PHYSICAL REVIEW B 15 FEBRUARY 1998-IVOLUME 57, NUMBER 7
Energy levels of quantum wires determined from magnetophonon resonance experiments
G. Ploner, J. Smoliner, G. Strasser, M. Hauser, and E. GornikInstitut fur Festkorperelektronik, Mikrostrukturzentrum der Technischen Universita¨t Wien, Floragasse 7, A-1040 Vienna, Austria
~Received 7 April 1997; revised manuscript received 4 August 1997!
We report the observation of magnetophonon resonances in quantum wires, fabricated by shallow etching onconventional high-mobility GaAs-AlxGa12xAs heterojunctions. It is shown that magnetophonon resonancemeasurements can be used as an alternative transport tool for the characterization of the subband spectrum ofquantum wires. The method proves to be particularly useful at low electron densities, when only a fewsubbands are occupied and the standard magnetic depopulation characterization technique can no longer beused. By variation of the electron density in the wires through illumination with a red light-emitting diode theone-dimensional~1D! subband spacing, as determined from magnetophonon resonance measurements, is foundto increase with decreasing electron density and to be systematically higher than the subband spacing obtainedfrom simultaneous magnetic depopulation measurements. We give an explanation of this observation, whichindicates that the combination of both transport characterization methods provides information on the actualshape of the underlying confinement potential. In addition, also the effective mass of the 1D electrons can bedetermined from the magnetophonon data. We observe a slight increase of the effective mass with decreasingcarrier density and a strikingly large enhancement ofm* with stronger confinement, achieved by deeperetching. We discuss several possibilities that could lead to this strong mass enhancement.@S0163-1829~98!08503-8#
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I. INTRODUCTION
LO-phonon-induced resonant scattering of electronstween Landau levels, also called themagnetophonon~MP!effect, has been widely used to obtain information on banstructure parameters, mainly the effective mass, and onelectron–LO-phonon interaction in bulk and twdimensional~2D! semiconductors. The magnetophononfect has proved to be a useful magnetotransport tool fordetermination of effective masses and is moreover partlarly suitable to carry out investigations of the temperatdependence ofm* in a temperature range that is commoninaccessible to other methods such as cyclotron resonmeasurements. A very recent example of this is given, ein Ref. 1, where high-precision effective mass valuesbulk n-GaAs are determined as a function of temperatureto 400 K using the magnetophonon effect.
In 2D systems such as quantum wells and single hetjunctions the magnetophonon effect has been used notto measure effective masses of 2D electrons2 but also toinvestigate the influence of various scattering mechanismthe high-temperature transport properties of these systeEvidence has been found that interface phonons are effein the MP resonant scattering in heterostructures and tfrequency has been determined.3 The self-consistent consideration of all scattering mechanisms in the calculation ofhigh-temperature magnetoresistance leads to the prediof the collapse of MP oscillations at high magnetic fields aexcellent agreement between theory and experiment hasobtained.4,5 The investigation of the magnetophonon effectextended to the vertical transport through short-period GaAlAs superlattices in a very recent work by Gassotet al.6
These authors were able to resolve the two series of matophonon resonances due to GaAs and AlAs LO phononinvestigating the MP effect at high hydrostatic pressure
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related the electric-field and temperature dependence ofMP oscillation to band-structure properties and carrier litimes.
In quasi-one-dimensional electron gases~Q1DEG’s! theinvestigation of magnetophonon resonances~MPR’s! is con-sidered as an alternative magnetotransport tool for the demination of the energy difference between adjacent 1D sbands. It can be shown that the lateral quantization leadsshift of the magnetic-field positions of the resonant extrein the high-temperature magnetoresistance. Under cerpremises this shift is proportional to the subband spacingE0
and provides a particularly simple method to determineE0 .This is particularly useful in those cases where, due to a vlow electron density or relatively high subband spacing, oa small number~less than, say, four! of subbands is occupiedin the quantum wire. In these cases the standard magndepopulation~MD! method, commonly used to estimate thsubband spacing, is no longer a suitable transport technfor the characterization of quantum wires. The MD methexploits the oscillatory structure in the low-temperature mnetoresistance occurring when the magnetoelectric hylevels are successively shifted across the Fermi level bycreasing the magnetic field. A small number of occupsubbands drastically reduces the oscillatory structureRxx(B) and the commonly used evaluation methods7,8 can nolonger be applied.
