z

V(z

)

Figure 7. Approximation of an arbitrary potential profile by a piecewise constant potential. Thej th section is located between zj−1 and zj . Note that the width of the single steps does not needto be equal for all subdivisions, but can be adjusted according to requirements.

arbitrary potential by a piecewise constant potential, it is especially well-suited to calculatethe transmission through such heterostructures which exhibit a band profile consisting ofrectangles, e.g. DBRTDs and superlattices.

For the following, consider a one-dimensional, arbitrary potential profile V (z). The givenpotential shape can be subdivided into several sections, where the potential and the effectivemass are assumed to be constant (see figure 7). Each potential change between two suchsections is taken as an ideal step function. Under these conditions, the Schrodinger equationfor each of these sections can be written down and solved easily. In the next step, those partialsolutions are connected together by using the continuity of the wave function and its derivativeto match the wave functions at each sampling point.

First, a band profile which is subdivided into n sections can be written as:

V (z) =n−1∑j=0

Vj with Vj ={

const for zj−1 < z < zj ,

0 elsewhere.(29)

The analogous description applies to the z-dependent effective mass, where mj is the (effective)mass in the j th region.

Using a stationary approach, the one-dimensional Schrodinger equation within the regionj (i.e. for a constant potential and a constant effective mass) is:

[− h2

2mj

d2

dz2+ Vj − Ez

]ψ(z) = 0. (30)

It must be pointed out that we first consider electrons with zero parallel momentum andtherefore with zero components of kinetic energy due to a motion parallel to the barriers. As inthis case the energy depends on kz only, it is, therefore, denoted as Ez. Electrons with non-zerokxy and electron refraction effects will be considered later in this section.

The general solution ψj in the region z ∈ [zj−1, zj ] is written as:

ψj(z) = Aj eikj z + Bj e−ikj z, (31)

with

kj =√

2mj(Ez − Vj )

h2 . (32)


The boundary conditions at the sampling point z = zj between the two adjoining sectionsj and j + 1 are:

ψj(zj ) = ψj+1(zj ), (33)

1

mj

∂ψj (z)

∂z

∣∣∣∣z=zj

= 1

mj+1

∂ψj+1(z)

∂z

∣∣∣∣z=zj

. (34)

For convenience, the following abbreviations are introduced:

uj (z) = Aj eikj z and vj (z) = Bj e−ikj z.

Now, the relations between uj and vj on the one hand and uj+1 and vj+1 on the other handcan be calculated from the boundary conditions (equation (33) and (34)), which read as:

uj (zj ) + vj (zj ) = uj+1(zj ) + vj+1(zj ), (35)

ikj

mj

uj (zj ) − ikj

mj

vj (zj ) = ikj+1

mj+1uj+1(zj ) − ikj+1

mj+1vj+1(zj ). (36)

Before we continue to describe the transfer matrix method, we have to discuss the influenceof non-zero kxy values on the relation between kj and kj+1.

If kxy is zero, kj and kj+1 are simply calculated by equation (32). For non-zero kxy ,the mass induced coupling between the parallel and vertical components of energy has to betaken into account. For better understanding, we first split the total energy E into a verticalcomponent Ez and an energy component parallel to the barriers, Exy . Thus, the total energyis: E = Exy + Ez. While kxy is always conserved, Exy is not. Calculating Exy in the regions(j ) and (j + 1) we get:

Exy,j = h2k2xy

2mj

, (37)

Exy,j+1 = h2k2xy

2mj+1. (38)

As the total energy has to be conserved, this simply means that Ez is no longer the samein the regions (j ) and (j + 1). The relation between Ez,j+1 and Ez,j is given by

Ez,j+1 = Ez,j + (Vj − Vj+1) + (Exy,j − Exy,j+1). (39)

From equation (39), the relation between kj and kj+1 for non-zero kxy can now be calculatedas earlier:

kj =√

2mj(Ez,j − Vj )

h2 , (40)

kj+1 =√

2mj+1(Ez,j+1 − Vj+1)

h2 . (41)

Having the relations between kj and kj+1, both for zero and non-zero parallel momentum,we can now write equation (36) in matrix form:(

uj (zj )

vj (zj )

)= M(j)

(uj+1(zj )

vj+1(zj )

), (42)

where

M(j) = 1

2

1 +kj+1mj

kjmj+11 − kj+1mj

kjmj+1

1 − kj+1mj

kjmj+11 +

kj+1mj

kjmj+1

. (43)


Note that the matrix M(j) does not depend on z. This information is solely contained in thevectors consisting of u and v.

Because the potential and the mass between two sampling points are constant, the wavefunction in this area can be written as:(

uj (zj )

vj (zj )

)= N(j+1)

(uj (zj+1)

vj (zj+1)

), (44)

where

N(j+1) =(

e−ikj+1�zj+1 00 eikj+1�zj+1

). (45)

The matrix N(j+1) depends only on the distance �zj+1 = zj+1 − zj between two samplingpoints, not on the location of a sampling point itself. The absolute position z is, again, onlycontained in the vectors consisting of u and v.

For the whole sequence of potential steps one can write:(u0(z0)

v0(z0)

)= M

(un(zn)

vn(zn)

), (46)

where M is the product matrix of all M(j) and N(j+1):

M = M(0) · N(1) · · · M(n−1) · N(n). (47)

Equation (46) links the waves incoming on the potential V (z) to the outgoing ones bymeans of a series of (2×2) matrices (equation (47)). The matrices Mj perform the connectionof two parts of the global wave function across an interface, while the matrices Nj+1 describethe propagation within a region of constant potential and constant (effective) mass.

Finally, to calculate the global transmission through the structure, one should demand,that vn = 0, which means, that on the outgoing side of the structure just a transmitted waveexists. With this, the global transmission can be simply described by

T (E) = |kn|m0

|k0|mn

|An|2|A0|2 = |kn|m0

|k0|mn

1

|M11|2 . (48)

Note that the transfer ratio of a barrier is defined as the ratio of the impinging and thetransmitted electron currents. As the current is a vector, and for non-zero kxy the current isnot impinging vertically onto the barrier, the ratio of the absolute values of the k-vectors,

|k| =√

k2z + k2

xy , has to be used in equation (48).

Further reading on the TMM and especially on its application to double-barrier structures,can be found, for instance, in [160, 161].

5. Measuring the hot electron energy distribution in HETs

5.1. Energy analysers in HET devices

The resolution of hot electron spectroscopy is directly related to the energetic width of the hotelectron beam generated at the emitter tunnelling barrier of a HET. In order to get a better insightinto the electron transport mechanisms in superlattices by hot electron spectroscopy [51,54] it isessential to resolve the individual states in a miniband separately. Therefore, the determinationand optimization of the energy distribution in a HET device as well as in BEEM is of greatestimportance.

To determine the energetic distribution in device-based hot electron spectroscopy, a specialHET device was designed, the conduction band profile of which is shown in figure 8. The layer


Figure 8. Conduction band diagram of a HET incorporating a highly optimized injector designbetween emitter and base, and a triple barrier RTD between base and collector as ‘energy filter’.

structure starts with a highly doped n+-GaAs collector contact layer (ND = 1 × 1018 cm−3)and the GaAs/AlGaAs-triple barrier, which is embedded between two drift regions. The driftregions are slightly doped (ND = 2×1014 cm−3) in order to avoid undesired band bending. Ascollector barrier, a triple barrier GaAlAs resonant tunnelling diode was chosen, which acts asband pass energy filter and analyser for the distribution of ballistic electrons. An advantage ofband pass filters is that the transfer ratio α = Ic/IE , which resembles the energy distribution,is obtained without using derivative techniques [1].

The layer structure continues with a 10 nm thin, highly doped n+-GaAs base contact layer(ND = 1×1018 cm−3), a 64 nm undoped GaAs spacer, and the AlGaAs injection barrier, whichwe also use as an etch stop layer. On top of the AlGaAs layer a 6 nm, undoped GaAs spacer isgrown, followed by a 80 nm n−-GaAs layer (ND = 1 × 1016 cm−3). The layer structure endswith a 300 nm thick, highly doped n+-GaAs emitter contact layer (ND = 1 × 1018 cm−3).

In this optimized structure, an energetically narrow and tunable electron beam is generatedat the tunnelling emitter barrier and reaches the energy filter after traversing the highly n-dopedGaAs base layer and the slightly n-doped drift region. Due to the transmission properties ofthe band pass filter the static transfer ratio α = Ic/IE is directly proportional to the energydistribution of the injected hot electrons.

