The Consequences of Bulk Compressibilityon the Magneto-Transport Properties

within the Quantized Hall State

Aylin Yildiz1, Deniz Eksi2, and Afif Siddiki2+

1Department of Physics, Faculty of Sciences, Dokuz Eylul University, Tinaztepe Campus, 35160-Buca, Izmir, Turkey2Department of Physics, Faculty of Sciences, Istanbul University, 34134-Vezneciler, Istanbul, Turkey

(Received August 31, 2013; accepted October 2, 2013; published online December 9, 2013)

We discuss the role of direct Coulomb interaction on the bulk insulator of the integer quantized Hall effect that bridgesthe topological insulators and the conductance quantization. We investigate the magneto-transport properties of a two-dimensional electron system in the bulk, numerically, utilizing the self-consistent Thomas–Fermi–Poisson screeningtheory. Topologically distorted Hall bars with and without potential fluctuations are considered that comprises twoidentical inner contacts. Although these contacts change the topology, we show that quantized Hall effect survives due toredistributed incompressible strips on account of interactions. It is shown phenomenologically that the impedancebetween these contacts can be obtained by a minimal transport model. An important prediction of our self-consistentapproach is a finite impedance between the inner contacts even in the plateau regime, where the maximum of theimpedance decreases with increasing temperature.

1. Introduction

Since the discovery of the integer quantized Hall effect1)

(IQHE) in a two dimensional electron system (2DES), itstopological properties are investigated intensively.2–5) Theguiding gauge invariance argument of Laughlin is funda-mental to the phenomena, which focusses on a cylinderconsidering a strong, time-dependent magnetic field Bthreaded to its surface in the normal direction.6) Each timethe field is increased by one magnetic flux quantum(�0 ¼ e=h, e the elementary charge and h being the Planckconstant), an electron is argued to be transferred adiabaticallyfrom one side of the cylinder to the other side, protectingthe geometrical phase. The result is a flow of electricalcurrent proportional to the electromotive driving force, witha precisely quantized proportionality coefficient- the Hallconductance. The topological aspects of the IQHE assumingperiodic boundary conditions was first discussed by Thoulesset al., where Hall conductance is re-introduced as thetopological invariant of the 2 + 1D system.2,3) In thesehighly appreciated works, the topological character of the2DES is described in terms of the incompressible bulk stateand an expression is obtained via Kubo formalism for theHall conductance, which is in turn determined by the Chernnumber.5) The topological arguments rely on the fact that theHall conductivity is an integer multiple of e2=h, if the Fermienergy EF is in between two Landau levels. In this situationthe 2DES is incompressible in the bulk, namely there are noavailable states at EF. Within the non-interacting picture,charge transport along the edges is described by dissipation-less 1D edge channels,7,8) which completes the analogybetween the topological insulators and IQHE. It is veryimportant to emphasize that all the above descriptions of theIQHE are in the momentum space, k. To be explicit, oneassumes periodicity in k-space, perform calculations relatedwith geometrical phase in this space and then due to thesymmetry arguments maps momentum space directly to realspace. For sure, such a mapping can be justified only if theelectronic system is periodic both in momentum and in realspace. For instance if one performs calculations on a closed

cylinder, with a radial B field. However, when considering areal size sample with finite length and perpendicular field thesymmetry argument is prone to fail. In addition, disorderemanating from remote donors provide localized states whichresult in finite width quantized Hall plateaus in certain B fieldintervals. Hence, the incompressibility of the bulk throughoutthis interval is preserved and is essential to observe IQHE.Namely, without disorder the plateaus would shrink to asingle B value for an unbounded 2DES, where the Landaulevel filling fraction ¯ is an integer, which is defined by� ¼ Nel=N�0

.9) Here, Nel is the number of electrons in acertain area A and N�0

is the number of flux quanta in A.The incontestable concept of insulator bulk state has beenchallenged by the inclusion of interactions starting fromearly 1990’s advocated by Chang.10) Later, Chklovskii andco-workers analytically showed that the direct Coulombinteraction results in a metal-like (compressible) bulk,whereas the insulating states can also be at the edgessuppressing back (or forward) scattering within the plateauinterval.11) In the last decade, the bulk and edge electrostaticproperties of the 2DES is investigated with the improvementof local probe techniques.12–14) Measurements show thatthe spatial distribution of the theoretically predicted edgeincompressible strips and the experimentally observed poorscreening regions coincide nearly perfect. However, with acrucial qualitative difference: the edge strips become trans-parent to Hall field whenever the widths of these incompres-sible strips become comparable with quantum mechanical orthermodynamical length scales. This discrepancy is removedwhen the electrostatic problem is solved also taking intoaccount quantum mechanical quantities, like the finite widthsof the wave function, together with the effects of finitetemperature and level broadening.15,16)