The theoretical models for MPR in Q1DEG proposedfar revealed the following. Vasilopouloset al.9 consideredthe model case of a relatively weak parabolic confinempotential and found that the main effect of a weak confinment is a shift of the position of the resonant magnetoretance minima to slightly smaller magnetic fields. The overshape of the oscillatory part ofrxx(B) is still given by thewell-known series of exponentially damped cosine osci
57 3967ENERGY LEVELS OF QUANTUM WIRES STUDIED BY . . .
TABLE I. Structure of the samples used in this work. Asterisk denotes after illumination.
SampleGaAs cap~Å!
~Si doped!Al xGa12xAs ~Å!
~Si doped!Al xGa12xAsspacer~Å!
Sheet density (cm22!~at 4.2 K!
Mobility ~cm2/V s)~at 4.2 K!
g166 100~undoped!
300~231018 cm23!
600 1.131011 * 1.63106 *
g73 200~231018 cm23
400~231018 cm23!
200 2.731011 6.53105
g79 200~231018 cm23!
400~231018 cm23!
200 2.931011 9.93105
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tions found in the bulk10 and two-dimensional11 cases.Later, Mori, Momose, and Hamaguchi12 pointed out that
in the case of arbitrarily strong confinement two differecontributions to the magnetoresistance have to be conered, one of which stems from the LO-phonon-induced hping motion of electrons localized on cyclotron orbits. Thsituation is already accounted for in the earlier model of R9. The second contribution is, in classical terms, due toimpediment of the electron’s skipping orbit motion along tlateral confining potential wells. This latter contributioshould be dominant in strongly confined systems and althe qualitative features of the high-temperature magnetsistance insofar asmaximashould occur in the oscillatorypart ofrxx at resonance. The most recent theoretical invegation of the MP effect in quantum wires is given by Rand O’Connell,13 who used a different quantum transpoconcept14 for their calculations and obtained basically tsame results as mentioned before.9,12 However, despite theexistence of detailed theoretical work, to our knowledthere is only very little experimental work reported oMPR’s in Q1DEG’s, which so far did not give unambiguoinformation on the parameters relevant for the charactertion of quantum wires.15
In this paper we present an experimental study ofmagnetophonon effect in quantum wires fabricated byshallow-etching technique. By comparison with MD data otained for the same samples, we show that MPR’s arsuitable method for the characterization of the energy sptrum of laterally confined electrons. Subsequently, MPRare used for a systematic study of the behavior of sublespacing and magnetophonon effective mass as a functioelectron density. This is achieved for electron densitieslow that only a small number~four or less! of subbands isoccupied. We interpret both the high- and low-temperatdata by model considerations that are commonly emploin the investigation of shallow-etched quantum-wsystems.8,16
II. EXPERIMENTAL DETAILS
The relevant structure parameters of the conventiomodulation-doped GaAs/AlxGa12xAs heterostructures usein this work are summarized in Table I. The MP effectstudied on samples of the type g166, whose structure isshown in the inset of Fig. 1~b!. The only special feature othis heterostructure is the low overall doping that allowsget appreciable lateral confinement of the electrons wonly very shallow etching is done. Preliminary experimeon samples with higher sheet density~see Table I! showed
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that this is essential in order to resolve the MPR oscillatioat the relatively moderate magnetic fields~maximally 10 T!available with our experimental setup. The samples g1have proved to give the most pronounced and detailed stture of the magnetophonon resonances. The two othererostructures~g73 and g79! also listed in the table are useonly for reference experiments.
On bar-shaped mesas arrays of forty 200-mm-long quan-tum wires were fabricated using laser holography andchemical etching. The period of the wire array was 475in all cases and the geometrical width of the etched wiwas determined from scanning electron micrographs to bthe order of 150 nm. The longitudinal magnetoresistancemeasured in a two-terminal constant-current configuratusing a cryostat equipped with a variable temperature inand superconducting magnet coils supplying field strengup to 10 T.
