Recent results of high magnetic field experiments on 2D systems: Localization and quatized hall resistance

Physica B 164 (1990) 50-58 North-Holland

RECENT RESULTS OF HIGH MAGNETIC FIELD EXPERIMENTS ON 2D SYSTEMS: LOCALIZATION AND QUANTIZED HALL RESISTANCE S. KAWAJI Department of Physics, Gakushuin

University, Mejiro, Toshima-ku,

Tokyo, 171, Japan

Hall conductivity crXvand diagonal conductivity ox,, of Si-MOSFETs were measured using the Hall current method in two magnetic fields of 15 and 25 T sweeping the gate voltage and at a fixed gate voltage sweeping the magnetic field up to 27 T. Temperature ranges were 1.5 K < T < 0.35 K in the 15 T experiment and 0.8 K < T < 0.05 K in the higher field experiment, respectively. The critical exponent of the localization in Landau subbands of a 2D system in Si inversion layers depends on the Landau quantum number and is in accordance with the numerical studies of localization. In high precision measurements of quantized Hall resistance R,(i) of Si-MOSFETs and a GaAs/AlGaAs heterostructre, the value of i x R,(i) depends on the device and the Landau quantum number at the level of 0.1 ppm. The result suggests possible deviations of i x R,(i) from the universal value hle’.

1. Introduction Two-dimensional (2D) electron systems at low temperatures and in high magnetic fields that quantize the 2D motion, show interesting transport properties arising from the discrete density of state. The transport properties depend strongly on the strength of the electron-impurity interaction and the strength of the electron-electron Coulomb interaction. When the electronimpurity interaction is stronger than the electron-electron Coulomb interaction, the 2D system shows the integral quantum Hall effect (IQHE) [l-3]. The fractional quantum Hall effect (FQHE) appears when the electron-electron Coulomb interaction is stronger than the electron-impurity interaction [4,5]. It is believed that a quantum liquid state arising from the electron-electron Coulomb interaction causes the FQHE [6]. A charge density wave state [7], which could be interpreted as the Wigner crystal [S], is another possible ground state of the system with the electron-electron Coulomb interaction when the system has a low density of electrons. However, not the Wigner crystal state but the FQHE has been observed at low densities down to v = l/7 where v is the filling factor of the lowest Landau subband ]9,101. Though the transport phenomena in 2D sys-

terns in high magnetic fields are interesting, the quantitative measurement of transport coefficients in the whole range of the Hall angle is a very difficult task, with the exception of the diagonal conductivity measurement using a Corbino disk electrode structure [ll, 121. In this paper, our recent results of experiments on 2D systems in high magnetic fields are presented. In the next section, results of localization studies on Si-MOSFETs performed in Gakushuin University (GU) using magnetic fields up to 15 T and performed in the Institute of Material Research (IMR), Tohoku University, in collaboration with IMR scientists, using magnetic fields up to 27 T at the High Field Laboratory for Superconducting Materials (HFLSM) are described. In section 3, results of high precision measurements of quantized Hall resistances of Si-MOSFETs and a GaAs/AlGaAs heterostructure performed at GU in collaboration with scientists from the Electrotechnical Laboratory (ETL) are described. In the final section, a discussion on these results is given.

2. Localization MOSFETs

in

Landau

subbands

of

Si-

The first systematic experimental study of electron localization in Landau subbands of 2D sys-

0921-4526/90/$03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)

51

S. Kawaji I Localization and quantized Hall resistance

terns was carried out for Si-MOSFETs by Kawaji and Wakabayashi [13]. We measured the diagonal conductivity ax1 at 1.4 K in magnetic fields up to 15 T and found that the sum of concentrations of localized electrons associated with the higher edge of the (N - l)th Landau subband and the lower edge of the Nth Landau subband, N being the Landau quantum number associated with radius of Landau orbit I, = (2N + 1)“*1,, I, = (he/B)“*, is approximately given by [2~ 1:(2N + 1)1-r. Here, the electron concentration means the surface density of the electron number. This result means that the electron wave functions in a Landau subband with index N become extended when the whole area of the inversion layer is covered by cyclotron orbits with radius I, at this temperature. The localization plays the most important role in the IQHE [14,15]. Our method of analysis of the localization is as follows. A classical expression for the Hall conductivity in 2D systems is given by ffXJ = - N,eIB + o-~~/o,~

