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Surface Science 170 (1986) 202 208 North-Holland, Amsterdam

E F F E C T O F F I N I T E E L E C T R I C FIELD O N T H E Q U A N T U M H A L L EFFECT T. T A K A M A S U and S. K O M I Y A M A Department of Pure and Applied Sciences, University of Tokyo, Komaba. Tokyo 153, Japan

and S. H I Y A M I Z U and S. SASA Semiconductor Materials Laboratory, Fujitsu Laboratories, Ltd., Ono, Atsugi 243-01, Japan Received 22 July 1985; accepted for publication 13 September 1985

Diagonal conductivity a~, of 2D electrons in G a A s - A I G a A s heterostructures is studied as a function of Hall electric field E, for i = 6, 4 and 2 HaL1 steps, and is compared with calculations taking account of electron heating. The breakdown of the q u a n t u m Hall effect is consistently explained for every Hall step in terms of an S-shaped characteristic in the ~,, versus E~ relations.

The integer quantum Hall (QH) effect breaks down at a critical current density (or a critical Hall electric field Ecr ) [1,2]. Several mechanisms have been proposed for this breakdown, such as an onset of acoustical phonon emissions analogous to the elementary excitation in a superfluid [3] and a reduction in the fraction of localized states due to applied electric fields [4]. However, the experimental values of Ecr are not quantitatively explained with these mechanisms [5,6]. On the other hand, it is well known that electron temperature in a two-dimensional electron system rises on application of electric fields whether in the absence [7,8] or in the presence [1,9] of a magnetic field. We have recently shown that particular mechanisms such as those quoted above are not necessary to account for the breakdown, but it can be explained quantitatively by an S-shaped current-voltage characteristic that is derived merely from the effect of electron heating [10]. In ref. [10] we have shown this for an i = 4 Hall step. In this report we examine the validity of our picture by describing the extended work including also i = 6 and i = 2 Hall steps. The samples are G a A s - A l ~ _ x G a x A s heterostructure devices of standard Hall geometry with an inter-Hall electrode distance of 100 # m and channel width of 50/~m, similar to those used in the previous work [10]. The electron 0039-6028/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) and Yamada Science Foundation

T. Takamasu et aL / Effect of finite electric field on QHE

"."

10-5

C~As - Al~_xC.,~xAs 1611 "

~

203

ns= 3.8x1011 c m -2

\ %~

.1 i

B(T)TEY('V/'cm)~

10-6

lol 2 Ill

0 ..IE

E 10-7

1013

l~ 10--8

I(~14

I

ri]

L..

1615 3

lO-9 1(~10 0

0.1

0.2

IIT

0.3

0.4

05

ld 16 O.6

( K "I)

Fig. 1. Diagonal conductivity o~x versus the inverse of lattice temperature 1 / T L (open circles), and the energy differences of the electron system [ - ( T e ) = --'-(T~)] calculated with T~- 1 = 2 K and F = 0.5 meV from eqs. (3) and (4) for i = 6, 4 and 2, as a function of 1/T~ (dotted lines).

densities are n s -- 3.8 x 10 tl c m -2 a n d the mobilities are/~ -- 4 x 10 5 c m 2 / V - s at 4.2 K. T h e o p e n circles in fig. 1 represent oxx at low E y for i = 6, 4 a n d 2 Hall steps as a function of lattice t e m p e r a t u r e T L. Each curve of In o~x versus 1 / T L has the following c o m m o n features: (1) In a lower range of T L, the curve is d e s c r i b e d b y a straight line o ~ ~x e x p [ - W ~ / k T L ] with activation energies Wi being 1.4, 2.6 a n d 5.8 meV, respectively, for i = 6, 4 a n d 2. (2) In an i n t e r m e d i a t e T L range the slope of the curve is steeper than that in the lower T L range. (3) W i t h increasing T L b e y o n d the i n t e r m e d i a t e range the slope decreases r a p i d l y to m a k e the curve nearly flat in a higher T L range. Figs. 2 a - 2 c d e p i c t the d e p e n d e n c e of o~x on H a l l field E>, at several different lattice t e m p e r a t u r e s for i = 6, 4 a n d 2 Hall steps. F o r each H a l l step, o ~ a b r u p t l y increases a n d the Q H effect b r e a k s d o w n at a critical field E>, = Ecr (Ecr = 24, 39 a n d 110 V / c m , respectively for i = 6, 4 a n d 2 H a l l steps at T L = 1.9 K). W i t h increasing Ev well b e y o n d Ecr at low TL's, the increase of

~

I

20 Ey

/~

/I~

/7il

I

40 (V/cm)

(o)

-,,6 ns : 3.Sx1011c:m-2 B = 2.56 T

o -A,,_xO xAS

60 o

10101

lO~1-

J

20

,~]

I

,

~

4'o 6'o Ey (V/cm)

?