The purpose of this paper is to discuss the role ofinteractions on the bulk insulator of the IQHE by solving theSchrödinger–Poisson equation in 2D coordinate-space, self-consistently. By implementing two inner contacts, weinvestigate the magneto-transport properties of the bulk ofa Hall bar by calculating the spatial redistribution of theinsulating states which can reside both at the edges or at

Journal of the Physical Society of Japan 83, 014704 (2014)

http://dx.doi.org/10.7566/JPSJ.83.014704

014704-1 ©2014 The Physical Society of Japan

the bulk. We find that, the inner contacts leave the QHEunaffected although the topology of the system is remarkablychanged from genus 0 to 2. Next, we explicitly includedisorder to our self-consistent screening calculations by anadditional impurity potential resulting from ionized donors. Itis observed that, the edge and bulk incompressible regionsmerge due to potential fluctuations resembling the topolog-ical bulk theory of the IQHE. We finalize our investigation bycalculating the impedance between the inner contacts by aminimal transport model, which predicts a measurablepotential difference between them even within the plateauinterval. This prediction imposes that the bulk of a Hall baris not an insulator throughout the plateau interval, hencechallenges the analogy between the topological insulators andthe IQHE. Our predictions can be easily tested in experimentstogether with the temperature dependency.

2. The Model

Assuming that the electrostatic quantities vary slowly onthe quantum mechanical scales such as magnetic length(‘ ¼ ffiffiffiffiffiffiffiffiffiffi

h�=eBp

), we employ the well appreciated Thomas–Fermi approximation to obtain the electron density andpotential distributions.15,17,18) Then the center coordinate X(¼ �‘2ky, where ky is the quasi-continuous momentum in y-)dependent eigenvalues can be calculated in the lowestorder perturbation as EnðXÞ � En þ VðXÞ; En ¼ h�!cðnþ1=2Þ, where !c ¼ eB=m� is the cyclotron frequency, VðXÞis the potential (energy) and n is the Landau index. Wecalculate the electron density nelðx; yÞ and the total potentialenergy Vtotðx; yÞ from the following self-consistent equations:

nelðx; yÞ ¼Z

dEDðEÞ

e½EþVtotðx;yÞ��?elch�=kBT þ 1

; ð1Þ

and

Vtotðx; yÞ ¼ Vextðx; yÞ þ Vintðx; yÞ; ð2Þwhere Vextðx; yÞ is the external and Vintðx; yÞ is the Hartreeinteraction potential (energy) between the electrons in thesystem is given by

Vintðx; yÞ ¼ 2e2

�

ZKðx; y; x0; y0Þnelðx0; y0Þ dx0dy0: ð3Þ

In above equations, DðEÞ is the density of states, �?elch is the

electrochemical potential being constant in the absence of anexternal current, � (� 12:4, for GaAs) is an average dielectricconstant and Kðx; y; x0; y0Þ is the solution of the Poissonequation considering periodic boundary conditions.19) Notethat the electrochemical potential in equilibrium is defined as�?elch ¼ �� jej�ðx; yÞ, where ® is the chemical potential and

�ðx; yÞ is the electrostatic potential. In addition, the chemicalpotential is determined by the statistical description assuminga grand canonical ensemble which is in contact with areservoir. Starting from T ¼ 0 and B ¼ 0 solutions, weobtain nelðx; yÞ iteratively in thermal equilibrium, keeping thedonor density n0 distribution fixed and average electrondensity constant, i.e., ® is constant and position independent.Since our results are independent of the particular nature ofthe single particle gap, we assume gs ¼ 2.

3. Results and Discussion

To discuss the role of interactions on the bulk insulator of

the quantized Hall effect, we will investigate the electrondistribution of a clean Hall bar (i.e., without potentialfluctuations resulting from ionized donors) and a disorderedHall bar both comprising two inner contacts. In both cases,we will focus on the formation of the incompressible regionsand investigate whether if they reside at the bulk or at theedge. In the presence of a strong disorder potential, we willshow that bulk and edge incompressible strips merge.