(1)

where N, is total electron concentration, B magnetic flux density, w, cyclotron angular frequency and r electron elastic scattering time. When an extreme quantum limit condition (k, T < r < fiw,), where r is the broadening of a Landau subband, is fulfilled and N, = ieBlh, where i is an integer, is fulfilled, we have a,.. = 0 because no electron scattering occurs in filled Landau subbands as well as between filled and empty Landau subbands. Then the Hall conductivity is quantized as u *Y =

-

ie*/h

(2)

Ando et al. [l] calculated for the first time the Hall conductivity of 2D systems in high magnetic fields based on Kubo’s formula. One of their results is that eq. (1) is valid in quantum transport if one replaces 1 /w,r by Tlhw,. Their important result is that eq. (2) is valid when ax, = 0 even if N, is not equal to ieBlh. Ando et al.? theoretical result of the quantized Hall conductivity was experimentally confirmed by Kawaji and Wakabayashi [2]. We

measured temperature dependences of a,, and mX,,of a Si-MOSFET in a field of 15 T at temperatures between 14 and 1.5 K and observed that the width of the gate voltage region where a,, is given by eq. (2) corresponds to mXx, = 0, increases with a decrease in temperature. It should be mentioned here that we employed the Hall current method [16] in the measurements of a,, and uXyin this work and the following experiments. Kawaji et al. [17] first tried to analyze the temperature dependence of a,, by an effective mobility edge model. We assumed first that the second term is much smaller than the first term in eq. (1). Our second assumption is that the Hall conductivity in a Landau subband with the filling factor v is given by uXy(v, T) = -(e*/h)n,(v,

T)IN,

(3)

where Nr,, is the total number of delocalized states and nh.I the number of electrons in the delocalized states. In addition to the two assumptions above, if one assumes that the density of states and the mobility edge E, are symmetric with respect to the center of the Landau subband and that the broadening r does not depend on V, it is easy to analyze the temperature dependence of EC/I’. The temperature dependence of the mobility edge arises from the temperature dependence of the localized length due to the inelastic scattering, which has been discussed by Moriyama and Kawaji [18] and Aoki and Ando [19]. We use the inelastic diffusion length as a cutoff length L, which destroys the localization in a Landau subband at finite temperatures. It is given by L, = ((2N + 1)(+)}“*1,,

,

(4)

where 7, is the inelastic scattering time, and the elastic scattering time T is related to the broadening r by r~ = fi. The inelastic scattering time has not been calculated in high magnetic fields. However, theories of the inelastic scattering in the absence of magnetic fields and results of negative magnetoresistance experiments for 2D systems have shown the relation 7/r, 0: TP [20].

52

S. Kawaji I Localization and quantized Hall resistance

The energy dependence of the localization length has been numerically studied by the Thouless number method by Ando [21] and by the finite size scaling method by Aoki and Ando [22] and Ando and Aoki [23]. Their results have shown a critical behaviour of the localization length as (5)

(Y(E) a PI”

where (Y is the inverse localization length. The critical exponent s of the localization length has been shown to be s 6 2 for the Landau quantum number N = 0 and s 6 4 for N = 1 [19]. When one combines eqs. (4) and (5), and assumes a relation r/r, a TP, the temperature dependence of the mobility edge is expected to be EC/l-w Tp’2s .

aO.& p”” 03-f 0.2 ‘g

31

9

b t t 4

3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 -Uv (d/h)

(6)

Wakabayashi et al. [24] used the Gaussian density of states given as 0.6 -

D(E) = (eB/h)(2/T)“2(l/r)