The experiments on the samples g166 are performethe following way. Immediately after the etching process,wires are completely depleted at 4.2 K due to the low todoping of the underlying wafer structure. In order to stuthe transport properties of the wire array at varying electdensity, charge carriers are generated by illumination witred light-emitting diode. The so-obtained carrier density cthen be gradually reduced by heating the samples to anpropriately high temperature. It turns out that the persistelectron density obtained after a first temperature cyclevery stable and could only be reduced by heating the samup to at least 180 K. In order to compare low- and higtemperature data at varying electron concentration, thelowing procedure was applied. After illumination, thsamples were slowly heated to a temperature high enougobserve the MP effect with satisfactory signal-to-noise ratypically between 100 and 140 K. After equilibrium ireached, i.e., when a stable sample resistance is observehigh-temperature magnetoresistance is recorded showindistinct structure due to the magnetophonon effect. In a sond step the samples are cooled down slowly to 2 K andagain the magnetoresistance is measured to get an estof the electron density from a standard MD experiment. Fure 1 shows typical magnetoresistance traces for thetemperature regimes taken on a sample of the type g1Since temperature cycles between 2 and 100 K give the sresults in the MPR as well the MD experiments, we assuthat the same electron density is effective in the MPR andsubsequent MD experiment. Before a MPR experiment aTnear 100 K is performed again, the sample is kept at 18for several hours to achieve a gradual reduction of the etron density.
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3968 57PLONER, SMOLINER, STRASSER, HAUSER, AND GORNIK
III. EXPERIMENTAL RESULTS
A. Determination of the 1D subband spacingby the magnetophonon effect
In the case of weakly confined electrons the longitudimagnetoresistanceRxx(B) is expected to have a componeDRosc, which has the form of an oscillatory function dampexponentially with decreasing magnetic field.9 Since themeasurements reported in this paper were restrictedmagnetic-field range with a maximal field strength of 10 Tis necessary to obtain good resolution of the magnetophooscillatory structure inRxx at fairly low magnetic fields.Consequently, when quantum wires are fabricated usingshallow-etching method, one has to take into accountconsiderable surface roughness induced by the etchingcess. This scattering mechanism gives rise to an additibroadening of the Landau levels, which makes the resoluof MP oscillations at low magnetic fields more difficult. Indeed, a strong dependence of the relative amplitude andlution of the MPR on sample structure and preparationrameters is found. The magnetophonon effect in Q1DEG’observed to be very well resolved at low magnetic fieldsthose samples where impurity scattering is reduced as fapossible by, e.g., large spacer layers and low integral dopThe latter sample property also keeps the influence of sur
FIG. 1. Two sets of typical data recorded for sample g116/8.~a!Magnetic depopulation trace taken atT52 K. Only the relevantpart of the magnetoresistance forB<4 T is shown for clarity.~b!Longitudinal magnetoresistance, measured atT5100 K, revealinga pronounced structure at high magnetic fields due to the matophonon effect. The inset shows the sample structure of g166
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roughness scattering within tolerable limits since apprecialateral confinement can be obtained in these cases byshallow etching.
The experimentally observed influence of sample strture and etching depth on the amplitude and resolution ofmagnetophonon oscillations is summarized in Fig. 2. In tfigure the oscillatory part of the magnetoresistanceDRosc,obtained from theRxx(B) traces after subtraction of thmonotonic background resistance, is reproduced for varisamples. Figure 2~a! showsDRosc for two samples with dif-ferent wafer structure. The dotted line corresponds tosample of the type g73~cf. Table I!, with a larger dopingregion and a smaller spacer layer. For this sample type csiderably deeper etching is necessary to get a sufficiestrong lateral confinement. The solid line is obtained fosample of the type g166. The broadening of the magneelectric hybrid levels induced by the enhanced surface rouness scattering and the larger contribution of the remotepurity scattering for g73 results in a reduction of amplituand resolution of the oscillations as compared to g166.
