The quantum Hall effect under the influence of a top-gate and integrating AC lock-in measurements
TThe quantum Hall effect under the influence of a top-gateand integrating AC lock-in measurements T OBIAS K RAMER , E
RIC
J. H
ELLER , AND R OBERT
E. P
ARROTT Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg,Germany Department of Physics, Harvard University, Cambridge, MA 02138, USA Department of Chemistry and Chemical Biology, Harvard University, Cambridge,MA 02138, USA School of Engineering and Applied Science, Harvard University, Cambridge,MA 02138, USA
May 20, 2009
Abstract
Low frequency AC-measurements are commonly used to determine the voltage and currents throughmesoscopic devices. We calculate the effect of the alternating Hall voltage on the recorded time-averagedvoltage in the presence of a top-gate covering a large part of the device. The gate is kept on a constantvoltage, while the Hall voltage is recorded using an integrating alternating-current lock-in technique. Theresulting Hall curves show inflection points at the arithmetic mean between two integer plateaus, whichare not necessarily related to the distribution of the density of states within a Landau level.
In Ref. [1], we reported evidence for inflection points of the Hall resistivity at half-integer filling factors,where the slope of ∂ρ xy /∂B goes through a local minimum. We attributed the existence of these inflectionpoints to features of the LDOS in the presence of a strong potential gradient near the injection corner ofthe device. In march 2009, new experiments in a different sample have shown additional features, whichrequire a new interpretation of the earlier data as well as of the new experimental results. In particular, aftermodelling the measurement protocol in every detail, we propose a different interpretation of the inflectionpoints shown in Ref. [1]. For an alternating current flow we predict oscillatory shifts of the Fermi energyin the two-dimensional electron system due to the presence of a non-zero Hall voltage. The oscillatoryshifts are quantitatively calculated in this manuscript and provide a different explanation of the data shownin Ref. [1]. We do not rule out the existence of inflection points caused by the LDOS in direct-currentmeasurements. In order to obtain a better signal-to-noise ratio, alternating-current (AC) measurements are often preferredto direct-current (DC) measurements. While there are different ways to record and process the AC signal,one commonly used protocol is to use a cos( ωt ) -AC wave-form for the current between source and drainof a Hall device I ( t ) = ˆ I cos( ωt ) = √ I rms cos( ωt ) , (1)1 a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y
60 −40 −20 2000.20.40.6 µ m E c [V] z [nm]cap doped n-AlGaAs spacer µ s − µ m = 0 → V g = 0 mV → n D = 2 . × m − E c ( z ) calculation for LM4640 µ s E (2 D ) F −60 −40 −20 2000.20.40.6 µ m E c [V] z [nm]cap doped n-AlGaAs spacer µ s − µ m = −
50 meV → V g = −
50 mV → n D = 2 × m − E c ( z ) calculation for LM4640 E (2 D ) F µ s Figure 1: Calculated conduction band profile E c ( z ) along the growth z -direction of the sample LM4640.Left panel: In the absence of a potential difference between gate potential µ m and 2DEG potential µ s , thedensity is n D = 2 . × m − and E DF = 8 . meV. Right panel: A potential difference of meVbetween µ m and µ s (by putting a negative voltage of − mV on the top-gate) shifts the quantum well andreduces the density to n D = 2 . × m − , while the Fermi energy shifts to E DF = 7 . meV.and to record the Hall voltage by integrating the instantaneous Hall voltage over one period of the cosinesignal. In order to amplify only signals matching the frequency of the current-modulation, the oscillating cos( ωt ) -signal is fed into the integration loop. Thus the read-out Hall voltage of the AC-lock-in amplifierbecomes V read H = c (cid:90) π/ω d t R xy [ I ( t )] I ( t ) cos( ωt ) , (2)where c denotes the normalization, determined experimentally by comparison with a known resistance. The top-gate is kept on a constant potential (the same potential as one of the Ohmic contacts), while thevoltage of the second Ohmic contact is oscillating. This leads to an oscillating AC current through thedevice. In the presence of a magnetic field the Hall voltage forms across the sample in response to thecurrent. The Hall voltage is also oscillating in phase with the current. The maximum Hall voltage duringthe oscillation period is given by ˆ V H = ˆ IR xy ( B ) , (3)and yields for I rms = 1 µ A at R xy = 19 k Ω a peak Hall voltage of ˆ V H = 27 mV. The range of variationof the Hall voltage is thus × mV = 54 mV. The measured sample LM4640 has been grown with thefollowing layer sequence:Material ThicknessAu/Ti gate on topUndoped GaAs cap nmSi doped Al . Ga . As ( N Si = 2 . × m − ) nmUndoped spacer Al . Ga . As nmUndoped GaAs µ mWe have calculated, based on [2] eq. (9.6), the conduction-band structure of the sample using the triangularwell approximation for the zero-point energy of the quantum well and under the assumption that a fractionof / Si atoms gets ionized. Part of the electrons will contribute to the 2DEG, while another part will2 R xy Magnetic field [Tesla] V g = − .