For a better understanding, the design of the energy filter has to be discussed in moredetail. During our work on double-barrier resonant tunnelling diodes, which was carried outoutside the field of this review, we have found that the transmission behaviour of resonanttunnelling structures can be shaped by adding an ‘antireflection coating’ for electrons [162].The simplest version of a resonant tunnelling structure with antireflection coating consists of atriple barrier structure, as it is shown in the left inset of figure 9. The shown TBRTD consists ofthree Al0.3Ga0.7As barriers and two GaAs wells. The two well widths are chosen to be 4.2 nmin order to get a first resonant state (for the isolated well) at E0 = 100 meV. To optimize thetransmission properties of the analyser the central barrier (8 nm) is chosen to be twice as thickas the neighbouring barriers (4 nm).

To illustrate the functionality of this antireflection coating, figure 9 shows a comparisonof the calculated transmission coefficients T (E) of the triple barrier RTD (solid line) and adouble-barrier RTD (dotted line) with the same full width at half maximum of 1 meV centredat E0 = 100 meV. The transmission coefficients T (E) were calculated using a simple transfermatrix method including nonparabolicity. This transmission function of the triple barrier RTD


Figure 9. Transmission coefficient of a triple barrier RTD (——) and a double-barrier RTD (- - - -)calculated using a transfer matrix method. The energy window of the TBRTD shows much steeperedges compared to that of the double-barrier RTD.

Figure 10. Bound states in the drift region of the HET calculated using a Schrodinger/Poissonsolver.

shows much steeper edges compared to that of the double-barrier RTD, which makes it moresuitable for the use as an energy filter. The rectangular shape of the transmission coefficientT (E) arises from the coupling of the two neighbouring wells. Due to tunnel splitting two quasi-bound resonant states are obtained (at energies E0,1 = 99.93 meV and E0,2 = 99.96 meV),which are delocalized over the whole structure and which are separated by 0.03 meV. Thesuperposition of the two corresponding transmission peaks leads to the special shape of thetransmission coefficient [162, 163].

5.2. Parasitic quantum states in the base region of the HET device

One very inconvenient property of HET devices is the existence of quantum states in the driftregion between the emitter barrier and the heterostructure between the base and the collector.The drift region embedded between the tunnelling barrier and the triple barrier RTD representsa wide AlGaAs–GaAs–AlGaAs-quantum well. Calculating the bound states of this quantumwell gives the discrete spectrum shown in figure 10. Due to the epitaxial interface between thebase and the collector region, and the fact that electron–electron scattering in the base even at


Figure 11. I /V characteristics of the emitter–base subdevice. The inset shows a magnified part ofthe emitter current curve.

highest doping levels will be much lower compared to the scattering rates in the Au-base layersnormally utilized for BEEM, parasitic quantum interferences between the emitter and the baseare almost inevitable. At emitter biases where the quasi-monoenergetic hot electron beam isaligned with a bound state of the drift region the emitter current is expected to be enhanced,whereas at biases where the hot electron beam lies energetically between two bound states IE

is expected to be reduced.In the case where the energy distribution of the ballistic electron beam is narrow enough,

those states can actually be observed experimentally as oscillations in the emitter current. Infigure 11 we have plotted the measured emitter current as a function of the applied emitter biasVE for a typical sample. The emitter current shows an exponential dependence on the emitterbias, which originates from the roughly symmetric structure of the emitter–base subdevice,where an AlGaAs tunnelling barrier is sandwiched between two highly doped contact layers.The inset of figure 11 shows a magnified part of the emitter current curve. In this plot aclear modulation of the current is observed, which is attributed to quantum interference effectsoccurring in the drift region of the HET [25]. However, if only the transfer ratio is considered,these parasitic interferences cancel out almost completely provided they are not too large.

Note that the problem of quantized states between the emitter barrier and the collectorbarrier does not exist in BEEM. First, possible quantum interferences within the base do notoccur, because the base is polycrystalline and therefore the scattering rates are high. Second,possible quantum interferences between the Schottky barrier and the buried heterostructure aresuppressed, because the interface between the base and the GaAs is not epitaxial. Thus, theobservation of quantum interference effects in the base or in the drift region is quite unlikelyin BEEM experiments.

5.3. Transfer ratio of the triple barrier RTD

To determine the transfer ratio of our HET device, the emitter and collector currents aremeasured as a function of negative emitter bias at T = 4.2 K using an HP-semiconductorparameter analyser. The spectrometer, implemented by the triple barrier RTD, is used underflat band condition (e.g. zero bias between base and collector). The experimental results areplotted in figure 12. Again, the modulation of the emitter current arises from electron coherenceeffects occurring in the drift region of the device.


Figure 12. Emitter and collector current as a function of negative emitter bias. Weak modulationsare due to quantum interference effects occurring in the drift region of the device. Transport ofballistic electrons and phonon replicas through the triple barrier RTD is indicated with arrows.

Below the energy of the first resonant state no collector current is observed, since theelectrons that are injected into the drift region are reflected by the spectrometer and collectedin the base contact. This also indicates that no significant leakage current occurs betweenbase and collector under flat band condition. The first peak indicates resonant tunnelling ofhot electrons through the TBRTD. The following peaks in the collector current are due toelectrons which have been scattered by longitudinal polar optical phonons while traversingthe drift region and which have, therefore, lost multiples of hωLO during the scatteringprocesses.

The collector current shows a similar modulation effect as the emitter current. Sincethe collector current is the portion of the emitter current which has been transported throughthe spectrometer, this modulation is just transferred through the TBRTD. The transfer ratioα = Ic/IE is shown in figure 13. The onset of the transfer ratio at VE = −100 mV indicatesthe beginning of hot electron tunnelling through the first resonant state of the analyser. Thefirst peak of the transfer ratio is proportional to the injected hot electron energy distribution,and its full width at half maximum was measured to be � = 17.25 mV. The shape of thedistributions is slightly asymmetric with its maximum at the high-energy side. The peakposition (VE,max = −107 meV) indicates the maximum of the vertical energy distribution.Note that the oscillations in the emitter and collector current due to the quantized states in thebase region do cancel out in the transfer ratio.

Besides the width of the vertical energy distribution it is essential to study its shape.Dividing the first peak at the maximum into two parts one can define a high energy tail and alow energy tail. In the transfer ratio the high energy tail occurs at less negative emitter biasas it reaches the energy filter first while increasing the absolute value of the emitter bias. Theexperimental results show that the width of the high energy tail is smaller than the width ofthe low energy tail: �1 = 5 and �2 = 12.25 meV. The shape of the high energy tail is due toelectrons generated at the emitter tunnelling barrier whereas the shape of the low energy tail isdue to elastic scattering processes such as charged impurity scattering and electron–electronscattering in the base of the device. The low-energy tail is calculated to decay exponentially.This decay is not completely observed in the measured transfer ratios due to the superpositionof the ballistic peak with the inelastic background formed by the phonon replicas.


Figure 13. Measured transfer ratios as a function of negative emitter bias. The first peak resemblesthe hot electron distribution of the injected hot electron beam.

6. Measuring the hot electron energy distribution in BEES

Due to the special design of the emitter structure in a HET device, the ballistic collector currentincreases when the bias between the emitter and the base is increased, whereas the energeticdistribution of ballistic electrons stays virtually unchanged. Thus, the energetic distributionof ballistic electrons can be mapped by sweeping the emitter energy over the energy windowof the analyser. In BEEM, which is operated in constant It mode, this is not the case andthe energetic distribution of injected electrons changes strongly with emitter bias. Thus, for amapping of the energy distribution the analysing collector electrode has to be swept in energy,while the emitter bias stays constant.

The principle of using a buried double-barrier resonant tunnelling diode (DBRTD) as ananalysing filter was first demonstrated by Sajoto et al [83]. They found, that a resonant levelgives rise to a linear increase in the ballistic current Ic(Vt ). As first pointed out by Smolineret al [86], this phenomenon originates in the electron refraction at the Au–GaAs interface.This linear behaviour was also confirmed by other groups in various experiments and is sincethen regarded as typical for buried RTDs. For instance, Strahberger et al [84] presented adetailed experimental and theoretical study on AlGaAs–GaAs DBRTDs, showing the effectsof the kxy momentum conservation on the transmission through buried and sub-surface RTDs,respectively. Smoliner et al [86] also demonstrated the effect of the electron refraction on thetransmission through buried AlGaAs–GaAs superlattices. Later, the transmission behaviourof buried superlattices under the influence of an external collector bias was investigated andseveral data points in the Ic(Vc) relation were deduced from conventionally measured BEEMspectra (i.e. Ic(Vt )) for different values of Vc [87].