Si-MOS NO001 72-17H53-33 (Ob-1 VSD= 0.5 mV Bzl5T O~AA &UT x

exp(-2(E/r)2)

o&

(7) to calculate eq. (3). Then, we investigated the temperature dependence of EC/T which reproduce d(-a,,) /dN, for the (0 & -) and the (1 t -) Landau subbands of an n-channel Si(O0 1) inversion layer measured at temperatures between 1.5 and 0.35 K in a magnetic field of 15 T. In the expression (N t J *), N is the Landau quantum number, t 4 is the spin index, and 4 is the valley index. These two Landau subbands were selected for the investigation beplots are symmetric accause their (ax*, -a,,) cording to Yamane et al.? study [25] as shown in fig. 1. The temperature dependence of EC/r are plotted on a log-log scale in fig. 2. In the evaluation of EC/r, calculated d( - a,,) / d N, reproduced well experimental results in the (0 & -) Landau subband. However, for the (1 t -) Landau subband, the simple model calculation could not reproduce the experimental d(-a,,)/dN, as a whole. Therefore, the EC/T value which can re-

0.2f

“4A

- AtA

Si-MOS N(100) 72-17H53-33 (lt-1

VSO= 0.5 mV B115T OaAA B=l4T x

l

l8A IA

At?. $0

O% .O 5.1 ’ 5.2 ’ 5.3 ’ 5.4’ 5.5 ’ 5.6 m 5.7 ’ 5.8 n 5.9 ncJ2 6.C Fig. 1. Diagonal conductivity Q,, vs. Hall conductivity -a,, in the (0 J -) Landau subband (a) and in the (1 t -) Landau subband (b), in a Si-MOSFET at temperatures of 1.5, 1.1, 0.87, 0.65, 0.50 and 0.35K in a magnetic field of 15T. Crosses represent (a,, , -a,,) in 14T at 0.35 K [25].

produce the lower energy shoulder of the d( - gX,,)/dN, curve at temperatures below 0.87 K is plotted in fig. 2. When the temperature dependences of EC/r are described as EC/I’ 0~T4 by the straight lines in fig. 2, we have q = 0.25 + 0.04 for the (0 J -) Landau subband and q = 0.15 + 0.04 for the (1 t -) Landau subband. In fig. 2, we have also plotted reciprocal maximum values {d(-a,,)ldN,},‘. Exponents of their temperature dependences are q = 0.18 ? 0.05 for the (0 & -) Landau subband and q = 0.09 2 0.05 for the (1 t -) Landau subband, respectively.

S. Kawaji

/ Localization

‘t’

Si4OS

N(100)

o_oj ...gfig)...,I,,, .-

10’

0.1

T:K) Fig. 2. Temperature dependence of the mobility edge EC normalized by the broadening I-, and of the reciprocal maximum of d(-u,,)ldN, in the (0 & -) and the (1 f -) Landau subbands in a Si-MOSFET in 15T [24]. (Ref. (241 contains E,lf only.)

Similar measurements were extended in magnetic held and temperature: the magnetic field up to 27 T and the temperature down to 50 mK [26]. Further, the measurements also included magnetic field sweep and gate voltage (V,) sweep measurements which is usually done for Si-MOSFETs. The measurements were carried out using the facilities at HFLSM, IMR, Tohoku University. Figures 3(a) and 3(b) show typical traces of V, dependence in a magnetic field of 25 T and B

and quantized

Hall resistance

53

dependence at V, = 21.4 V of uXXand uXxyat three temperatures, respectively. Figure 3(a) shows six o;, peaks associated with the (0 t +), (0 t -), (O& +), (0 1 -), (I t +) and (I t -) Landau subband from left to right and a,, plateaus between them. Figure 3(b) shows three aXXpeaks associated with the (0 J -), (1 t +) and (1 t -) Landau subband from right to left and a,, plateaus between them. In order to investigate the temperature dependence of EC/T, the temperature dependences of FWHM of d(-a,,)ldhT, and d(-a,,)ldB derived from the (0 J -) and (1 t -) Landau subbands are plotted in a log-log scale in fig. 4. We have confirmed that the temperature dependence of {d(-a,,)/dN,},’ is the same as that of the FWHM for the (0 J -) Landau subband. The filled symbols show d( - o,,) ldN, and open symbols show d(-c=,)/dB. The number of temperature points for d(-a,,) /dB is smaller than that for d(-cXY) IdN,, but we can see that the temperature dependence of the open symbols is almost the same as that of the filled symbols with the same Landau quantum number. At temperatures lower than 0.2 K, all the data show saturation in the temperature dependence. This saturation arises probably from the increase of the electron temperature. In the high temperature region, the FWHM data of d(- m*,,)ldN, give the exponent q of the temperature dependences q = 0.29 & 0.10 for the (0 1 -) subband and q =

