Scanning Tunneling Microscopy

Graham Baker, Oğuzhan Can, and Étienne Lantagne-Hurtubise(Dated: November 26, 2016)

We present the basics of the Scanning Tunneling Microscopy (STM) technique, along with somerecent applications to research on quantum materials.

I. INTRODUCTION

A Scanning Tunneling Microscope (STM) is an instru-ment used to image and study the electronic propertiesof surfaces at the atomic scale. It was pioneered by Bin-nig, Röhrer and their groups in the early 1980s1, earningthem half of the 1986 Nobel Prize in Physics.

In an STM, an atomically sharp conducting tip isbrought very close to the surface of the sample understudy – the distance d between the two is on the or-der of a nanometer. The space between the tip and thesample, which is usually vacuum, forms a potential bar-rier. When a bias voltage V is applied between the tipand the sample, a tunneling current It flows between thetwo. The tunneling current depends on the tip-to-sampledistance d(r) = z−z0(r) where z0(r) is the profile of thesample’s surface. It also depends on the local density ofstates (LDOS) of the sample over the energies within eVof the Fermi energy. STM is thus a powerful experimen-tal technique for measuring the surface topography andLDOS of materials.

The following report is divided as follows: in Sec. II,we present the different data acquisition modes used withan STM, while Sec. III addresses some common experi-mental considerations. In Sec. IV, we explain the theoryof electronic tunneling in the context of STM, and Sec.V concludes by presenting modern applications of STMin quantum materials.

II. MODES OF OPERATION

Figure 1 shows a schematic diagram of the experimen-tal apparatus. Piezoelectric transducers are used to con-trol the lateral position r = (x, y) of the tip as well as itsheight z. The piezoelectric transducer controlling heightis also connected to a feedback control unit which de-pends on the tunneling current It. Usually, the tip isgrounded and the bias voltage is the voltage −Vs appliedto the sample (Vs > 0 in the following, without loss ofgenerality). Not shown in Figure 1 is the cryostat, whichis used to cool the experiment.

Therefore, there are four (non-independent) tunableparameters during measurement: the tip height z, the tipposition r = (x, y), the bias voltage −Vs, and the tun-neling current It. Through controlling this set of param-eters, one can operate a STM in several different modes:topography, spectroscopy and spectroscopic imaging2. InFigure 2, we show a representative example of a data settaken in each of these modes.

Figure 1. Schematic diagram of an STM. Figure taken fromRef. 3.

A. Topography

The first way in which an STM is used is to measure thetopography of a sample’s surface. There are two ways inwhich this can be done. In both cases, the sample voltage−Vs is kept constant.

In constant current mode, the tip is scanned acrossthe surface of the sample and the height z of the tip iscontrolled by a feedback loop in such a way as to maintaina constant tunneling current It. In this case, the heightof the tip z(r) = z0(r)+d records the profile z0(r) of thesample’s surface which contains the corrugations of theatomic lattice.

In constant height mode the tip is again scanned acrossthe surface of the sample, this time at a constant heightz, and the tunneling current It(r) is measured at eachposition. Using the dependence of the current on the tip-to-sample distance d(r) = z− z0(r), It(r) can be used todeduce the surface profile z0(r) of the sample.

B. Spectroscopy

Scanning Tunneling Spectroscopy (STS) measures theelectronic LDOS of the sample, by exploiting the factthat

dItdVs

= C(d)× ρs(r, ε = EF − eVs) (1)

where C(d) is an exponential function of the tip-sampledistance d, as shown in Sec. IV. Therefore, the LDOS

2

Figure 2. Left: Topographic image of LiFeAs with impurities in bright contrast. Center: STS taken at location indicated by 1in left panel. Right: Tunneling conductance map measured over same area as in left panel. Figure taken from Ref. 3.

can be measured by keeping the tip at a constant heightand position and measuring the tunneling conductancedIt/dVs while sweeping Vs. There are two ways of mea-suring dIt/dVs.