However, a direct measurement of the refraction, by mapping the energetic distribution ofthe ballistic electrons after their transfer through the interface, is extremely difficult in BEEMfor several technical reasons. Rather severe restrictions exist on the leakage currents and themechanical and electrical drift stability of the measurement set-up. A special sample design wasutilized, which allows both the application of a bias voltage between the base and the collectorand the performance of an energy spectroscopy by utilizing an AlAs–GaAs-AlAs DBRTD asa narrow energy analyser. The decision to use AlAs instead of AlGaAs barriers (which aremore common in BEEM on the Ga/Al/As material system) was based on the experience that at


Figure 14. Sketch of the experimental principle together with the � conduction band profile ofa typical flat band AlAs–GaAs-DBRTD sample. Vt denotes the tunnel voltage, Vc the collectorvoltage, It the tunnel current, Ic the ballistic current, and Ef the Fermi energy. The resonant levelof the double-barrier structure is schematically indicated by the horizontal bar within the RTD.Also illustrated is the decrease in the velocity component parallel to the z-axis from vAu

z to themuch smaller vGaAs

z and the consequential energy loss in Ez.

cryogenic temperatures the ballistic transmission behaviour of an AlAs–GaAs heterostructureis mainly determined by the � valley. This leads to a large barrier height of the AlAs barriersand, therefore, to good energetic filtering capabilities of the RTD combined with the advantageof low leakage currents.

In figure 14, a sketch of the experimental set-up together with the � valley conductionband profile of the sample is shown. The resonant level within the RTD is indicated by ahorizontal bar. As usual, a tunnel voltage Vt between the STM tip and the sample surface(i.e. the metallic base) is used to inject electrons into the sample, whereby the tunnel currentIt is kept constant during the measurements. Those electrons, which travel ballistically intothe semiconductor heterostructure and can either energetically surmount or tunnel throughthe RTD are measured via Ic. Thus, the onset in the ballistic current on this sample will bedetermined by the energetic position, where the Fermi level in the STM tip is aligned with thefirst resonant level inside the DBRTD.

For the experiments presented here, the tunnel voltage was always chosen in such a waythat the Fermi level in the STM tip, representing the maximum possible energy Ez of anyinjected electron, is still well below the barrier height of the RTD. Therefore, only electronswith an Ez suitable for a transmission through the resonant level of the RTD will be able toreach the collector and to contribute to Ic. Consequently, the RTD acts as an energy filter in Ez.

Although the sketch in figure 14 does show the additional voltage source which was usedduring the experiment for biasing the sample, the band structure depicted corresponds to acollector voltage of Vc = 0 V. Actually applying a collector voltage Vc �= 0 means to shiftthe Fermi level in the collector region and leads to a tilt in the band profile. With this tilt alsothe energetic position of the resonant level is shifted, and, therefore, the energy filter can beadjusted to select a certain Ez. By varying Vc continuously and recording the correspondingcollector current Ic, one can ‘scan’ the electron distribution in dependence of Ez.

Also illustrated in figure 14 is the impact of the electron refraction on the energy Ez.As discussed in section 4, during the transition of electrons through the Au–GaAs interfacethe momentum component kxy is conserved while the effective mass is changing. Since the


Figure 15. (a) Curve 1 (left axis): � valley conduction band profile of the sample used, calculatedfor Vc = 0 V. Curve 2 (right axis): effective mass profile. The sample surface is on the left,z-values <0 indicate the region of the Au base layer. The collector region (on the right side) isnot shown in full depth. (b) Transmission through the RTD shown in (a), calculated for kxy = 0,i.e. for electrons at the � point.

momentum is the product of effective mass and velocity, a constant momentum componentwith a changing mass will lead to a change in the velocity component parallel to the interface,as indicated in figure 14. Because the effective mass decreases at the Au–GaAs interface, theelectrons will gain velocity perpendicular to the growth axis. This, in turn, means a gain inthe energy component associated with the xy-component of the velocity, Exy . Because thetotal energy is, of course, conserved, the velocity parallel to the growth axis and the energyassociated with it, Ez, must change accordingly, i.e. Ez will decrease. One can show that thez-associated component of the energy in the GaAs is given by (see (5)):

EGaAsz = EAu

z − EAuxy

(m0

m∗ − 1)

− eVb0. (49)

Note that the first subtrahend in this expression represents the decrease in Ez due to thechange in effective mass, while the second one originates from the potential step between theAu and the GaAs, which, via this term, also contributes to the refraction. In summary thismeans, that a conversion of Ez into Exy takes place at the Au–GaAs interface. As a finalconsequence, those electrons which initially have too high a value of Ez to be transmittedthrough the resonant level of the RTD can lose enough Ez at the Au–GaAs interface so thatthey finally match with the position of the resonant level. Of course, those electrons whichhave exactly a perpendicular angle of incidence at the interface, i.e. kxy = 0, will not undergoany refraction.

To illustrate the sample design in a more quantitative way, curve 1 in figure 15(a) showsthe conduction band profile in the � valley of the heterostructure, including also the regionof the gold base layer. It was calculated using a program by Snider [164], which is based ona self-consistent solving of Poisson’s equation. As one can see clearly, the region betweenthe Au–GaAs interface on the one hand and the δ-doping (at z = 750 Å) on the other hand isvirtually flat. Behind the δ-doping starts a gradual potential decrease down to the level of thecollector region.

In curve 2 of figure 15(a), the size of the effective electron mass is plotted along the growthaxis. In the metallic base layer the effective mass is equal to the free electron mass, while inthe semiconductor region it is much smaller (0.067m0 in GaAs, 0.162m0 in AlAs).

In figure 15(b) we have plotted the calculated transmission through the � band profile ofthe semiconductor heterostructure for electrons with kxy = 0, i.e. with an angle of incidence


Figure 16. Ballistic spectra for different collector voltages. T = 10 K, It = 2 nA. Curves 1–5were measured at Vc = +50 mV, 0 mV, −50 mV, −100 mV, and −150 mV, respectively. A verticaloffset is added for better viewing. Line ‘a’ indicates the shift of the lowest resonant level in the �

valley, line ‘b’ the one of the L valley resonant levels. (Note that for technical reasons Vt is thevoltage at the sample base.)

which is perpendicular to the interface. The calculation was carried out employing the transfermatrix method (TMM) within the parabolic band effective mass approximation, as describedin section 4.5. The first resonant level of the RTD is represented by a quite sharp peak, whichlies at an energy of 1.2 eV, that means 200 meV above the GaAs conduction band edge.

The next resonant level has its maximum transmission approximately 900 meV abovethe GaAs conduction band edge, and is, therefore, well out of the range of interest for thisexperiment. Although this second peak is, mainly due to its close vicinity to the top of the AlAsbarrier, noticeably broadened, a significant increase in the calculated transmission does notappear until ≈1.85 eV. This excellent energetic isolation of the first resonant level together withthe possibility of applying an external bias voltage Vc to the heterostructure makes this sampledesign very well suited as a tunable energy filter for ballistic electron emission spectroscopy.

Note, as we have shown in our previous work [156], that the � band profile is indeed therelevant one for the BEEM experiment presented here and the amount of electrons comingfrom an overshoot of the AlAs X valley barrier can be neglected at low temperatures. The onlyvisible feature in the measured BEEM spectra at higher Vt is a second current onset which isdue to the resonant levels existing in the L valley of the DBRTD. However, this feature is alsoenergetically well isolated from the first resonant level in � and, moreover, less pronouncedthan the typical resonance of a level in the � valley.

6.1. Influence of Vc on the BEEM Spectra

We now study the dependence of the BEEM spectra on the collector voltage. As discussedearlier, applying an external voltage to the collector will tilt the band structure and thereforeshift the energetic position of the resonant level. On the other hand, the onset of the ballisticspectrum is determined by the energetic position, where the Fermi level in the STM tip isaligned with the first resonant level inside the DBRTD. Therefore, the onset in the BEEMspectra will shift accordingly to the applied collector voltage.

In figure 16, the influence of the external collector voltage Vc is illustrated by showingBEEM spectra for different values of Vc. Line ‘a’ in the figure is a guide to the eye to illustratethe onset shift more clearly. For positive values of Vc the onset voltage in the BEEM spectrais shifted to lower values of Vt , because in this case the resonant level is lowered in energy.


Figure 17. Ballistic current measured as a function of collector voltage Vc for a constant tunnelvoltage Vt . T = 10 K, It = 2 nA. Curves 1–4 were measured at Vt = 1.05 V, 1.10 V, 1.15 V,and 1.20 V, respectively. A vertical offset is added for better viewing. The straight line is a guideto the eye to indicate the current onset positions. Because negative collector voltages correspondto higher energetic positions of the resonant level, the horizontal axis is plotted from positive tonegative values of Vc .

For negative values of Vc the energetic position of the resonant level rises and therefore thecurrent onset is shifted to higher emitter bias values.

Though it is more difficult to perceive, because the onset for the transport through resonantlevels in the L valley RTD is only weakly pronounced, the resonant levels existing in the Lband do shift parallel to the � resonant level, as expected (line ‘b’ in figure 16).