Fig. 3. Diagonal conductivity a, and Hall conductivity -oxY as a function of gate voltage, V,, at 25 T (a), and as a function of magnetic field, B, at V, = 21.4 V, at three temperatures, respectively 1261.

S. Kawaji

54

$81 2 0.6 al -

(I?-) .

.

.

.

.

l

a05

0.1

q

,a4-

LA_ 0.11

I Localization

.A. 0.2 -f (K)

Q5

1

Fig. 4. Temperature dependence of full width of the half maximum (FWHM) of d(- a,,) /dN, (filled symbols) and d(-u,“)ldB (open symbols). Circles are for the (0 1 -) and squares for the (1 7 -) Landau subband [26].

0.16 + 0.02 for the (1 t -) subband. These exponents are close to the exponents for the (0 & -) subband and the (1 t -) subband obtained in the measurements at 15 T 1241, respectively. Then, let us use the exponents of the temperature dependence of EC/T obtained at 15 T for further discussions: q(N = 0) = 0.25 2 0.04 and q(N = 1) = 0.15 t 0.04. As shown in eq. (6), q = p/2s. Following the results of numerical studies [19, 21-231, when we use s < 2 for the Landau subband with N = 0, we have p G 1.0 ? 0.16. For the Landau subband with N = 1, we have p s 1.2 * 0.32 when we use s d 4. Results on the temperature dependence of r/7, so far studied by negative magnetoresistance experiments in Si (00 1) inversion layers have shown that p = 1 - 1.8 depending on N, and T [20,27]. The present results of p in high magnetic fields are in close agreement with the value of p in weak magnetic fields. Next, let us examine the numerical value of 7/r,. By use of numerical results in fig. 1 of ref. [19], the present result of EC/T = 0.15 in the (0 4 -) Landau subband at 1 K in fig. 1 gives for short-range scatterers and c”(E)/,, ~0.01 i= 0.05 for long-range scatterers. The prea(,%, sent result of EC/T = 0.5 in the (1 t -) Landau subband at T = 1 K in fig. 2 gives a(E = 0.01 for short-range scatterers. When we use eq. (4) for the Landau subband with N = 0, we have = 0.01 and (r,/r) = 400 (T,/T) = lo4 for a(E for (x(E)l,, = 0.05. We have r?,/~ 1: 3 X 10’ for the

and quantized

Hall resistance

Landau subband with N = 1. The value of T,/T depends on the range of the scatterers. However, (T,/T) = 400 is about 10 times the value measured in weak magnetic fields. Therefore, the values of the inelastic scattering time in high magnetic fields evaluated in the present work is not unreasonable. One significant difference observed in d(-a,,)ldN, and d(-a,,,)ldB between the (0 4 -) and the (1 f -) Landau subbands is that they show a peaked shape in the (0 4 -) Landau subband while they show a flat top trapezoid in the (1 t -) Landau subbands in 25 T. The similar difference in the shape of d(-mX,,)ldN, is observed in 15 T though the shape in the (1 t -) Landau subband is a sloped top trapezoid. The flat top trapezoid is well reproduced by eqs. (3) and (7). The experimental results described above are consistent with the critical behaviour of localization obtained by numerical studies where the critical exponent s depends on Landau quantum number N (19, 21-231. The present results are not in accordance with Wei et al. [28]. They reported that the exponent of temperature dependence of the maximum value of dpX,ldB, where p,, is the Hall resistivity, of an InGaAs/InP heterostructure was independent of the Landau quantum number in the range of 0.1 K < T < 4.2 K. Pruisken [29] presented a conjecture which explains Wei et al.‘s experimental results by introducing a scaling function. Quite recently, Wei et al. [30] extended their experiment on InGaAs/InP heterostructures to examine the temperature dependence of dzpX,ldB2 and d”pX,/dB” in addition to dp,,ldB and found a universal relation. Further, Engel et al. [31] found the same power-law dependence on T of the maximum value of dpX,/dB in the FQHE of an AlGaAsiGaAs heterostructure.