The first way is to measure It − Vs data and performnumerical differentiation. This technique offers a highdata acquisition rate however the process of numericaldifferentiation introduces noise to the measurement. Thesecond method is to use a lock-in amplifier to modulateVs by a small voltage V ACs sin(ωt). Using the conditionthat V ACs � V DCs and expanding It in series one findsthat

It = IDCt +dItdVs

V ACs sin(ωt) +O((V ACs

)2)(2)

so that the signal at ω is proportional to dIt/dVs4,5. Thistechnique can improve measurement sensitivity if ω isselected so that it is far from the frequencies of sources ofnoise. However, the time constant of the lock-in amplifieris on the order of 10 milliseconds, which when repeatedfor each data point can significantly decrease the dataacquisition rate.

C. Spectroscopic Imaging

In Spectroscopic Imaging STM (SI-STM) the topogra-phy and STS modes are combined. As in the topographicmode of operation, the tip is scanned across the sample’ssurface at constant Vs while either d or It is held constantand the variation in the other is recorded. This time, ateach pixel in the topographic map the tip freezes its po-sition (z, r) and sweeps Vs while measuring dIt/dVs. Byplotting dIt/dVs at a given Vs as a function of lateralposition r, the spatial variation of the LDOS is mapped.

III. EXPERIMENTAL CONSIDERATIONS

A. Temperature

The lower the temperature at which STM measure-ments are performed, the better the resolution. This isbecause as temperature decreases, so does the thermalbroadening of the Fermi-Dirac distribution. The lowerlimit to energy resolution due to thermal broadening isgiven by ∆Ethermal = 7

2kBT where kB is Boltzmann’sconstant6. Typically, STMs are cooled by helium which,depending on the pressure and which isotope is used,gives access to temperatures from 300 mK to 4.2 K. Byusing a dilution refrigerator, temperatures below 300 mKcan be accessed. With the push to examine phenomenathat occur at ever smaller energy scales, the design andrealization of dilution refrigerator STMs has been thesource of much recent experimental effort6–8.

B. Vibration Isolation

Because of the exponential dependence of the tunnel-ing current on the tip-to-sample distance d, STMs are ex-tremely sensitive to the effects of vibrations, which can besignificant sources of noise in measurements. Vibrationisolation is thus a critical consideration in STM experi-mental design. Since the corrugation of an atomically flatsample surface can be as low as 10 pm, it is commonlyconcluded that d should change no more than 1 pm dueto the influence of vibrations2,9,10. Vibration isolationtakes place at multiple stages of the STM apparatus: atthe STM head (which contains the tip and sample), be-tween the head and the cryostat, and between the entireapparatus and the external environment.

STM heads are designed to be extremely rigid so that,when exposed to vibrations, the tip and sample movetogether and the tip-to-sample distance d is preserved.

For helium temperature STMs, the head is mounted

3

within the cryostat by eddy-current-damped spring sus-pensions to achieve vibration isolation. However, in di-lution refrigerator STMs, the spring suspensions formtoo poor a thermal link between the refrigerator and thehead. To operate at such low temperatures, it is neces-sary to compromise on the vibration isolation that canbe achieved at this stage of the apparatus.

For high-performance STM experiments, extensive ef-fort goes into the design of ultra-low vibration facilities.In a typical facility of this type, the STM is mounted ona pneumatically suspended inertia block weighing on theorder of 103−104 kg. The entire experiment is housed inan acoustically-absorbing enclosure, and rests on foun-dation which is isolated from that of the rest of thebuilding11.

IV. THEORY

A. Tunneling current and DOS

In this section, we derive how the LDOS is related tothe tunneling current measured by an STM.

Applying a bias voltage −Vs to the sample (while thetip is grounded) shifts the energy of the electrons in thesample by +eVs with respect to the electrons in the tip,as shown in Figure 3. Tunneling can occur through thevacuum in between the tip and the sample – the tun-neling current measured will be I = −eΓ, where Γ isthe transition rate (probability of transitioning from ini-tial to final state per unit time). Using time-dependentperturbation theory (Fermi’s golden rule), the tunnelingcurrents are found to be:

Isample→tip =− 2e2π

~|M(ε)|2n(ε)ρs(r, ε)

× [1− n(ε+ eVs)] ρt(ε+ eVs) (3)

and

Itip→sample =− 2e2π

~|M(ε)|2n(ε+ eVs)ρt(ε+ eVs)