From the amount of the onset shift one can determine which sample regions are affected bythe application of a collector bias Vc. As described earlier, the total thickness between the sam-ple surface and the collector is 1250 Å, while the DBRTD itself lies 600 Å below the surface.Thus, if the applied collector voltage dropped uniformly over the whole sample, the energylevels inside the RTD would be shifted by approximately eVc/2, where e is the electron charge.

However, the data in figure 16 indicate that the onsets of the spectra measured atVc = +50 mV and Vc = −150 mV are shifted by about 170 mV with respect to each other.This suggests, that the main part of the applied collector voltage does actually drop over theregion between the sample surface and the DBRTD, while the region between the δ-dopingand the collector plays just a minor role for the biasing.

6.2. Dependence of the ballistic current on Vc at constant Vt

For a direct mapping of the energy distribution of the ballistic electrons the ballistic currentis measured while the tunnel voltage Vt is kept constant and the collector voltage Vc is sweptover a certain voltage range.

As already mentioned above, varying Vc shifts the resonant level in the DBRTD in itsenergetic position Ez and, therefore, can be used to scan an energy range, detecting the numberof transmitted electrons in dependence of Ez. In contrast to superlattices, the transmissioncoefficient of an RTD remains almost unchanged when a collector bias is applied. Thus, for aconstant tunnel voltage, the ballistic current measured as a function of collector voltage directlyrepresents the energetic distribution in Ez of the ballistic electrons in the GaAs.

In figure 17 the result of this measurement is shown. As one can see, for high energeticpositions of the resonant level, i.e. for a large negative collector bias, the tunnel voltage is notsufficient to exceed the resonant level and, therefore, to cause a signal in the Ic(Vc) curves.


Figure 18. Calculated ballistic current (more precisely: (dIc/dEz)�Ez) as a function of Ez fora constant tunnel voltage Vt = 1.15 V. (a) Energy distribution of the ballistic electrons in the Aubase (within the acceptance cone). (b) Energy distribution of the ballistic electrons in the GaAs.

Decreasing the energetic position of the resonant level, i.e. going to more positive valuesof Vc, leads to a detectable ballistic current as soon as the resonant level is aligned with theFermi level in the STM tip, which is defined by the tunnel voltage. Thus, the onset in anIc(Vc) curve depends on the specific tunnel voltage used for the recording of the curve. This isillustrated in figure 17 by a comparison of curves for different Vt . As expected, higher valuesof Vt shift the current onset to more negative values of Vc.

Following the line of a single Ic(Vc) curve again, the ballistic current increases for morepositive values of Vc and finally saturates. Due to the limited range available in Vc just thebeginning of a saturation can be seen in the graph.

To provide a comparison with the experimental data and to facilitate a more detaileddiscussion, the amount of the ballistic current which flows in a small, fixed energy interval,(dIc/dEz)�Ez, was also determined by calculation. As already mentioned above, this quantityreflects the energetic distribution of the ballistic electrons as a function of Ez. In the experiment,this corresponds directly to the Ic(Vc) curves, where the energy interval �Ez is provided bythe width of the resonant level of the RTD.

For the calculation, an extended Bell–Kaiser model [64, 65], which fully includes thedescription of the electron ‘refraction’ [86, 153], was used (see section 4.3). To keep thecomputing time within reasonable limits, all scattering effects in the metal base as well asin the GaAs region in front of the DBRTD were neglected. To allow a comparison with theexperiment over the whole range of interest, a tunnel voltage of 1.15 V and a scanned energyrange from 0.9 to 1.2 eV were chosen as input parameters for the model. In figure 18 one cansee the results of the calculation, both for the electron distribution in the gold base and for theone in the GaAs, just after crossing the Au–GaAs interface.

In figure 18(a) one can see the current distribution from those electrons in the gold basewhich are able to enter the semiconductor, that is, the electrons within the so-called acceptancecone. Due to the influence of the transmission coefficient of the vacuum barrier between thetip and the Au base layer and due to the large difference in effective mass at the Au–GaAsinterface, which leads to a total reflection of all electrons outside the rather small acceptancecone, this distribution is quite narrow. Most of the ballistic current flows close to the Fermienergy in the STM tip, which is at 1.15 eV for the chosen parameters.

As figure 18(b) shows, the ballistic current distribution in the GaAs looks quite differentfrom the one in the Au base. Because of the refraction at the Au–GaAs interface, a largenumber of the ballistic electrons are transferred from high values of energy Ez in the Au filminto lower values of Ez in the GaAs, just as discussed earlier. As a result, the ballistic currentdistribution in the GaAs becomes extremely broad, with a maximum at rather low energies


and just a long slope which decreases slowly with increasing Ez, finally reaching zero at theFermi energy.

With decreasing energy, the ballistic current per energy interval first increases, thensaturates at Ez = 1.05 eV, and finally decays quickly for energies Ez below the Schottkybarrier height at the Au–GaAs interface (eVb = 1.0 eV). The ballistic current below theSchottky barrier originates from electrons which initially have an Ez > eVb (and, therefore,can surmount the Schottky barrier), but are transferred via refraction into energy regions beloweVb when crossing the Au–GaAs interface. As long as the band profile of the biased sampleis tilted downward, the conduction band minimum after the interface lies below the Schottkybarrier height, and thus provides the necessary free states at Ez < eVb, where the electronscan be refracted into.

A comparison of the measured Ic(Vc) curve for Vt = 1.15 V (curve 3, figure 17) withthe calculated current distribution in the GaAs (figure 18(b)) shows that the long slope of thecalculated distribution for increasing Ez is appropriately reproduced by the experiment. So isthe saturation behaviour for more positive Vc, which corresponds to lower energies in Ez. Onthe other hand, the calculated current decay for electron energies below the Au–GaAs Schottkybarrier height is not observed in the experiment.

This discrepancy is most probably due to the simplicity of the used model. For instance,all scattering effects in the base as well as scattering in the 600 Å drift region between theAu–GaAs interface and the DBRTD were neglected. However, scattering processes in themetal base were found to be important by several groups [165–167].

Furthermore, Vc does not drop linearly over the drift region for the whole bias range, whichmay also contribute to the observed deviation between the experimental and the calculated data.Although one can see from the shift of the current onsets in the ballistic spectra (figure 16) thatVc drops almost completely over the drift region for −150 mV < Vc < +50 mV, further exper-iments gave evidence to the fact that this is not the case for higher values of Vc. For higher Vc,indications of a nonlinear shift of the current onset position in dependence of Vc are observed.This implies that the total amount of the applied collector voltage is no longer effective in low-ering the energetic position of the resonant level and that, therefore, very low energies cannotbe accessed by mapping the current distribution using the RTD, which acts as the analysingfilter in this experiment. Further, due to the increased leakage current, the measured spectrarapidly become more noisy in this region, so that the data are no longer reliable enough.

7. Hot electron spectroscopy on superlattices

7.1. BEEM results

To compare the spectroscopic capabilities of BEEM and hot electron spectroscopy with HETdevices, we now discuss some spectroscopic results obtained on GaAs–AlGaAs superlattices.In our group, STM based ballistic transport through the lowest miniband of a buried GaAs–AlGaAs superlattice was studied already some time ago [155, 86]. In this work, we had shownthat the miniband transport results in a BEEM current threshold clearly below the height ofthe AlGaAs barriers, and that the measured and calculated miniband position in the GaAs–AlGaAs superlattice are in very good agreement [155]. Like the DBRTs discussed in theprevious sections, the miniband in the superlattice also acts as tunable energy filter, whereboth the position and the transmission of the miniband are a function of the applied biasvoltage Vc. The data are in excellent agreement with the model of Smith and Kogan [153],which indicates that the electron transport through the superlattice is highly coherent. As asample for this BEEM experiment we used a 10 period 25 Å (Al0.4Ga0.6As)/30 Å (GaAs) MBE


Figure 19. Conduction band diagram of the sample and schematic view of the experimental set-up.Ef is the Fermi energy, Vc the collector voltage.

Figure 20. (a) BEEM spectra of the superlattice sample recorded at Vc values of −200 mV, 0 mVand +200 mV, respectively. A vertical offset was added for clarity. The arrows indicate the onsetvoltage of the BEEM spectrum. (b) Second derivative of the BEEM current plotted for Vc voltagesof +200 mV, +100 mV, 0 mV, −100 mV, and −200 mV, respectively. The second derivative wasobtained numerically from the experimental data.

grown superlattice on top of 600 Å of undoped GaAs and a highly doped n-type collectorregion. The superlattice was followed by 300 Å of undoped GaAs before finally capping itwith an Au base layer. In order to provide ‘flatband’ conditions at the Au–GaAs interface, ap-type δ-doping (NA = 1.4 × 1012 cm−2) was inserted between the superlattice and the highlydoped collector region. The flatband condition in the superlattice regime is essential for theformation of a miniband. If the electric field across the superlattice is too high, the minibandbreaks up into localized states and ballistic transport through the miniband will be inhibited.This flatband approach was first used by Sajoto et al [83] as well as by O’Shea et al [170] whostudied single and double AlGaAs barriers buried in the semiconductor.