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FIG. 2. ~a! Oscillatory partDRosc, obtained after subtraction othe monotonic background from the measured magnetoresistafor two samples fabricated from different heterostructures. The dted line corresponds to a sample of the type g73. The solidshows DRosc for a sample of the type g166.~b! DRosc for twosamples g166/1~lower trace, left ordinate scale! and g166/8~uppertrace, right ordinate scale!. On g166/1 slightly deeper etching thaon g166/8 is done. The electron densities present in the two samare of comparable magnitude. A slight increase of the etching deleads to a reduction of the oscillation amplitude by almost an orof magnitude~note the differenty-axis scales for the two samples!.
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57 3969ENERGY LEVELS OF QUANTUM WIRES STUDIED BY . . .
Fig. 2~b! the effect of different etching depth on the amptude ofDRosc is illustrated for two samples of the type g16one (g166/1) being slightly deeper etched than the o(g166/8). The result is again a reduction of the magnetosistance oscillation amplitude for the deeper etched samg166/1 by almost an order of magnitude. Thus, to obsesufficiently pronounced MPR, we had to restrict our invesgations to the case of relatively weak confinement, i.e., wdisplaying subband energy differences in the range omeV, and fairly low electron densities.
The main purpose of this work is to verify that the manetophonon effect can be used as a tool for the charactetion of Q1DEG’s, i.e., for the determination of subband eergies of quantum wires with small numbers of subbaoccupied. We therefore compare in Fig. 3 the results ohigh-temperature measurement with those of a magnetic
FIG. 3. ~a! Landau plot of the Landau level indexN8 versus theinverse magnetic-field position of the magnetic depopulatminima in the magnetoresistance measured atT52 K. The solidline interpolates the values obtained from a fit according tomodel of Ref. 8. The deviation from a straight line indicates thequantization of the system. The estimated sublevel spacing ismeV. ~b! Plot of the squared magnetic-field position of the resonminima in DRosc versus 1/N2 according to Eq.~1! for the samesample and electron density as in~a!. The corresponding value othe subband spacing extracted from the intersection of the resustraight line with theB2 axis is 1.6 meV.
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population measurement atT52 K, both performed on thesame sample g166/8. The electron density has been adjuusing the procedure described in Sec. II. Figure 3~a! shows aLandau plot obtained from the MD experiment. The subleindexN8 is plotted versus the inverse magnetic field positiof the MD minima in the magnetoresistance traces recorat T52 K. The deviation of the plot from a straight linshows clearly the 1D behavior of the laterally confined eltrons. The solid line interpolates between calculated pofitted to the data according to the model of Berggren, Roand van Houten,8 which assumes a parabolic form of thconfinement potential. The resulting subband spacing isE051.160.2 meV.
In Fig. 3~b! the squared magnetic-field positions of thresonant minima in the oscillatory part ofRxx ~cf. Fig. 2!,measured atT5100 K, are plotted versus an inverse squarindexN that is defined according to the generalized MP renance conditionN\veff5ELO , with ELO536.6 meV beingthe LO-phonon energy in bulk GaAs. If parabolic confinment is assumed,\veff is given by (\veff)
25(\vcyclotron)2
1E02, where E0 is the subband spacing of the 1D wire
Using this expression, the resonance condition can be rewten in the form
B25S m*
e\ D 2 ELO2
N2 2S m*
e\ D 2
E02. ~1!