3.1 The clean Hall barWe consider a d ¼ 7µm wide Hall bar without potential

fluctuations induced by disorder on which two identicalsquare contacts (of size ³1µm2) are defined in the interior.The properties of the heterostructure and the details of 3Dself-consistent calculations are given elsewhere.20) In Fig. 1,we show the electron density distribution [or equivalentlylocal filling factors, �ðx; yÞ ¼ 2�‘2nelðx; yÞ] as a function ofspatial coordinates, where the color gradient depicts com-pressible and black regions correspond to incompressibleregions with constant electron density. The 2DES effectivelyvanishes beneath the inner contacts and at the edges (whiteareas), which is bordered by broken (red) lines. Along theincompressible regions the electrostatic potential varies dueto poor screening and is approximately flat when the regionis compressible, i.e., �ðx; yÞ varies. Since, the formation ofincompressible regions is shown to be sensitive to temper-ature and the strength of the magnetic field, we will considersituations where the electron temperature is sufficiently lowso that the incompressible regions are well developed.Namely, if kT=h�!c & 0:05 the incompressible region issmeared out.16,21) Here, we present results only consideringthe quantized Hall plateau interval. In Fig. 1(a), we show asituation where two incompressible strips reside along theedges and two encircling the inner contacts. The inset depictscuts along the x-direction, where one sees that encircling onesare narrower than that of the edge incompressible strips. Laterwe will argue that, once the incompressible regions becomenarrower than thermodynamical length scales they becomeevanescent and unable to decouple inner contacts. Bydecoupling, we mean that electron transport between twocompressible regions is hindered by the incompressibleregion. Namely, scattering or tunneling is not possible acrossthe stirp. At a higher B field, we again observe that the 2DEScomprises two edge incompressible strips parallel to they-axis together with circular strips surrounding inner contacts[Fig. 1(b)], however the strips become wider. In this situationthe bulk is compressible, in contrast, the inner contacts aredecoupled from each other by the encircling strips, known asthe Corbino effect.22) At �c ¼ 1:45, the incompressible stripsmerge and the sample becomes approximately incompres-sible, as depicted in Fig. 1(c). Increasing B furthermoreresults in an electron distribution where the incompressibleregions are disconnected around the contacts, as shown inFig. 1(d), however, are percolating along the edges. In thissituation a dissipative transport between inner contacts ispossible, which will be described by Ohm’s law.

3.2 Hall bar with disorderThe above situation is altered if one considers disorder. To

analyze the disorder effects on the electron distribution, weembodied charged impurities to donor layer resulting in

J. Phys. Soc. Jpn. 83, 014704 (2014) A. Yildiz et al.

014704-2 ©2014 The Physical Society of Japan

potential fluctuations. Following a recent investigation,23) weconsidered sufficiently high number of impurities (³2000).Note that, disorder is always included to our calculationsthat determines Landau level broadening and conductivities.Inclusion of the disorder, apparently enlarges the edgeincompressible strips as shown in Fig. 2(a). At a higher Bthe two edge strips merge with the encircling strips as shownin Fig. 2(b). By the inclusion of disorder, we observe that thebulk becomes incompressible for larger B intervals, hencewider quantized Hall plateaus are expected in accordancewith typical experiments.24) A characteristic filling factordistribution is shown in Fig. 2(c). Even grippingly, at aslightly higher B field the bulk incompressible region startsto become disconnected around the inner contacts, allowingdissipative current paths between them.

4. Transport between Inner Contacts

As mentioned, incompressible bulk is a key concept indeveloping the analogy between topological insulators andconductance quantization. However, we have shown that itis possible to have a compressible metal-like bulk withinthe plateau interval due to interactions. Next, we investigatethe transport between the inner contacts by calculating theimpedance when a finite potential difference is appliedbetween the inner contacts. In our model we assume that thetransport can be well described by a local version of theOhm’s law15) and the impedance is composed of resistiveR and capacitive C terms as, Z ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R2 þ aC2

p. Here, a is a

constant depending on the frequency of the imposedexcitation, which we take to be of the order of unity. Thecalculation of the (quantum) capacitance is somewhat trivial,since it is only proportional to the thermodynamic densityof states at EF, namely C ¼ e2DTðEFÞ. If there exists an

incompressible region [i.e., DTðEFÞ ¼ 0], which decouplesinner contacts then the impedance reads to infinity for a pureHall bar at T ¼ 0. However, for a realistic Hall bar withdisorder, i.e., with localized states at EF, and at finitetemperatures capacitance also becomes finite. To decidewhether if the inner contacts are decoupled or not by theencircling incompressible strips, our first task is to determinethe widths of the incompressible regions depending on B,which we calculate self-consistently and show in Fig. 3(a).Once the strip widths are wider than the thermodynamicallength scale, i.e., Fermi wavelength �F, one can assume that