× [1− n(ε)] ρs(r, ε) (4)

where the factor of 2 in front accounts for spin, M(ε)is the matrix element of the tunneling process, which ingeneral depends on the energy, ρs(r, ε) and ρt(ε) are thelocal electronic density of states (LDOS) of the sampleand the tip, respectively, and

n(ε) =1

eβ(ε−EF ) + 1

is the fermion number density in a state of energy ε, EFis the Fermi energy and β ≡ 1/kBT is the inverse tem-perature. The fermion number densities in Eqs. (3) and(4) appear because tunneling can only occur if the initialstate is occupied and the final state is empty. The nettunneling current It is comprised of the current from the

Figure 3. Density of states (DOS) as a function of energyfor a generic STM sample (left) and tip (right). When a biasvoltage −V is applied, the Fermi energy of the tip is shifted by−eV with respect to the Fermi energy EF of the sample. Thegreen regions represent occupied states at low temperatures,and the solid arrow shows the main tunneling channel fromthe sample to the tip. Figure taken from Ref. 12

sample to the tip, minus the current from the tip to thesample, integrated over all energies:

It = −4πe

~

∫ ∞0

dε|M(ε)|2ρs(r, ε)ρt(ε+ eVs)

×{n(ε) [1− n(ε+ eVs)]− n(ε+ eVs) [1− n(ε)]

}(5)

where we have taken the lowest electronic eigenstate tolie at ε = 0. At low temperatures, the Fermi-Dirac dis-tribution is very sharp, so we can suppose that statesin the sample are filled up to ε = EF and states in thetip are filled up to EF − eVs. In that setting, electronsin the tip have nowhere to tunnel through (all states inthe sample with corresponding energy are already occu-pied), whereas sample electrons with energy in the range[EF − eVs, EF ] can tunnel to the tip and contribute tothe integral. Furthermore, in order to extract informa-tion on the DOS of the sample electrons, we choose a tipmade of a material with a constant DOS over the rangeof energies of interest. Thus, the integral simplifies to:

It = −4πe

~ρt(EF )

∫ EF

EF−eVs

dε|M(ε)|2ρs(r, ε) . (6)

The last step is to figure out the matrix element forthe process. A useful approximation in this case is tomodel the potential barrier as a square well with heightφ and width d (the sample-tip distance). The tunnel-ing through such a potential can be estimated using theWKB method13, leading to |M |2 = e−2γ , where

γ =d

h

√2mφ

4

which is independent of the energy of the electron, sothat

It(r, Vs) = −4πe

~ρt(EF )e−d

√8mφ/h

∫ EF

EF−eVs

dερs(r, ε) .

(7)

The total tunneling current is proportional, under ourpresent assumptions, to the integral of the sample elec-tronic DOS on the interval [EF − eVs, EF ]. Therefore,

ρs(r, ε = EF − eVs) ∝dIt(r, Vs)

dVs. (8)

B. Quasiparticle Interference

Impurities on the surface of a material lead to Friedeloscillations. These are oscillations in the electronic den-sity of states at the surface, caused by the scattering andinterference of the electronic surface states on the impu-rities. Therefore, Friedel oscillations can be seen in STMimages as oscillations in the local density of states nearthe Fermi level14. This modulation pattern in the localdensity of states is referred to as quasiparticle interfer-ence, and contains information about the wave vectorsconnecting surface electronic states near the Fermi level.The Fourier Transform of a tunneling conductance mapgives us the Fermi contour in momentum space becausethe modulation waves in LDOS are composed of surfacestate wave vectors near the Fermi level. Figure 4 is anexample of the surface state of Be(0001), which is a goodcandidate for the experimental realization of a free elec-tron gas14.

In the figure, we see the tunneling conductance map(a) and its Fourier Transform (b). As we expect from afree electron gas, the Fermi contour can be seen as a cir-cle. This technique allows us to study the surface statesin both real and reciprocal space. However, ARPES canonly access occupied states and cannot obtain real spaceinformation. With STM, LDOS oscillations in real spaceobserved through tunneling conductance maps allow usto delve deeper into understanding disorder and scatter-ing processes on material surfaces.