05 V V g = 0 Vnotice the shift of the plateaus due to -50 mVgate voltageTheory for the sample LM4640 with top-gate: effect of putting the gate on -50 mV R clxy [ V g = − .
05] V R clxy [ V g = 0] Figure 2: Effect of applying a fixed voltage to the top-gate (theory). A negative gate-voltage reduces thedensity and leads to a steeper Hall curve.contribute to the surface states. We assume that the surface states lead to an offset of . V. The resultingconduction band diagram for zero gate voltage V g and for a negative gate voltage of V g = − mV isshown in Fig. 1. Note that the gate voltage is defined as the difference in potentials of the metallic top-gate µ m and the potential of the two-dimensional electron gas (2DEG) µ s : V g := µ m − µ s e . (4)The calculated values agree very well with the measured densities inferred from Hall experiments at gatevoltages and − mV, shown in Fig 2. The possibility to control the density of the 2DEG by applying agate-voltage and shifting µ m is well-known.Interestingly, eq. (4) shows that the “gate-voltage” can be set by either changing µ m or changing µ s .A change in µ s does not usually occur in a 2DEG, but the Hall effect is an important exception from thisrule. In a Hall device, the 2DEG is not on a single potential across the device, rather the potential of the2DEG drops from one side to the other side of the device by the Hall potential µ H . Thus the presence ofthe Hall potential generates a non-zero difference between the metallic gate at potential µ m and the 2DEGat potential µ s + µ H . The Hall potential increases for fixed magnetic field with increasing current. Fromeq. (4) we deduce that a change of the potential of the 2DEG is equivalent to applying a gate voltage andthus adjusting the density and the average slope of R xy ( B ) .The last observation bears important consequences for the interpretation of AC-experiments at non-zerocurrents, which induce a possibly large Hall voltage: the recorded Hall voltage is a superposition of all Hallvoltages present during one period of the integration. The AC-experiment in the presence of a top-gate,which is kept on a constant potential µ m , records the (weighted) mean of many Hall-curves, where eachindividual Hall curve is effectively measured at a different gate voltage, since the Hall potential oscillatesin phase with the current: µ = µ s + µ H ( t ) . (5)The effect of the oscillations of µ s + µ H ( t ) on the quantized Hall curves is precisely the same as puttingan oscillatory voltage on the top-gate during the integration process of the lock-in measurement, see Fig. 3.3 R xy Magnetic field [Tesla] V g = 0 , V H = −
25 mV V g = 0 , V H = +25 mVTheory for the sample LM4640 with top-gate: oscillating Hall voltagein this region a plateau survives averagingin this region an AC measurementrecords the arithmetic mean of two plateaus Figure 3: Effect of the oscillating Hall voltage underneath top-gate kept on a constant potential (theory).Integrating over one period of the AC-lock-in amplitude records the (weighted) mean of different Hallcurves. Note that the figure is schematic, since V H is dependent on the magnetic field and is thus thechange in slope of the Hall curve is less pronounced at lower magnetic fields. The complete simulation,which takes the Hall-voltage dependence into account, is shown in the appendix.In Fig. 3 extended regions are visible, where the lock-in integration will lead to a read-out voltage corre-sponding to the arithmetic mean of two adjacent Hall plateaus (in LM4640 this occurs around B = 6 T, B = 4 T, and B = 3 T).The implications for experiments are that inflection points are induced by the AC-protocol at values of R inflection xy = 25812 Ω 12 (cid:18) n + 1 n + 1 (cid:19) , n = 1 , , , . . . , (6)corresponding to the arithmetic mean of two adjacent Hall plateaus. Converted to filling factors, we expectAC-inflection points at ν = (cid:20) (cid:18) (cid:19)(cid:21) − = 4 / (7) ν = (cid:20) (cid:18)
