PHYSICAL REVIEW B VOLUME 35, NUMBER 3 15 JANUARY 1987-II

Quantized Hall effect in quasi-three-dimensional systems

Tao Pang and C. E. CampbellSchool of Physics and Astronomy, University of Minnesotal, l6 Church Street SE, Minneapolis, Minnesota 55455

(Received 22 September 1986)

We apply the scattering theory of the quantized Hall effect to three-dimensional systems to ex-plain the recent experimental observations of this effect in superlattices, and also to determine thegeneral criteria for the existence of this effect in such systems.

Recent experimental observations of the quantization ofthe Hall effect in a superlattice system' provide evidencethat the integral quantum Hall effect (IQHE) is not re-stricted to purely two-dimensional (2D) systems. In par-ticular, the recent experiment is on a 30-periodGaAs/(A1Ga)As superlattice containing barrier heightsand widths which permit substantial tunneling in thedirection normal to the 2D planes of the junctions, i.e.,parallel to the applied magnetic field B=Bz. A plateauwas observed in the Hall conductivity at a value ofcr„y =48e /h to within five parts in 10 . The interpreta-tion of this measurement was that o„y =ije /h, where i =2is the Landau level index (corresponding to the lowestLandau level when the Zeeman splitting is unresolved),and j =24 is the number of occupied spatial states in thelowest band arising from the coupled layer structure. Thefact that j falls short of the number of periods was attri-buted to a possible depletion of several top and bottom lay-ers due to pinning of the Fermi level.

A plausible explanation of this quantization given byStormer et al. is that, while bands are developed in theexcitation spectrum due to the layering, thereby lifting thej-fold degeneracy which would otherwise be present ifthere were no tunneling or hopping between the stack ofheterojunctions, the quantization will be retained as longas the resulting first minibandwidth 8' is exceeded bythe gap between Landau levels (W, ( h, co„whereco, =eB/m„yc is the cyclotron frequency for xy effectivemass m„y) so that the Fermi level lies in the localizedstates.

Theoretical support for this interpretation was recentlygiven by Vasilopoulos. However, the unit of quantizationwas found to be shifted to (1+I )e /h, where I arises fromthe scattering of electrons between quantum wells by inho-mogeneities (in this case optical phonons). The estimateof I for the experimental system described in Ref. 2 is ap-proximately 1.5/10, and thus exceeds somewhat the es-timated error in the experiment. This apparent disagree-ment between theory and experiment on the unit of quant-ization leaves open the question of whether this quantumunit may be weakly sensitive to the dispersion in the zdirection.

The theoretical approach used in Ref. 4 makes use ofcertain approximations which lend some uncertainty to thefinal conclusion. Since there have been several theoreticalapproaches to the IQHE in 2D which have providedrigorous proofs of the quantization and the precise quan-tum unit e /h, it is important to see if these approaches

can be generalized to the quasi-3D systems along with aprediction of the unit of quantization. In this note we be-gin with a general model of the 3D systems and use thescattering method to find the criteria for the IQHE inthese systems.

The scattering approach examines the role of randomimpurities in establishing the IQHE. The result of thisanalysis is that the Hall current carried by a band of ex-tended states originating from a Landau level is unaffectedby a random potential generated by an arbitrary numberof impurities. The principal result of our extension ofthis analysis to quasi-3D systems is that this conclusion,including the unit of quantization and the values of 8where the plateaus are found, is unaffected by the three-dimensional nature of the system under certain well-defined conditions. We find that the Hall conductivity isquantized in the form

o'„y =jie z/h

where i is the number of Landau levels occupied (includ-ing the spin states) and j is the number of spatial states inthe lowest occupied z band. If a superlattice consists of nwells, j will be an integer with j ~ n.

The criteria for these results are best described in termsof the energy-level structure of the system. For a superlat-tice system, which has a periodic potential Vp(z ) over a fi-nite number of periods, the criteria for the IQHE are thatW, & Aco, & h,„where 5, is the gap between the first andsecond minibands in e, (k, ) (given below); andA. co, )&kgT. If, for example, the magnetic field is de-creased until hm, & 8'„ the quantization will be washedout. Thus the maximum observed plateau provides abound on the minibandwidth. For the minibandwidth es-timated in Ref. 2 this would correspond to 8 & 1.4 T, cor-responding to i & 10. This band overlap may also accountfor some of the unusual low-field behavior found in Ref. 2.

Our analysis follows closely that of Ref. 8. We considera 3D sample with size (L„,Ly,L, ) in a strong magneticfield 8 in the z direction and a weak applied electric fieldin the x direction. The Hamiltonian can then be written as

H =Hp+ V(x,y,z ),H p

= [p„,py +eB (x +a )/c l1

2mxy

p 2

+ ' +V,(z)+W(x),2mz

1459 1987 The American Physical Society

1460 TAO PANG AND C. E. CAMPBELL

where W(x) is the potential of the applied field and a po-tential confining the electrons to 0» x» L„, Vo(z) is az-dependent potential such as a periodic potential in a su-perlattice system, V(x,y,z) is the scattering potentialwithin the system, and the magnetic field is included viathe Landau gauge. An additional gauge field A~ =aB withgauge momentum g =eBa/c is introduced as a generatorof the y component of the average velocity v y in eigenstate~a& of H:

&~~1~(

BH)Bg BPy

(4)

For the purpose of setting boundary conditions suitable fora theory of the current in the y direction, the sample is ex-tended to y =+ ~ and periodic boundary conditions areimposed in that direction, with the further requirementthat the impurity potential V vanish in the periodic exten-sion. In that case the asymptotic eigenstates are alsoeigenstates of Ho. Thus for y «0 the extended eigenstatesmay be written as

4'(x,y,z)=co(x,y,z) =y(z)e" ' "(x ~p, v) .

When W(x ) varies more slowly than the magnetic lengthI =dhc/eB, the eigenvalue spectrum is given by

s„(k„p)=@co,(v+ —,' )+W(xy)+s, (k, ),

where xy =a —pc/eB is the center of the harmonic oscilla-tor and e, (k, ) is the single-body eigenvalue in the z direc-tion with quantum number k, .

The effect of the impurity region on these extendedstates is to introduce a phase shift from one side of the im-purity region to the other. For the simple case when s(p)is a monotonic function of p, the boundary conditions in yrequire that

p+6(p, k )/Ly g =hn/Ly

where e is the phase shift and n is an integer. Since theonly dependence of s„upon g is through x~, one finds thatthe current density may be written as

~y= e g favnk, Uevn/c, y/(LxLy )

o vnk,

ac„(k„p„)favnk,crvnk,

(8)

(9)

where f, is the occupation probability for state a. Theonly terms which enter into the sum are those withnonzero velocity, namely, those corresponding to extendedstates. Since the k, dependence, the v dependence, andthe n dependence enter e separately, the o, v, and k, sumscan be done immediately, giving

ij e r)s(pn)

L„h „r)n (10)

Following Ref. 8 closely, Eq. (10) can be averaged overone flux quantum, with the result that

We would like to acknowledge stimulating discussionswith E. D. Dahlberg. This work was supported in part bythe Microelectronic and Information Sciences Center ofthe University of Minnesota, and by the National ScienceFoundation Grant No. DMR-8406553.

where p (p ) is the momentum at the upper (lower) edge ofa Landau band. This energy difference is identified as eU,where U is the applied voltage. Therefore the quantizedresult is obtained as

cy y=Jy/E =JyLX/U =jie /h

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