A self consistently calculated conduction band profile of this sample is shown in figure 19together with a sketch of the experimental set-up. Due to the sample design, only the lowestminiband lies energetically in the superlattice. All other minibands are energetically in thecontinuum above the AlGaAs barriers. By applying an additional voltage Vc between thecollector and the base contact, the electric field in the superlattice regime can be changed.

In figure 20(a) we have plotted typical BEEM spectra measured at collector voltages ofVc = 0 mV, +200 mV and −200 mV, respectively. Two features are evident: first, the onset


Figure 21. (a) Transmission coefficient of the superlattice calculated by the transfer matrixformalism, plotted on a semi-logarithmic scale. As typical examples we have plotted thetransmission for collector voltages of +200 mV, 0 mV, and −200 mV, respectively (curve (1), (2),(3)). (b) Superlattice transmission integrated over the miniband width. For better comparison, thecalculated curves were scaled with respect to the experimental data. The dashed line is just a guideto the eye.

voltage for BEEM current detection is shifted with collector bias. For positive collector bias,the onset is shifted to lower STM bias voltages, for negative collector bias the onset is shiftedto higher STM bias voltages. As a second feature, the BEEM current is reduced as soon as acollector bias voltage is applied, independent of the bias polarity. At low temperatures, wherethe Fermi distribution function is sharp enough, the onset voltage of the BEEM spectrum marksthe energetic position, where the Fermi energy in the STM tip is aligned with the bottom of thesuperlattice miniband. Thus, the onset voltage is expected to change, if the miniband is shiftedin energy by applying a collector voltage to the sample, as already dicussed in section 5. Toanalyse the data more quantitatively, we used the model of Smith and Kogan [83]. A detaileddiscussion of the spectral features of these BEEM data is published in [86].

As the main result of the Smith–Kogan model, it is found that the second derivative ofthe BEEM current is directly proportional to the transmission coefficient of a sub-surfacesuperlattice. This can be used for a direct determination of the miniband position fromthe transmission as a function of collector voltage. In figure 20(b) the second derivative of theBEEM current is plotted for collector voltages of Vc = −200 mV, −100 mV, 0 mV, +100 mVand +200 mV, respectively. Close to the onset voltage of the BEEM current, in the region wherethe Fermi level in the STM tip is aligned with the superlattice miniband, the d2IBEEM/dV 2

curves show a well-resolved peak. As expected, the peak in the d2IBEEM/dV 2 curves shiftsto lower voltages for positive collector bias and to higher voltages for negative collector bias.In addition, the peak amplitudes decrease for both collector bias polarities. Note that thed2IBEEM/dV 2 curves completely go to zero outside the energetic regime of the miniband.Besides the decreasing d2IBEEM/dV 2 amplitudes for both collector bias polarities, this isanother indication that the energetic distribution of the ballistic electrons is close to idealityand that inelastic scattering by LO phonons can be neglected. In the equivalent experimentcarried out on HET devices, this is not the case. As we show later, the transfer ratio of HETdevices does not go to zero outside the energetic regime of the superlattice miniband due tothe strong influence of phonon replicas on the energetic distribution of ballistic electrons.

To analyse this behaviour quantitatively, we have calculated the miniband transmission asa function of electron energy using the transfer matrix method. The result is plotted on a semi-logarithmic scale in figure 21(a) for Vc values of −200 mV, 0 mV and +200 mV, respectively.In contrast to the experiment, the individual states of the miniband are clearly visible, which isdue to the fact that scattering was not included into the calculation. The calculated miniband


Figure 22. Calculated conduction band profile of a HET incorporating a 5-period superlattice.The hot electron energy distribution is used as a probing tool to map the energetic structure of thesuperlattice. The energy of the hot electron beam can be tuned independently of the superlatticebias condition. (a) Sample A—old injector, (b) sample B—optimized injector.

positions are quantitatively in good agreement with the experimentally measured peaks in thed2IBEEM/dV 2 curves, which indicates that the self-consistent calculation of the conductionband profile is adequate and that flatband conditions are really achieved at zero collector bias.Although, due to the simplicity of the model, the measured decrease of the transmission is notreproduced, the behaviour of the total transmission, i.e. the transmission integrated over theminiband width, is nicely described by the model. This can be seen in figure 21(b). The solidline represents the calculation, the dashed line connecting the experimental data is just a guideto the eye. In the integrated miniband transmission, the calculated collector bias dependenceis well-reproduced by the experimental data. The observed decrease in the total transmissionfor both collector voltage polarities and the good agreement with the calculated transmissionis an indication that inelastic scattering in the superlattice by phonons can be neglected. In thecase of scattering, for positive collector voltage the scattered electrons would be driven intothe collector by the external electric field, while for negative collector voltage they would bedriven back into the Au-base. Therefore, scattering would lead to much higher integratedtransmissions for Vc > 0 than for Vc < 0.

As this is not observed, we think that inelastic scattering within the finite superlattice playsa minor role in this experiment and that the transport through the superlattice is genuinelyballistic. This is further supported by the fact that the transfer time through the superlattice isbelow 0.1 ps, and thus below the typical timescale for LO phonon scattering.

7.2. Results obtained with the HET device

To give a brief and concise demonstration of the strength of HET devices, we will restrictourselves to the results obtained on short-period superlattices.

In the following, three different samples are presented. The first one (sample A) representsthe original injector designed by Rauch et al [51,54] applied to a 5-period superlattice, whichis placed between the base and the collector. Figure 22(a) shows the conduction band profileof this HET. The superlattice consists of 3.5 nm Al0.3Ga0.7As barriers and 3 nm GaAs wells.In figure 23(a) the conduction band profile of the superlattice is shown in detail. The firstminiband is positioned between 122 and 158 meV, which results in a miniband width of� = 36 meV. The energetic position of the lower edge of the miniband is chosen to lieabove 100 meV in order to avoid undesired leakage current through the structure. The spacing


Figure 23. Superlattices between the base and the collector of a HET device. The left sides in bothfigures show a schematic conduction band profile, the right sides the corresponding transmissioncoefficients in the energy range of the first miniband calculated using a transfer matrix method.Sharp transmission peaks indicate the energy positions of individual states. (a) 5-period superlattice(samples A and B), (b) 4-period superlattice (sample C).

of the individual miniband states (6–11 meV) was designed in a way to be comparable to theenergetic width of the hot electron beam generated by the improved electron injector. Thesecond miniband already lies above the AlGaAs barrier height.

In the course of our work the injector design was optimized for energetic resolution. Theimprovements achieved can be seen in the samples B and C. Sample B (see figure 22(b))consists of the optimized injector design together with the the same 5-period superlattice as insample A.

Sample C consists of the optimized injector design and a 4-period superlattice, which isshown in figure 23(b). It consists of 4 nm Al0.3Ga0.7As barriers and 3.2 nm GaAs wells. Thefirst miniband is positioned between 122 and 143 meV. As a result the miniband width equals� = 21 meV and the spacings between the individual states (4–9 meV) are smaller than in the5-period superlattice.

Using two types of superlattices (samples B and C) providing different numbers of periodsand different separation of the individual miniband states allows us to study the resolution ofthis spectroscopy method as well as the hot electron transport inside the superlattices. Theconduction band profile of sample B is shown in figure 22(b). Using the superlattices underflatband conditions (Vc = 0), the transfer ratios of samples A, B, and C are plotted in figure 24.Below the energies of the first miniband, no collector current is observed, since the ballisticelectrons are reflected at the superlattice. The onsets at VE ≈ −120 mV indicate the beginningof hot electron tunnelling through the first miniband. Comparing the transfer ratio of sample Ato the transfer ratios of samples B and C, the increase of the resolution is clearly evident.The transfer ratio of sample A does not show any feature in the energy range of the firstminiband whereas, due to the optimized injector design, the transfer ratios of samples B and Cshow detailed features which can be attributed to hot electron transport through the individualminiband states. At energies above the first miniband, the ballistic electrons are reflected at thesuperlattice due to the minigap which separates the first and the second miniband. However,the transfer ratio does not drop to zero in this energy range due to the overlap with phononreplicas generated in the GaAs drift region.