According to this equation, theB2 values plotted versus1/N2 should lie on a straight line whose slope is a measurethe polaron effective mass in the quantum wire, whereasintersection with theB2 axis can be used for the determintion of the subband spacingE0 . It should be emphasized thathis simple relationship is valid only if the confining potetial can be approximated by a parabolic form. The expemental relationship betweenB2 and 1/N2 plotted in Fig. 3~b!is almost perfectly linear, which shows that the assumptof a parabolic confinement potential is a good approximtion. The latter statement is also confirmed by the resultsself-consistent calculations of the lateral confinement pformed for shallow-etched quantum wires of the type invtigated in this work.16,17 In the analysis according to Eq.~1!the LO-phonon energy for bulk GaAs is used, neglecting ainfluence of the lateral confinement on the LO-phonon sptrum as well as of interface phonon modes. With thesesumptions we obtain from the data shown in Fig. 3~b! amagnetophonon effective mass of (0.06960.007)me and asubband spacingE0 of 1.660.3 meV. E0 is enhanced incomparison to the corresponding value obtained from a ssequent magnetic depopulation [email protected] meV,Fig. 3~a!#. This enhancement was found in all investigatsamples where a direct comparison was possible and theband spacings determined from MPR lie some 30–50above those extracted from magnetic depopulation expments.
To explain the difference in theE0 values obtained at lowand high temperatures, respectively, we first note that thmal effects cannot be made responsible for this observatAt elevated temperatures the electrons are thermally rranged among the 1D sublevels. If this rearrangementaken into account self-consistently, it can be shown15 that itwill indeed lead to a slight enhancement of the subba
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3970 57PLONER, SMOLINER, STRASSER, HAUSER, AND GORNIK
spacing. However, these changes inE0 are small ~of theorder 10%! unless one assumes considerable rechargingreordering of the electrically active impurities, which is novery realistic assumption.
It rather turns out that the observed difference canconsistently explained if one takes into consideration thatactual confinement potential for shallow etched quantwires is not exactly parabolic but sinusoidal.16 This meansthat the results obtained from the MD experiments usinparabolic model potential have to be interpreted with socare. In the following it is qualitatively outlined how thshape of the potential leads to a difference of the subbspacings determined from MD and MPR experiments. Tquantitative details of these considerations will be discuselsewhere.
As shown in Fig. 4~a!, a sinusoidal potential leads to nonequidistant 1D subbands, the highest occupied subbanding more closely spaced than the low-lying ones. Now
FIG. 4. ~a! Cosine-shaped potential together with its loweenergy levels as obtained from the numerical solution of the cosponding one-dimensional Schro¨dinger equation. Note that the situation shown is only a qualitative account of the experimensituation in the text. The parameters chosen for the potential, exfor its width, are not intended to reproduce the experimental sition. The dashed line shows schematically the parabolic potethat reproduces best the positions of the highest, closely spsubbands.~b! Sum of an electrostatic cosine potential and a supimposed magnetic confinement according to Eq.~2!. For the cosinepotential the same parameters as in~a! are used. The magnetic fielis varied in 0.25-T steps. Already at a magnetic-field strength ofthe magnetic confinement dominates. At still higher magnetic fieit is therefore a good approximation to assume a parabolic formthe total potential.
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MD data are analyzed in the way indicated above, the minformation on the subband spacingE0 is extracted fromthose depopulation minima inRxx that occur at low magneticfields. These correspond to the depopulation of the hilying closely spaced energy levels of the wire and aresponsible for the characteristic deviation of the 1D Landplot from linearity. Thus, what is actually done in the evalation according to Fig. 3~a! is to approximate the upper nonparabolic part of the confinement by a parabolic potenwhose confining frequency reproduces best the subbspacing of the high-lying levels. This is indicated schemacally by the second parabola~dashed line! in Fig. 4~a!. Inother words, the subband energies extracted by usingindicated procedure correspond to those levels lying nearupper edge of the sinusoidal confinement potential.