3. Quantized Hall resistance: device dependence and Landau

quantum

number

dependence

Since the first precision measurements of the quantized Hall resistances R”(i), where i is an integer, by Von Klitzing et al. [3], a number of

S. Kawaji I Localization and quantized Hall resistance

groups around the world have worked towards the realization of a resistance standard or the accurate determination of the fine structure constant by means of the quantized Hall resistance. In fact, in October 1988, the Comite International des Poids et Mesures (CIPM) adopted a recommendation on the representation of the Ohm by means of the quantum Hall effect which will be used from 1st January 1990 [32] according to a report of the ComitC Consultatif d’Electricite (CCE) (18th Meeting, 1988) [33]. The CCE recommended a value of 25 812.807 R for the Von Klitzing constant, R, with a one-standarddeviation uncertainty with respect to the Ohm to be 0.2 ppm. This R, is defined by R, = i x R&i) where i is an integer and the quantized Hall resistance of the ith plateau RH(i) is the quotient of the Hall potential difference of the ith plateau and current in the quantum Hall effect [33]. The report contains a note on R, mentioning that this symbol is not intended to represent the combination of fundamental constants hle2. Figure 5 shows several R, values in the report to the CCE from the Working Group of the Quantum Hall Effect (B. N. Taylor, NBS, Coordinator) [34]. Our result of Si-MOSFETs based on 1 R determined by a calculable capacitor in CSIRO [35] is shown as CSIRO/GU in fig. 5. I

I

I

I

I

CSIRO/ GU

I

4

CSIRWBlF'l.4 CSIRO

I

I

l-s--l NBS

I

I

I

[(RK/

25812.8fib

0.3

I

1

0.4

I

0.5

I

I

0.6

l]xlOs

Fig. 5. R, = i x R, values reported in 1988 from several laboratories. C’(ae) is the value calculated from the fine structure constant derived from the electron anomalous moment by QED theory (from ref. [34]).

55

Other R, values were obtained by GaAs/ AlGaAs heterostructure devices. An R, value calculated by assuming R, = h/e2 = poca -’ from the reciprocal fine structure constant (Y-’ which was obtained by the anomalous magnetic moment of the electron and the QED calculation is also included in fig. 5. In our paper [35], we reported a preliminary result of the comparison of the R,(4) values of a Si-MOSFET with that of a GaAs/AlGaAs heterostructure device measured at 0.5 K. The result was that the value of the Si-MOSFET was always larger than that of the heterostructure and the difference was (0.09 + 0.06) ppm. Since then, we have made several improvements on the measurement system at GU and carried out more precise comparison measurements of Rn(4, GaAs), R,(4, Si) and Rn(2, GaAs). Recent results are briefly described here. Details of the measurement system and experimental results are to appear elsewhere [36]. Quantized Hall resistances R,(4) and R,(2) were compared with reference resistors R,(4) and R,(2) whose nominal values are 6453.20 and 12 906.4 R, respectively. Reference resistors were calibrated against a lOOn resistor by a self-balancing resistance-ratio bridge using a cryogenic current comparator [37]. All measurements were carried out with a channel current of 10 PA at 0.3 or 0.5 K and in magnetic fields of 15 T for Si-MOSFETs, around 5.1 T for Rn(4, GaAs) and around 10.4T for Ru(2, GaAs). Two types of automated resistance comparison measurement systems gave the same results for the comparison of two R,(4)s with those obtained by the self-balancing resistance ratio bridge within an uncertainty less than 0.01 ppm. Figure 6 shows differences between i x R”(i) and 4 X Ru(4, GaAs). The results show that R, obtained as i x R”(i) is not a simple constant but depends on the device and the Landau quantum number. In connection with our results, we refer to other laboratories’ results as follows. Delahaye et al. [38] reported that 4 X R,(4, GaAs) and 2 X R,(2, GaAs) measured using a channel current of 30 FA agrees with each other and the agreement is better than +7 x 10m9. They also