Table 1 presents the results for the 5- and 4-period superlattices. Comparing the calculatedenergies of the individual states with the measured peak positions shows that the experimentalresults exceed the calculated energies up to 25 meV. This can be explained by considering(i) the energy offset �E = eVE − eVE,max between the applied emitter bias and the maximum


Figure 24. Measured transfer ratios of the 5- and 4-period superlattices using the original injectorin sample A and the improved injector in samples B and C. Hot electron transport through theindividual states is indicated with arrows. At higher emitter bias, the phonon replicas contribute asinelastic background.

Table 1. Calculated (E) and measured (−eVE ) energy positions of the individual states in the firstminiband of the 5- and 4-period superlattice.

5-period SL 4-period SL

Index E (meV) −eVE (meV) Index E (meV) −eVE (meV)

1 122 130 1 122 1362 128 138 2 126 1453 138 150 3 135 1574 149 162 4 143 1685 158 173

of the hot electron distribution and (ii) a possible deviation between the nominal and realsuperlattice parameters due to the growth accuracy of the molecular beam epitaxy (about onemonolayer).

Since �E is supposed to be constant in the energy range of the first miniband, it cancelsout when deriving the energy spacings

Eij = (Ei + �E) − (Ej + �E) = Ei − Ej . (50)

The measured energy spacings between the individual states fit best to calculated spacingsusing superlattice parameters of 3.3 nm AlGaAs barriers separated by 2.9 nm GaAs wellsfor the 5-period superlattice and 3.7 nm AlGaAs barriers with 3 nm GaAs wells for the4-period superlattice, respectively. These deviations to the nominal superlattice parameterslie within one monolayer for GaAs and AlGaAs. The reduced barrier and well widths resultin broader minibands which are shifted to higher energies. The calculated first miniband,using these modified parameters, is positioned between 123 meV and 166 meV (� = 43 meV)for the 5-period superlattice and between 124 and 156 meV (� = 32 meV) for the 4-periodsuperlattice. The smallest energy spacing resolved in the experiments is E21 = 8 meV in the4-period superlattice. Thus, the resolution of the device is found to be �E � 8 meV.

After the energetic structure of the superlattice miniband has been determined, we can nowtry to reconstruct the energetic distribution of those electrons which are transmitted throughthe superlattice coherently. For this purpose, we first split the transmission coefficient of the


superlattice into a coherent and an incoherent component:

T (Ez) = Tcoh(Ez) + Tincoh(Ez). (51)

As is common in HET spectroscopy, here ‘coherent’ means without any scattering whileincoherent refers to electrons affected by scattering processes.

In a HET the hot electron energy distribution in the emitter region is much broadercompared to the very sharp transmission peaks of the resonant quantum states in the superlattice.Therefore, it is valid to replace the sharp transmission peaks of the superlattice states by a sumof weighted δ-functions

T (Ez) =N∑

i=1

(cicoh + ci

incoh)δ(Ez − Eiz), (52)

which are positioned at the eigenenergies Eiz. The differences in the peak areas between the

δ-functions (peak area = 1) and the individual transmission peaks are taken into account byintroducing the coefficients

cicoh =

∫ ∞

0T i

coh(Ez) dEz (53)

and

ciincoh =

∫ ∞

0T i

incoh(Ez) dEz. (54)

Inserting equations (52), (53), and (54) into (27) the following result is obtained:

α(Ez) =∫ ∞

0F(E′

z − Ez)T (E′z) dE′

z

=∫ ∞

0F(E′

z − Ez)

N∑i=1

(cicoh + ci

incoh)δ(E′z − Ei

z) dE′z

=N∑

i=1

(cicoh + ci

incoh)F (Eiz − Ez). (55)

This means, that the total transmission per state ci = cicoh +ci

incoh can be extracted from themeasured transfer ratios in the form of the corresponding peak amplitudes. These are gainedby applying multi-peak fit procedures using the measured energy distributions of the ballisticelectron beam and of the phonon replicas.

In figure 25 the measured transfer ratio of a 4-period superlattice is compared to theconvolution of the vertical energy distribution of the hot electron beam with the coherenttransmission coefficient. As one can see, the agreement between the transfer ratio and thecalculated ballistic electron distribution is excellent in the range where the contribution of thephonon replica peaks can be neglected.

As demonstrated in this section, HETs provide an excellent energetic resolution for hotelectron spectroscopy. Furthermore, they also facilitate tuning the energy of the injectedelectrons independently of the electric field applied to the superlattice, providing a situationanalogous to the BEEM experiments discussed in section 7.1. For the superlattices shown infigure 23, the individual Wannier–Stark states in the first miniband have been resolved up toelectric fields of 27.6 kV cm−1. The results on this are presented in [169]. The basic transportthrough the Wannier–Stark states in such short-period superlattices was found to be coherent.By tuning the Wannier–Stark state splitting into the optical phonon energy, the opening of LOphonon mediated transport paths was observed.


Figure 25. Measured transfer ratio of the 4-period superlattice (——) compared to the convolutionof the vertical energy distribution of the hot electron beam with the coherent transmission coefficientof the 4-period superlattice. The convolution equals the superposition of 4 hot electron energydistributions of amplitudes ci

coh with i = 1, 2, 3, 4.

8. BEEM studies of single impurities in the regime of the Mott transition

In the last section of this article, we would like to discuss some BEEM studies on singleimpurities embedded in an AlAs barrier [171], an experiment which benefits both from thespectroscopic capabilities of BEEM as well as its imaging capabilities.

Doping studies of heterostructures have always been of interest, since in heterostructures,thresholds for the Mott transition for thin layers can vary [172] compared to those in bulkmaterials, and even small doping concentrations [173] or delta-doping layers [81, 174] canhave a huge impact on the transport properties of such a system. With the devices becomingsmaller and smaller, the local influence of doping atoms and unintentional impurities on theelectrical behaviour of a device becomes more and more critical. This immediately leads toBEEM/BEES as a promising instrument for the investigation of doping in nanoscale devices.

To investigate the influence of impurities on the transmission of ballistic electrons througha heterostructure in a most direct way, samples with a single, Si-doped AlAs barrier, embeddedin a GaAs matrix, were used. According to values from the literature for bulk AlAs,isolated silicon atoms cause an impurity level 70 meV below the AlAs conduction band edge(X valley) [176]. In order to judge the feasibility of imaging single impurities with BEEM onsuch samples as well as to choose a suitable scan range for this purpose, a rough estimation ofthe expected ‘size’ of the impurities was conducted using the simple hydrogen model [175].In a material with a dielectric constant εrel and an effective electron mass m∗ the effective Bohrradius a∗ is obtained from the Bohr radius a0 by

a∗ = m0

m∗ εrela0. (56)

This yields an effective Bohr radius of about 104 Å for bulk GaAs and a value of about 20 Åfor bulk AlAs. Those values are within the limits of the spatial resolution of a BEEM experiment(≈10 Å) and indicate that, in principle, it should be possible to depict single impurities inGaAs/AlAs with BEEM. Furthermore, the estimated size of the impurities suggests a scanrange of, e.g., 100 × 100 nm2 to be suitable for the search for impurities.

Another key point is the impurity concentration level. In order to actually conductmeasurements of impurities with a local probe, the choice of a useful doping concentration is


Figure 26. Schematic view of the experimental set-up together with the conduction band profile ofthe sample. The conduction band edge is determined by the � valley in GaAs and by the X valleyin AlAs. (The sketch shows also the � valley in the AlAs barrier.) Vt denotes the tunnel voltage,It the tunnel current, Ic the ballistic current and Ef the Fermi energy.

constrained by two limits: on the one hand, the doping should be high enough to provide atleast one impurity within the chosen scan range (100 × 100 nm2); on the other hand it shouldbe low enough to avoid an electrostatic distortion of the conduction band profile due to acharge buildup by the donor ions. Calculating the band profile by a self-consistent solving ofPoisson’s equation [164] for various doping concentrations shows, that a doping concentrationof ND = 1×1017 cm−3 still causes no noticeable change in the conduction band profile, whilefor ND = 1 × 1018 cm−3 the band bending can no longer be neglected and therefore changesin the current onset directly due to the impurities cannot be distinguished from band bendingeffects.

Besides a reference sample with no intentional doping in the barrier of the heterostructure,the doping concentrations actually chosen for the experiment were ND = 3 × 1016 cm−3 andND = 1 × 1017 cm−3, respectively. Making the simplifying assumption that all doping atomsare electrically active, the mean distance between two impurities can quickly be estimatedusing the following consideration: for a doping concentration of ND , the average volumewhere exactly one impurity can be found will be 1/ND . Assuming this volume to be a spherewith the impurity in its centre, this leads to a radius of r = 3

√3/4πND . In this simple model,

the average distance between the centres of two impurities can be approximated by 2r . ForND = 3 × 1016 cm−3 this yields 400 Å, while for ND = 1 × 1017 cm−3 the average distanceis 267 Å. Therefore, in both cases the probability of finding at least one impurity within thechosen scan range is sufficient to conduct this sort of measurements.