On the other hand, the magnetophonon resonances inhigh-T magnetoresistance occur at the opposite end ofmagnetic field scale. In Fig. 4~b! we plot the sum of thesinusoidal electrostatic and the parabolic magnetic confiment for a series of magnetic fields according to
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In the figure, a zero-center coordinatex0 ~Ref. 8! has beenchosen for simplicity. According to this, the magnetic cofinement strongly dominates over the electrostatic onethose magnetic-field strengths where the MP resonancesobserved. The total potential is therefore parabolic to a gapproximation with the nonparabolic parts ofVe(y) only en-tering as a weak perturbation that can be neglected atmagnetic fields. Moreover, the magnetophonon resonanin Rxx correspond mainly to LO-phonon-induced transitiomainly from the lowest magnetoelectric hybrid statesthose lying 36.6 meV higher in energy, which correspondthose subbands lying deep in the electrostatic potentialother words, what is probed by the magnetophonon effectthe lowest-energy levels of the 1D confining potential, whihave, in the case of a sinusoidal electrostatic confinemenlarger spacing than the high-lying ones. This simple conseration not only explains the observed difference in the sband spacings but also suggests that a combination oftwo transport characterization methods, as employed inpresent work, potentially allows one to distinguish wheththe original electrostatic potential is of a sinusoidal~para-bolic! or rather of a square well form. In the latter case owould expect the subband spacings, as determined frommagnetophonon method, to be smaller than those extrafrom the MD measurement.
B. Dependence of the 1D subband spacing on electron density
Having shown the usefulness of MPR for the determintion of sublevel spacings or quantization energiesQ1DEG’s, one may use the method for a systematic studthe dependence ofE0 on the electron density by employinthe procedure described in Sec. II. This allows us to extthe investigation to regions where the electron density islow that the relevant information cannot be obtained froMD experiments alone.
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57 3971ENERGY LEVELS OF QUANTUM WIRES STUDIED BY . . .
Figure 5~a! shows the subband spacingE0 as a function ofthe 1D electron density for the two samples g166/1 ag166/8. Open symbols result from an analysis of the Mmeasurements atT5120 K ~for g166/1! and T5100 K ~forg166/8!, respectively. Solid circles show the results of manetic depopulation experiments performed for sample g16at T52 K. The latter could be obtained only in those caswhere a sufficient number of subbands were occupied toply the evaluation method proposed by Berggren, Roos,van Houten.8 It is for the same reason that correspondiMD values of E0 cannot be given for the deeper-etchsample g166/1 due to its considerably higher-sublevel sping in the displayed range ofn1D . The solid lines drawn inFig. 5~a! stem from a free linear fit to the MPR data~opensymbols! and are only intended as a guideline to the eye. Telectron densities underlying the plot in Fig. 5~a! were ob-tained from the MD data using8
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FIG. 5. ~a! Subband spacing as a function of the 1D electrdensity for samples g166/1~upper line! and g166/8~lower line!.Open symbols indicate the values obtained from MPR experimeThe solid circles show results of MD experiments performedsample g166/8. The horizontal arrow indicates the low-electrdensity range where an analysis of the subband spectrum of g1by MD is not possible.~b! Results of a low-temperature referenmeasurement on a sample of the type g79, using only the magdepopulation method. The electron density, which is considerahigher than in Fig. 4~b!, was varied by applying a back gate voltagThe overall sample geometry is exactly the same as for g166/1g166/8.
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which asserts that the high-field values in theN8 vs 1/B~Landau! plot should lie on a straight line having a slopproportional to (n1DE0)2/3. Exploiting this linearity of theN8(1/B) relation, the corresponding values for the ondimensional electron densities are extracted, taking forsubband spacingE0 and the effective massm* ~see below!the values obtained from the MPR data. This procedslightly underestimates the effective 1D channel densisince it uses the spacing of the subbands at the bottom oconfinement potential.
The observed increase of the subband spacing withcreasing electron density has been predicted by sconsistent calculations16 and can be qualitatively explaineas follows. The confining potential is formed by a periodmodulation of the donor distribution induced by the etchiprocess. If the presence of conduction electrons in thedefined 1D channels is taken into account self-consistenthe bare potential of this positive space charge is partiscreened out. Therefore, the confinement will be stronwith fewer electrons present in the wire. The same behaof E0 as a function ofn1D is also observed for variousamples with different wafer structure when the electron dsity is varied by alternative methods, e.g., by applying a bgate voltage. An example of this is shown in Fig. 5~b! for asample of the type g79. The higher electron channel denin these samples allowed the determination of the abfunctional dependence from MD experiments alone. Nthat the higher overall doping in these samples requires csiderably deeper etching in order to achieve appreciable cfinement. According to the discussion at the beginningSec. II, it is therefore not possible to achieve a sufficieresolution of the small-amplitude MPR oscillations. Therfore, no MPR data can be given for comparison in Fig. 5~b!.