The electronic properties of surfaces are crucial for theinvestigation of materials such as topological materialsand high temperature superconductors.

V. MODERN APPLICATIONS OF STM

A. Topological Materials

Topological insulators are materials with a gap in thebulk and symmetry protected gapless states on the sur-face which can be described by a massless Dirac FermionHamiltonian. In the presence of certain symmetries, aslong as the gap in the bulk does not close, the topo-logical insulator phase is robust against disorder. With

their peculiar surface states, topological materials havepromising applications ranging from spintronics to quan-tum computation.

One of the hallmark features of these surface states isthe absence of scattering between states of opposite spinand momentum (see Ref. 16 and 17 for a comprehensivereview).

In Ref. 18, the authors used STM – combined withARPES – to image the gapless surface states of com-pound Bi1−xSbx, and probe the scattering of the edgestates on defects created by random alloying. They showthat back-scattering is absent even for strong atomic-scale disorder, a result consistent with the theory of 3Dtopological insulators. Around the same time, anothergroup reported similar conclusions by applying STM tothe topological insulator Bi2Te319.

Weyl semimetals are another kind of topological mate-rial with interesting surface properties (see Ref. 20 for anintroduction). A peculiar property of the surface statesof Weyl semimetals are the Fermi arcs. Fermi arcs areFermi contours that are not closed loops and can only ex-ist on the surface as a boundary of a 3D material. Weylsemimetals have also been investigated using STM withQPI methods21. Although Fermi arcs are not directly ob-servable on the momentum space QPI images, signaturesof Fermi arcs can be identified22. It is an important mile-stone that evidence for Fermi arcs has been shown for thefirst time independently from and agreeing with ARPESresults.

B. Vortices on Superconductors

STM has been an important tool in probing supercon-ductors. For instance, the superconducting gap can beobserved by tunneling spectroscopy if we plot LDOS ver-sus bias −Vs (See Figure 5 – bottom curve) The gap iswhere the LDOS vanishes. STM has also played an im-portant role in the experimental verification of Abrikosovvortices.

Abrikosov vortices are topological defects that occur intype-II superconductors when an external magnetic fieldabove a critical value is applied (in fact, they are thedefining property for this phase). In this regime, the mag-netic field pierces through the superconductor instead ofbeing expelled by it. Each vortex has a magnetic fluxquantum Φ0 and these vortices can form lattices (Fig-ure 4.c) that are accessible to an STM15. The electronicstructure of these vortices and their spatial distributionprovide insight into fundamental properties of the super-conducting state. For example, the size of the vortexstates can be used to measure the superconducting co-herence length23.

STM studies not only allowed the experimentalobservation of these vortices with high spatial resolution,but also played a crucial role in exploring them further.The superconducting gap closes at these vortices whichshow no superconducting behaviour. It was expected

5

Figure 4. (a) Tunneling conductance map of Be(0001) showing LDOS oscillations in real space. (b) Fourier Transform of (a).Figures taken from Ref. 14. (c) Topographic image of an Abrikosov vortex lattice on the surface of NbSe2. Figure taken fromRef. 15

Figure 5. Density of states vs bias voltage for NbSe2 nearan Abrikosov vortex. Three measurements were made fordifferent distances from the vortex: the core center (top),75 Å from the vortex (middle) and far away from a vortex(bottom). Note the superconducting gap at zero bias voltagein the bottom curve. Figure from Ref. 15.

that these vortices would be metallic since the super-conducting gap is closed and there are states in placeof the gap for electrons to tunnel into. Therefore, thesuperconducting gap would not be seen, instead theobserved conductance versus bias voltage would be a flatsignal when the tip goes through the vortex. However,instead of a flat spectrum Hess et al. found a peak15 atthe zero bias voltage (Figure 5), indicating the existenceof localized vortex core states. This discovery lead tofurther interest in the study of Abrikosov lattices.

VI. CONCLUSION

Scanning Tunneling Microscopy is a powerful experi-mental technique that has allowed to investigate the sur-face electronic states of a large array of materials withatomic resolution. It is likely that it will play a pivotalrole in the research and development of the next genera-tion of quantum materials.

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