12 + 13 (cid:19)(cid:21) − = 12 / (8) ν = (cid:20) (cid:18)
13 + 14 (cid:19)(cid:21) − = 24 / (9)Indeed a close inspection of the experimental record shows that in an AC-measurement inflection pointsoccur at the arithmetic mean R inflection xy Experiment [Ω] ν at 19359 Ω Ω ( ν = 1 . ) ν at 10755 Ω Ω ( ν = 2 . ) ν at 7528 Ω Ω ( ν = 3 . )4he LDOS signature analyzed in Ref. [1] predicted inflection points at R inflection, LDOS xy = 25812 Ω 1 n + 1 / , n = 2 , , (10)corresponding to the following filling factors ν LDOS = 52 (11) ν LDOS = 72 (12)and values of R xy : R inflection, LDOS xy Experimentnot expected (only intersection point at 17208 Ω ) 19320 Ω ν = 5 / at 10325 Ω Ω ν = 7 / at 7375 Ω Ω The new experimental data and the detailed model of the AC-measurement protocol suggest a new inter-pretation of the data presented in Ref. [1]:1. AC averaging explains an additionally observed feature at ν = 4 / , not contained in the LDOStheory.2. The observed inflection points may not be the signature of the LDOS. The observed inflection pointsare closer to the values ν = 4 / , ν = 12 / , and ν = 24 / , predicted by the AC-model of thetop-gate, than to the values ν = 5 / and ν = 7 / , predicted by the LDOS theory.3. The overall agreement of the AC averaging model with respect to the width of the Hall plateaus andthe slope of the Hall curve is very good (see comparison in the Appendix).It is interesting to note that the gate keeps the two-dimensional Fermi-energy fixed, and no assumptionof disorder is required to describe the physics of the device and to calculate the Hall curves shown in theappendix. Thus we do not rule out the possibility to construct a modified injection model of the QHE, whichincorporates the current injection process and the Hall field in its foundations. Further DC experiments areunder discussion to reveal the shape of the LDOS in the injection region. References [1] Tobias Kramer, Eric J. Heller, Robert E. Parrott, Chi-Te Liang, C. F. Huang, Kuang Yao Chen, Li-HungLin, Jau-Yang Wu, and Sheng-Di Lin. Half integer features in the quantum Hall effect: experimentand theory. 2008. Online: http://arxiv.org/abs/0811.3572v1 .[2] J. H. Davies.
The physics of low-dimensional semiconductors . Cambridge University Press, 1998.
A Comparison of Hall curves
On the next pages we compare the theoretical prediction of the LDOS theory of Ref. [1], and the AC-averaging theory given in the present manuscript. 5 ρ x y B [Tesla] ν = ν =
100 nA400 nA800 nA1250 nA1750 nA2500 nA
AC-lock-in amplifier (cosine-wave), sample with top-gate
Figure 4: Theory: AC-model with a featureless Gaussian broadened DOS. The legend denotes the value of I rms . The calculation takes the change of the Hall voltage with magnetic field into account and the effectivegate-voltage caused by the Hall potential, even though the gate is kept at a fixed potential. The curves aredirectly comparable to the experimentally recorded ratio V read H /I rms obtained with an AC-lock-in amplifier.6 = ν =
100 nA400 nA800 nA1250 nA1750 nA2500 nA
Theory (RC) LM4640 B [Tesla]01000200030004000500060007000800090001000011000120001300014000150001600017000180001900020000210002200023000240002500026000 ρ x yy