Fig. 1. (Color online) Filling factor distribution in the Hall bar for characteristic �c ¼ h�!c=E0F values, where E0

F is the Fermi energy in the bulk calculatedfor a typical electron density of 3� 1011 cm¹2. (a) �c ¼ 0:81, (b) 1.31, (c) 1.45, and (d) 1.5, calculated at sufficiently low temperatures, � ¼ kT=E0

F ¼ 0:016.The inset depicts the spatial variation of the density along x-axis denoted by horizontal lines. Here W2 is the width of the incompressible strips with � ¼ 2.

Fig. 2. (Color online) Same as Fig. 1, however, with additional potential fluctuations emanating from ionized donors. The electron density is thereforeslightly (� 20%) increased, resulting in field strengths of (a) �c ¼ 1:36, (b) 1.46, (c) 1.49, and (d) 1.52 calculated at default temperature.

0,4 0,6 0,8 1,0 1,2 1,40,0

0,1

0,20

100

200

300

0,6 0,8 1,0 0,5 1,0

0

1

2

3

Zin

(a.u

.)

Ωc

c)

b)

Ωc

w2

(nm

)

τ0.00810.02430.04050.0567

a)

λF

DT

/D0

μ / E0

F

Fig. 3. (Color online) (a) Calculated widths of incompressible strips atdifferent temperatures and (b) corresponding thermodynamical density ofstates in units of D0 ¼ 2:8� 1010 meV¹1 cm¹2. (c) Resulting impedancebetween inner contacts without (thick lines) and with potential fluctuations(thin lines).

J. Phys. Soc. Jpn. 83, 014704 (2014) A. Yildiz et al.

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the inner contacts are decoupled. Hence, impedance isdominated by the capacitive term. Fig. 3(b) depicts DTðEÞ,obtained from Gaussian broadened Landau levels at differenttemperatures. The corresponding impedance is shown inFig. 3(c), where we observe that Z is constant [/ 1=DTðEFÞ]if incompressible regions decouple inner contacts. Otherwise,i.e., if the width of the incompressible strip becomes narrowerthan the Fermi wavelength, transport is determined by theresistive term which we assume again a Gaussian levelbroadening to calculate RðBÞ, see Fig. 3(b). The narrowincompressible strips are called evanescent25) and areexperimentally observed, recently.26) By increasing thetemperature the incompressible strips become narrower,hence, the constant Z interval shrinks while its amplitudedecreases due to larger available states at EF. We alsodepicted the disordered situation by thin lines in Fig. 3(c)where the bulk incompressible region is extended in B asshown previously. The result is wider constant Z intervals,which decrease in amplitude and shrink in width similar tothe pure Hall bar case by increasing the temperature.

5. Conclusion

Our numerical calculations show that the bulk of theelectronic system is not insulating for all magnetic fieldstrengths throughout the plateau interval and IQHE isinsensitive to topological distortion in real space. Thisfinding is in well agreement with existing local probeexperiments, however, is in contrast with the single particletheories of the IQHE. The discrepancy is removed by theconsideration of the disorder effects. In this work wequantitatively demonstrated that the edge and bulk picturesof the IQHE merge, which are both the limiting cases of thescreening theory.

In light of our calculations we propose to measure Hallresistance of a 2DES that embodies two inner contacts in thebulk and simultaneously measure the potential differencebetween them. A constant high impedance B subinterval ispredicted within the plateau regime, in contrast to singleparticle theories for which one expects the constant Z intervalto be spread all over the plateau regime. We claim that thevarying impedance evidences a compressible bulk. It is alsoexpected that the amplitude of the constant Z will decreasewith increasing temperature. Different from the early experi-ments27) we suggest to consider a relatively narrow Hall bar(d < 20µm) to be defined on a high mobility (>3� 106

cm2V¹1 s¹1) wafer.

Acknowledgments

We would like to thank J. Jain and K. von Klitzing forinitiating the idea of inner contacts. This work is supportedby TÜBİTAK under grant 112T264 and 211T148, IU-BAP:6970 and 22662. A.S acknowledges support fromBilim Akademisi—The Science Academy, Turkey under theBAGEP program. We also thank to ITAP, where the finalversion of the paper was edited.

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