S. Kawaji I Localization and quantized Hall resistance

56 1

Si-MOS GaAs No.1

No.2

- 0.1

-

*.

-

1

-

-

72-17H53-32-L2 EPF 234 /7

1

GaAs(2)

Gad4)

0 i*Rw(i) 4 * Rn(4,GaPb)

’

SI (4)

w

GaAs(4) -

-

4

Gals(Z) 0.2

0.1 -1

(ppm)

Fig. 6. Relative values of 4 x R,(4) measured in a SiMOSFET, 2 x R,(2) measured in a GaAslAlGaAs heterostructure to 4 x R,(4) value measured in the same heterostructure device [36].

found a relative difference of (3 -+ 5) x 10m8 between 4 x R,(4, Si) measured with a channel current of 10 FA at 0.5 K in a magnetic flux density 13T and R, evaluated with the GaAs heterostructure. Small et al. [39] reported that 2 X Ru(2, GaAs) measured around 7.5T and 4 X R,(4, GaAs) measured around 6 T at 1.3 K with channel current of 25 FA (10 ~_LLA in some measurements) agrees within 0.02 ppm.

4. Discussion 4.1. Localization in Landau subbands Wei et al.‘s [29] experimental results, which presented that the exponent K of temperature dependence of (dp,,ldB)““” is universal and does not depend on the Landau quantum number N, are very interesting. The exponent K is related to the localization exponent v and the inelastic scattering rate exponent p through the ratio K = p/2v in their expression. Our experimental results which obtained by dmX,,ldN, and du,,/d B of Si-MOSFETs, however, showed that the exponent q of the temperature dependence of EC/r and maximum value or the FWHM of du,,ldN, and du,..ldB depend on the Landau quantum number. Moreover, the value of q, which presumably has the same physical meaning as K, is much smaller than the value of

K = 0.42. At present, it is difficult to understand the origin of such a significant difference. It is very possible that Wei et al.? results are not associated with localization but associated with other phenomena such as electron-electron interaction effects. Another possibility is that the resistivity tensors measured by Wei et al. may not be correct quantities to be used in the present type of quantitative study in the dissipative region. We remark here that eqs. (3) and (7) lead to the following relation:

W(-~,,VdN,l,,,,

T=O)-’ =

(hle2)(eBlh)(8/~)“2(E,lT)

.

(8)

The temperature dependences of {[d(- cX,,)/ dNs],,,}-’ at finite temperatures plotted in fig. 2 shows that the relation between EC/T and {[d(-a,,)/dNs],,,}-’ given by eq. (9) holds fairly well though the temperature is finite. This fact suggests that the effective mobility edge model we employed describes the experimental result. The temperature dependence of the derivative (dp,,ldB) maxdoes not have a sound base associated with localization. We comment here that one should reflect upon the method of transport coefficient measurement of 2D systems in high magnetic fields. In such systems, the diagonal and the Hall conductivities can not be obtained with quantitative accuracy by usual tensor transformation from measured resistivity tensors as a,p,l(pf, + pi,) except for the case when p,, = 0. Englert and Von Klitzing [40] evaluated ox,, and oXYof Si-MOSFETs from p,, and pXYby tensor transformation. The peak values of uX,, they obtained are much smaller than the values measured directly by Corbino disk samples. Kawaji [ll] and Wakabayashi and Kawaji [12] describe a similar difficulty from a different point of view. The fact that the correct conductivity tensors can not be obtained from measured resistivity tensors shows that the resistivity tensors measured by use of long Hall bars do not represent the correct values for those of 2D electron systems. The only way to measure both o;-, and aXYsimultaneously is the Hall cur-