Another effect of the reduction of the 1D electron densis the enhancement of the MP oscillation amplitude~see Fig.6, whereDRosc is shown for g166/8 at two different electrodensities!. This effect is also observed in 2D systems18 and isinterpreted by Moriet al.11 as a strong dependence of thLandau level broadening on the concentration of charremote impurities, an explanation that is also applicablethe present situation.
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FIG. 6. DRosc for sample g166/8 at two different 1D electrodensities, demonstrating the drastic increase of the oscillationplitude when the carrier density is reduced.
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3972 57PLONER, SMOLINER, STRASSER, HAUSER, AND GORNIK
C. Magnetophonon effective mass of 1D electrons
As shown above, the use of a parabolic magnetoelecconfinement potential in Eq.~1! is a good approximationFrom the slope of the linearB2(1/N2) plot @Fig. 3~b!#, there-fore, the magnetophonon effective massm* can be deter-mined. Figure 7 showsm* in units of the free-electron masme for the two samples g166/1 and g166/8 as a functionn1D . Again, the solid lines in the plots result from a frelinear fit to the data and serve only as a guide to the eyeshould be noted that no corrections for the conduction-bnonparabolicity or the resonant polaron effect have bmade in Eq.~1! since the so-achieved gain in accuracy lwithin the statistical error of 10% of the linear fit to thmagnetophonon resonant minima’s positions, an examplwhich is shown in Fig. 3~b!. The accuracy of the so-obtainemass values is nevertheless sufficient to make two mainservations.
As can be seen from Fig. 7,m* increases slightly withdecreasing carrier density. Whereas this observation is sowhat obscured for g166/8 due to the experimental error,more clearly revealed for the mass data of g166/1. Theond interesting fact is that for the more strongly confinelectrons of sample g166/1 a considerably larger effecmass is obtained than for sample g166/8. Currently, we hno conclusive explanation for this surprisingly large ehancement ofm* , but we are able to rule out several posbilities commonly considered to lead to a mass enhancemof Q1D electrons when the confinement strength increaFirst of all, we verified that the observed mass enhancemis not an artifact arising from the application of a parabomodel to an actually sinusoidal confinement potential. Wcalculated the energy spectrum of a quantum wire in a mnetic field using different model functions for the electrstatic confinement potentialVe(y) such as, e.g., a cosineshapedVe(y) as in Eq.~2! or a square-well potential with‘‘soft’’ walls. We simulated the application of the parabolmodel to these cases and found that the extracted mass
FIG. 7. Magnetophonon effective mass in units of the freelectron mass as a function of electron density for samples g1and g166/8. Sample g166/1 is characterized by a somewhat stroconfinement than for g166/8. The solid lines result from a flinear fit to the data and serve only as a guide to the eye. Tindicate a slight increase of the effective mass with decreasing etron density. Note the considerably enhanced mass values of g1compared to those of g166/8.
ic
f
Itdn
of
b-
e-isc-
eve--nts.nt
eg-
al-
ues are insensitive to the actual shape of the potential.evaluatedm* turned out to be simply the same cyclotroeffective mass, which was used in the calculations asinput parameter in the expression ofvc . Only if the ampli-tude of the confining potential is chosen unrealistically hiare the extracted mass values found todecreasefrom thatinput value. Thus the particular shape of the confining pottial is not responsible for the observed behavior of the efftive mass.