All samples used in this project were fabricated by molecular beam epitaxy, using semi-insulating GaAs(100) wafers as a substrate. After a 1000 Å thick GaAs buffer layer andseveral smoothing layers, the actual heterostructure starts with a highly n-doped GaAs region(Si, ND ≈ 1.5 × 1018 cm−3), which provides the collector for the ballistic current. Thethickness of this region is 6500 Å for the samples with the undoped barrier and 5000 Å forthe samples with the doped barriers. The collector region is followed by 1500 Å of undopedGaAs, which serves as a spacer. On top of this layer a 100 Å thick AlAs barrier is grown, eithernominally undoped (i.e. ND < 1×1014 cm−3) or with a silicon doping of ND = 3×1016 cm−3

or ND = 1 × 1017 cm−3, respectively. Finally, all barriers are covered with an undoped GaAscap layer (100 Å) for protection purposes. In figure 26 a sketch of the band profile togetherwith the principle of the experimental set-up is shown. The low temperature measurementswere carried out in He exchange gas at temperatures of 180 K and 10 K, respectively. The


Figure 27. Typical BEEM spectra measured at T = 180 K and It = 2 nA. For better comparisonwith the spectra of the doped samples, a reference spectrum obtained on the undoped barrier is shownin both, (a) and (b). (Note that, for technical reasons, Vt is applied to the base and therefore positivein our experiment.) (a) Undoped barrier compared with a barrier doping of ND = 3 × 1016 cm−3

(vertical offset added for more clarity), (b) undoped barrier compared with a barrier doping ofND = 1 × 1017 cm−3.

tunnel current used for the presented spectra as well as the BEEM images was It = 2 nA.Etched tungsten tips with a sputtered gold covering were employed as STM tips for goodspatial resolution and spectroscopic stability.

8.1. The influence of doping in AlAs single barriers on the current onset in BEES

To investigate the actual influence of a certain amount of doping on the transport behaviour ofthe barrier, a nominally undoped AlAs barrier (ND < 1 × 1014 cm−3) was used as a referencesample. For this sample, the current onset in the BEEM spectra is given by the position of theAlAs X valley.

As can be seen from figure 27, at T = 180 K the current onset in a BEEM spectrumfor an undoped AlAs barrier is typically at 1.19 V. Reference measurements on a GaAs bulksample yielded an Au–GaAs Schottky barrier height of 0.96 eV at T = 180 K. Taking intoaccount that the AlAs barrier lies below a 100 Å thick GaAs cap layer and the band structureis slightly tilted (see figure 26), this means that the AlAs X valley lies about 280 meV abovethe conduction band edge in the GaAs (i.e. the GaAs � valley).

This measured value is in good agreement with data from other BEEM experiments (see,e.g. [69,97]) as well as with results from tight-binding calculations, as, for instance, providedby Jancu et al [177].

One might notice that the size of the ballistic current in figure 27 is considerably less thanthe typical BEEM current known from measurements at room temperature (up to several pA,see, e.g. [82]). This suppression of �–X transport at low temperatures was also observed inprevious experiments [99] as well as by other groups [69].

The doping concentration in the AlAs barrier of the first doped sample was chosen to beND = 3 × 1016 cm−3. As mentioned earlier, this leads to a mean distance of approximately40 nm between two impurity centres, and gives just a sufficiently high probability of havingat least one impurity within a scan range of 100 × 100 nm2. In figure 27(a) a typical BEEMspectrum measured on this sample is shown in comparison with one of the undoped barrier.The two spectra are almost identical, showing both the same onset position in Ic(Vt ) and aboutthe same amount of transmitted ballistic current.


Figure 28. Histograms of Ic current onsets extracted from BEEM spectra which were measured atarbitrarily distributed locations on a sample. (Vt is applied to the base and therefore positive.) Datameasured at T = 180 K: (a) undoped barrier, (b) barrier doping ND = 3 × 1016 cm−3, (c) barrierdoping ND = 1 × 1017 cm−3.

This fact is confirmed by further measurements of BEEM spectra taken at various positionson both samples. All spectra look very similar, whether they are acquired on the sample withthe moderately doped barrier or on the undoped reference sample. To get an overview of thetransmission behaviour, the minimum tunnel voltage necessary to detect a ballistic current(= onset voltage) is extracted from all spectra. In figure 28(a) we have plotted the distributionof these onsets for the nominally undoped barrier, while figure 28(b) shows the result forthe barrier with ND = 3 × 1016 cm−3. To provide a convenient way for a quick comparison,a simple Gaussian fit to the histograms was utilized. This yields a mean current onset of 1.19 Vfor both barrier types, showing no significant influence of this amount of doping. The onlyvisible difference between the two samples is that the width of the distribution for the undopedbarrier is only 23 mV, while the medium doped barrier has a noticeably wider distribution of38 mV. Apart from this broadening, the onset distribution of this medium doped barrier doesnot reveal any local effects of the impurities on the current onset.


For the next type of sample the doping within the AlAs barrier was chosen to beND = 1 × 1017 cm−3. A quick estimate yields a mean distance of approximately 27 nmbetween two impurity centres for this doping concentration. In figure 27(b) a typical BEEMspectrum of this sample in comparison with a spectrum of the undoped barrier is shown.As can be seen clearly, the doping now causes a significant lowering of the onset voltage forthe ballistic current.

For a more thorough comparison, several BEEM spectra were also measured at variouspositions on this ‘highly’ doped sample and used for a systematic determination of thecorresponding onset voltages. The result, which can be seen in figure 28(c), showsunambiguously a general lowering of the onset voltage compared to the undoped barrier.For a better comparison, again a Gaussian curve was fitted to the distribution. The mean onsetvoltage for ND = 1 × 1017 cm−3, as provided by the Gaussian fit, is 1.08 mV. This is 110 mVless than for the undoped barrier and clearly indicates that the doping opens an additionaltransmission channel which lies below the AlAs X valley.

Although the onset distribution for the highly doped barrier is broader (31 mV) than theone for the undoped barrier (23 mV), it is still peaked around a single maximum. Again, asfor the moderately doped sample, from the onset voltage distribution no local influence of theimpurities on the current onset is apparent.

This general decrease in the current onset voltage for the highly doped barrier indicatesthat in this case the regions influenced by individual impurities are actually overlapping,and thus cause not a punctual but a wide-ranging alteration of the transmission behaviour.Therefore, these measurement data lead to the conclusion that already a doping concentrationof ND = 1×1017 cm−3 is sufficient to form a continuous impurity band. This can also explainthe fact that the onset measured for this doped sample is 110 meV below the AlAs X valley,while the value found in the literature for the energetic position of a silicon impurity in AlAsis only 70 meV below the AlAs X valley. The latter is denoted to be valid for an isolatedimpurity with zero-dimensional characteristics, while in an impurity band the lower level ofconfinement is expected to lead to a decrease in the energetic position.

Although the onset distributions of the doped barriers do not show any obvious patternwhich would allow one to decide whether the data had been measured within or outside of thelocal range of influence of an impurity, the ballistic spectra do reveal an interesting detail athigher tunnel voltages. Figure 29 shows plots with a series of ballistic spectra for each sample.

As expected, the spectra of the undoped barrier show just some dispersion around anaverage behaviour for the whole range of Vt . On the contrary, the spectra of the doped barrierssplit into two groups when the tunnel voltage increases. Apparently, despite that the currentonset for each sample is preferentially found around a single value of Vt (see figure 28), in theregion of higher tunnel voltages there are distinctive areas of enhanced current transmissionon the doped samples.

This effect is seen for the sample with a barrier doping of ND = 1 × 1017 cm−3 as well asfor the one with ND = 3 × 1016 cm−3. Therefore, this observation suggests that the impuritiesdiminish the effective barrier thickness in their wider surroundings, even in the regime wherethe doping concentration is too low to form a continuous impurity band and the barrier heightis generally still determined by the AlAs X valley.

8.2. Imaging of impurities in AlAs single barriers

The finding of a locally enhanced transmission at higher tunnel voltages for the doped barriersprovided new encouragement to conduct a search for single impurities in the imaging modeof BEEM. In figure 30 BEEM images for all three barrier types can be seen. All images


Figure 29. Ballistic spectra taken at arbitrarily distributed locations on a sample. All data weremeasured at T = 180 K using a tunnel current of It = 2 nA: (a) undoped barrier, (b) barrier dopingND = 3 × 1016 cm−3, (c) barrier doping ND = 1 × 1017 cm−3. (Note that, for technical reasons,Vt is applied to the base and therefore positive in our experiment.)

were recorded with the same tunnelling conditions. To facilitate a direct comparison, thesame colour code for the z-scale was used in all three BEEM images. In order to differentiatebetween topography related transmission variations and any features in the BEEM imageswhich are not caused by topographic effects, simultaneously to the BEEM images also thecorresponding STM images were measured. Those surface topographies are also depicted infigure 30 and show just the usual corrugation of the evaporated Au base layer for all samples.