S. Kawaji

I Localization

rent method [15] which is used in our experiments. Further studies are needed to settle the controversy. 4.2. Quantized

Hall resistance

Figure 5 shows that the measured R, values are systematically larger than those calculated from the reciprocal fine structure constant. Even the R, value closest to the CY -’ value disagrees by much more than one standard deviation uncertainty from it. Our recent results in fig. 6 show that the R, value depends on the material or device and the Landau quantum number. Various theoretical studies on the quantized Hall conductivity have been made by many authors [14]. Aoki and Ando [41] have derived a mathematically rigorous result in a general case based on the Kubo formula. Their result shows that the exact quantization of oXy= -ie’/h is a direct consequence of the localization and the mathematical theorem that a topological invariant called the winding number is an integer. Several studies on the topological aspect of the quantized Hall conductivity are cited in ref. [41]. However, these theoretical studies have treated a system without boundary at zero temperature. Recently, Ishikawa [42] claimed that the topological nature of the quantized Hall conductivity is valid when one includes the boundary effect. Several theoretical studies have been made on corrections to the quantized Hall conductivity arising from the boundary effect. Niu and Thouless [43] examined the effects of long distance behaviours of the Green function on a strip geometry. Their result is that the boundary correction to the exact quantization is exponentially small when the system size L is large compared with the magnetic length I,. Their result is too small to explain our experimental results. Ohtsuki and Ono [44] analyzed numerically the electronic states and the quantized Hall conductivity in an infinitely long narrow channel and on a cylinder surface with finite width. In both cases, the correction to the quantization is the order of l/L, which is too large to explain our experimental results.

and quantized

Hall resistance

57

The origin of the small but appreciable dependences of the quantized Hall resistance, i x R”(i), on the device made of different materials and on the Landau quantum number in the identical device is not understood at present. A possibility of its explanation is the nature of contacts between the 2D electron systems and potential electrodes [45,46]. To solve this problem, further careful experimental studies are needed. Acknowledgements

The author thanks his colleagues in GU, IMR, and ETL. He thanks Dr. A. Yagi, Sony Corporation, for providing Si-MOSFETs, Dr. W. Schwitz, OFM, for providing a heterostructure device and Prof. Y. Muto and the staff of HFLSM, IMR, Tohoku University, for the use of the high magnetic field facilities. He is indebted to Dr. T. Ando for helpful discussions. This work is partially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture and Science.

References 111 T. Ando, Y. Matsumoto and Y. Uemura, J. Phys. Sot. Jpn. 39 (1975) 279. PI S. Kawaji and J. Wakabayshi, Physics in High Magnetic Fields (Proc. Oji Int. Seminar, Hakone, 1980) S. Chikazumi and N. Miura, eds. (Springer, Berlin 1981) p. 284. 131 K. von Khtzing, G. Dorda and M. Pepper, Phys. Rev. Lett. 45 (1980) 449. 141 D.C. Tsui, H.L. Stormer and A.C. Gossard, Phys. Rev. Lett. 48 (1982) 1559. PI H.L. Stormer, A. Chang, D.C. Tsui, J.C.M. Hwang, A.C. Gossard and W. Wiegmann, Phys. Rev. Lett. 50 (1983) 1393. (61 R.B. Laughlin, Phys. Rev. Lett. 50 (1983) 1395. 171 H. Fukuyama, P.M. Platzman and P.W. Anderson, Phys. Rev. B 19 (1979) 5211. PI D. Yoshioka and H. Fukuyama, J. Phys. Sot. Jpn. 47 (1979) 394. [9] V.J. Goldman, M. Shayegan and D.C. Tsui, Phys. Rev. Lett. 61 (1988) 881. [lo] J. Wakabayasdi, A. Fukano, S. Kawaji, K. Hirakawa, H. Sakaki, Y. Koike and T. Fukase, J. Phys. Sot. Jpn. 57 (1988) 3678.

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S. Kawaji

I Localization

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and quantized

Hall resistance

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