Aside from this, there are two possible reasons for a menhancement with increasing confinement strength:~i!nonparabolicity effects19 on the effective mass due to an increasing energetic distance of the 1D electron levels fromGaAs conduction band bottom and~ii ! an enhancement othe polaronic mass correction compared to the pure 2D cdue to the modified electron–LO-phonon interaction in qutum wires.20 However, the influence of nonparabolicity othe effective mass can be neglected in the present situasince it is appreciable only when the lateral dimensionsthe wire are reduced drastically.19 For example, in order toexplain an effective mass of 0.088me , which is the largestmass value obtained for sample g166/1, the lateral size owire had to be of the order 10 nm or even less. Howevercan be seen from the subband spacings in Fig. 5~a!, whichare not very different for the two samples, there should bedramatic decrease in wire width for sample g166/1. A rouestimate of the electrical or effective wire widths for the twsamples g116/1 and g166/8 can be obtained from an anaof the low-field magnetosize peak in the low-temperatumagnetoresistance.21 In this way it is found that the widthsfor g166/1 and g166/8 vary between 90 nm at an electdensity of 1.13108 m21 and 140 nm at 23108 m21. Theelectrical width for the deeper etched g166/1 is, at comrable electron density, some 10 nm smaller than the cosponding width of g166/8 but still of the same order of manitude. Thus the dramatic change in the effective mass isaccompanied by a sufficiently strong decrease of the wwidth in order to make a nonparabolicity argument plausibAlso the predicted enhancement of the polaronic mcorrection20 is appreciable only at wire widths below 10 nand does not give the correct order of magnitude to expour observations.
Therefore, other mechanisms will have to be considein order to explain our data. As has been shown fromlarge effect of an increasing etching depth on the MP oslation amplitude@cf. Fig. 2~b!#, the surface roughness is considerably more pronounced for sample g166/1 thang166/8. Since this is the only property that marks a signcant difference between the two samples, one could thina disorder-induced effect on the effective mass orelectron-phonon interaction in Q1DEG’s. This would alallow a simple interpretation of the fact that the effectimass values increase slightly when the electron densitreduced, which could then be attributed to a reduction ofscreening of that hypothetical disorder effect.
IV. SUMMARY
In summary, we have demonstrated that the magtophonon effect can be used for the investigation of theergy spectrum of quasi-one-dimensional electron system
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57 3973ENERGY LEVELS OF QUANTUM WIRES STUDIED BY . . .
systematic study of the dependence of the 1D sublevel sing on the channel electron density was performed usmagnetophonon resonance measurements as an alternastandard magnetic depopulation measurements in the cavery low electron density. The magnetophonon effect proto be particularly useful for the transport characterizationQ1DEG’s when only small numbers of subbands are ocpied at low electron densities. In this case a magneticpopulation experiment does not provide sufficient informtion on the relevant wire parameters. Moreover, it was shothat the combination of the two transport methods provideparticularly simple way to obtain direct information on thshape of the underlying electrostatic confinement potenThe second important parameter that could be determfrom our MP data is the effective massm* of the Q1Delectrons. It was found thatm* increases slightly when th1D carrier density is reduced and that stronger confinemobtained by deeper etching of the wires, enhances the mnetophonon effective mass considerably. At present therno conclusive explanation for this strong mass enhancemThe usual mechanisms known to lead to an enhanced etive mass in very narrow quantum wires either do not apto the present case or do not yield the correct order of mnitude of the observed effect. The shallow-etching methadvantageous though it is in certain respects, does not aa systematic variation of all the wire parameters neces
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for a systematic study of the mass enhancement with incring confinement strength. Therefore, in order to studyobserved phenomena in greater detail, alternative methfor wire fabrication will be applied in forthcoming expermental work. A promising technique is provided by the cobination of deep mesa etching and the deposition of sgates on the sides of the etched narrow channel~side gatetechnique; see, e.g., Ref. 22!. The achievement of relativelyhigh subband spacings is reported using this fabrication tenique and it is expected that it will also facilitate the studythe transition from the weak to the strong confinement cas well as the very special MP features predicted forlatter.
ACKNOWLEDGMENTS
The authors are indebted to Frank Stern for performself-consistent calculations of confinement potentials aelectron density distributions and for helpful discussionsthe subject. Thanks are also due to J. C. Portal, Y. LevinsJ. Allen, P. Streda, and W. Boxleitner for numerous discsions. This work was partly supported by the Bundesminterium fur Verkehr, Wissenschaft and Kunst, the Geseschaft fur Mikroelektronik, and the U.S. Army ResearcOffice Grant No. 34978PH.
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