As can be seen from figure 30(a), the BEEM image of the undoped barrier displays a quiteuniform transmission profile and reflects just the topographic features also visible in the STMimage, primarily the granular structure of the gold base layer.

Quite a different picture is exhibited for the barrier with the intermediate dopingconcentration, which can be seen in figure 30(b): superimposed on the typical granularity ofthe gold base layer is a larger structure in the transmission profile. In particular, two brighterspots protrude from a plane with a generally low transmission. The simultaneously measuredSTM surface topography shows just the usual corrugation of the evaporated gold base layer,as for the other samples. This confirms that those bright areas in the BEEM image of themedium doped sample are not caused by any topographic irregularities but do originate in alocal modification of the transmission in the sub-surface heterostructure due to the impuritiesin the AlAs barrier.


Figure 30. Left column: BEEM images of the three different sample types, recorded atT = 180 K, It = 2 nA, Vt = 1.4 V. Right column: corresponding STM surface topographies(recorded simultaneously with the BEEM images). The maximum z-range in the topographiesis 20 Å: (a) undoped barrier, (b) barrier doping ND = 3 × 1016 cm−3, (c) barrier dopingND = 1 × 1017 cm−3.

Although the diffuse shape of these spots makes their size rather hard to determine, arough estimate of the mean radius shows that it is about 10–13 nm. Using the hydrogen modelof impurities this size is more like the effective Bohr radius of an impurity in GaAs (≈10 nm)than the one of an impurity in AlAs (≈2 nm). Of course, the observed radius is only a veryrough estimation and the experimental difficulties can have a distorting effect on the estimatedsize. On the other hand, one should not forget, that the AlAs barrier in the heterostructures iscomparatively thin and embedded in a GaAs matrix. Therefore, although the doping was doneonly during the growth of the AlAs layer, the behaviour of impurities in this heterostructurecan differ significantly from the behaviour of impurities in bulk AlAs.

One should also keep in mind, that the distribution of the impurities within the barrier isstochastic. That means, an impurity might be located in the centre of the barrier as well as besitting right at the AlAs–GaAs interface. The actual position of an impurity will most probably


influence its effect on the local transmission and therefore also determine how it will appearin the BEEM image.

Finally, looking at the barrier with a doping of ND = 1 × 1017 cm−3, a rather uniformtransmission profile can be seen again. This is shown in figure 30(c). The contrast of thisBEEM image is similar to the one of the undoped barrier. The most apparent distinction ofthis sample compared to the undoped barrier is that the overall transmission is higher. In theleft edge of the image there seems to be a region with an even slightly higher transmissionthan the rest of the picture, but, in general, the contrast is rather weak. This lack of contrast,compared to the image of the sample with ND = 3 × 1016 cm−3, was observed not only in thedepicted image but also in all other BEEM images taken on this sample, which minimizes theprobability that this was, per chance, just an area with no impurity in it.

The fact that this highly doped sample does not exhibit well-defined areas of hightransmission which could be allocated to impurities can be explained by the closer proximityof the doping atoms. If one considers the diameter of the high-transmission areas observedin figure 30(b) (20–26 nm) together with the mean distance between two impurity centresestimated for the high doping concentration (≈27 nm) it is clear, that for the highly dopedsample the regions of enhanced transmission will partially overlap, thus leading to little contrastand a generally higher transmission. This also corresponds very well with the data fromthe onset measurements, which suggest a continuous impurity band for a barrier doping ofND = 1 × 1017 cm−3.

9. Summary

In the previous sections, we have compared the basic features of BEEM/S and hot electronspectroscopy with HET devices. After discussing the spectroscopic strengths and weaknessesof both techniques, we hope to have demonstrated that BEEM is not just a local versionof device-based hot electron spectroscopy, but a complementary method which can yieldsupplementary results.

At a first glance, the strength of BEEM is the high spatial resolution. In spectroscopic modethe energetic resolution is in the order of 20–50 meV at low temperatures. In contrast to this,HET-based spectroscopy offers an energetic resolution below 10 meV and short measurementtimes. The main drawback of HET spectroscopy is the lack of spatial resolution. Of course,as pointed out in this review, this description is quite simplified. Each of the two techniqueshas additional features, which are not available in the other one. Therefore, a simple decisionabout which spectroscopic technique is the superior one cannot be made. Rather, the moresuitable method has to be chosen according to the particular requirements of a given problem.

In BEEM, the refraction of the ballistic electrons at the metal–semiconductor interfaceleads to a relatively broad and bias dependent energetic distribution. On the one hand, this isof disadvantage for spectroscopic resolution; on the other hand, this enables electron injectioninto higher conduction band valleys, which are not accessible in HET devices without aviolation of momentum conservation rules (see, e.g. the experiments on L valley RTDs in [99]).Furthermore, these electron refraction effects lead to the existence of the so-called ‘acceptancecone’, which filters out all scattered electrons in the base. Thus, the energetic distribution ofballistic electrons in BEEM is always close to ideality. As a further advantage in BEEM/S,problems related to quantized states in the drift region between the emitter and the collectorare negligible. In BEEM samples any quantum interferences would have to occur either in theAu base layer or in the drift region in front of the collector barrier. However, possible quantuminterferences within the base do not occur, because the base is polycrystalline and, therefore, thescattering rates are high. Furthermore, possible quantum interferences between the Schottky


barrier and the buried heterostructure are suppressed because the interface between the baseand the GaAs is non-epitaxial. Thus, the observation of quantum interference effects in thebase and the drift region is quite unlikely in BEEM experiments.

In addition to this, inelastic scattering by LO phonons can usually be neglected, since fora typical BEEM sample the classical electron transfer time through the active region is usuallywell below the LO phonon scattering time. Due to these almost ideal conditions, the ballisticcurrent is surprisingly well described by the simple Bell–Kaiser formula in a tunnel voltagerange of up to ≈200 mV above the onset, and in many samples even up to tunnel voltages of≈600 mV above the onset. At higher tunnel voltages numerous scattering processes occur andenergy dependent influences become important, so that more complex models have to be used.Considering BEEM/S on GaAs–AlGaAs superlattices, the above features of BEEM/S allow theclear observation of well-separated superlattice minibands as well as bias dependent transportstudies without the disturbing influence of phonons. Single states within the superlattice,however, cannot be resolved with BEEM.

In hot electron spectroscopy with HET devices, the excellent energetic resolution (≈8 mV)allows single states within a superlattice to be resolved. Due to the special design of theemitter structure in a HET device, the energetic distribution of ballistic electrons is virtuallyindependent of the applied bias. As only transfer ratios of the emitter current and the collectorcurrent are of interest, the emitter and collector currents need not be calculated explicitly.This, in principle, simplifies the data analysis for HET devices. In practice, however, the dataanalysis in HET devices becomes quite complicated due to the strong influence of LO phononscattering and the influence of parasitic electron interference effects in the drift region of thedevice. Other than in BEEM, parasitic quantum interferences in the drift region can becomequite dominant in HET devices. Here, the interface between the base and collector region isepitaxial, so that no significant interface scattering occurs. Further, even at very high dopinglevels, electron–electron scattering in the base is much lower compared to the scattering rates inthe Au base layers normally utilized for BEEM. Thus, parasitic quantum interferences betweenemitter and base are almost inevitable. To reduce the influence of the interference effects, thelength of the drift region can be increased. As a consequence, the classical electron transfertimes through the active region of HET devices easily reach values comparable to the phononemission times. Thus, the initially narrow distribution of ballistic electrons is broadened by LOphonon replicas. These replicas reduce the spectroscopically useful energy range significantly,since the overlapping peaks obscure the spectroscopic features of interest, especially at higherenergies.

Acknowledgments

This work was sponsored by the Austrian Science Foundation (FWF), projects P16337, SFB-F016 and Z24, and by the Gesellschaft fur Mikro- und Nanoelektronik (GME). Substantialcontributions to this work came from C Eder, R Heer, C Pacher, and C Rauch. All MBEsamples used in this work were provided by G Strasser. The authors greatly appreciate thescientific input provided by P Vogl, C Strahberger, and A Wacker. In addition, we would like tothank B Basnar and W Brezna for valuable discussions as well as H Schenold for the technicalassistance. Finally, we wish to thank E Gornik for his continuous support over so many years.

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