~ ~i *

/

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~

/ 4.3S

7 , ~-~

~s,~,

(b)

8o

1(]11

1010

10.9

10-8

10.7

~

;

Ey

2.9 ~ 1.9

o

~

o

~

3.5

o

%

o

~ ~

°

~K

4.2~

9.6

10- 6 - ~ ~ - - j '

(c)

B-- 7 . 5 7 1

(Wcm)

~

'

Fig. 2. Conductivity o~, versus Hall electric field E , at ith Hall steps. Broad arrows in (c) indicate the regions where o , , exhibits hysteresis.

10-8

~o -~

10-s

e~

,-.,.

T. Takamasu et al. / Effect of finite electric field on QHE

205

o, X tends to be saturated. The saturated values of o~, are close to the saturation values at higher T e in the sweep of T L with low E,, for i = 6 and 4 (Fig. 1), while that for i = 2 is by more than one order of magnitude smaller than the high temperature values at low E,," As T e rises, the variation of o,, with E~, becomes less remarkable; a~:, is almost independent of E v for i = 6 and 4, while for i = 2 it decreases with E~'" It is also noted that o,.,. at low T L for i = 2 decreases with E~ in several limited intervals of E,,, a few of which being indicated with broad arrows in fig. 2c. These decreases of o~., were always accompanied by hysteresis when the direction of the E v sweep is inverted. (The data shown are those for the upward sweep of E~..) Additional measurements utilizing different pairs of the Hall electrodes revealed that the spatial distribution of o~,. for i = 2 is highly inhomogeneous over the whole range of E~' above Ecr, while it is so only in the vicinity of E~.-- Ecr for i = 6 and 4. These data for i = 6 and 4 Hall steps strongly indicate that the electron system at low T L abruptly changes from a low-electron-temperature state to a high-electron-temperature state when E~, reaches E~r. The data for the i = 2 Hall step, on the other hand, suggest an additional mechanism to cooperate with the electron heating effect. We consider the electron heating effect starting from the energy balance equation, similarly as in ref. [10]: [.%-(Te) - . ~ ( T L ) ] / T e = oxxE~2.

(1)

Here,

1/r~ = CT~2

(2)

is the energy relaxation time of the electron system at an electron temperature T~ with a constant C, and

is the energy of the electron system per unit area with the Fermi energy e F and the Fermi distribution function f ( e ) = 1 / { 1 + e x p [ ( e - eF)/kT]}, where the Gaussian density of states D~(e) is given by

D,(e) = (2eB/h)(2rr)-l/ZF

' exp[-(e-

e,) 2/ 2 F 2 ],

(4)

with the level broadening parameter F, and e, = (n + ½)he%. Then, if the parameters C and P are given, we can determine T~ and o ~ as a function of T L and E~. by assuming ox:, to be uniquely determined by T~. We have applied P = 0.5 meV in c o m m o n for the calculation of all the Hall steps, as this value is supposed to be reasonable for the present system [10]. The values of C are so determined that the calculated values of oxx for the highest E~. and the lowest T e studied equal the measured values of oxx (in the case of i = 6 and 4) or a high temperature value of O~x at low E~' (in the case of i = 2).

i0-7

0

10"10[

'C16.i~K-2s-1 t 10-8

20 40 Ey ( V/cm )

//

711 n

2o Ey

h

,O+l~~o

40 ~ ( V/cm )

I

8o

0

1l~12 ~

" 5o

loo Ey

:

i

<,)1 +114.2_.~L.--~ I

/~ l

.

11+7 10-9 ~

c:1.49×1o7K2~ I-" i0";m4eV ~

Colculation

i

(c)

i 150 2oo 2~ (V/cm)

I

Cr--o.smeV : 1.18x107K-2s-1

Iculation

,06~ : 1 + ~ ~

Fig. 3. Conductivity o,, versus E,, calculated from the energy balance equation for ith Hall steps. The dotted parts of the lines specify the region for the S-shaped negative differential conductivity.

0

10-8

X10-7~

"~ 10"6

1o+~

K3

e~

eb

,2

T. Takamasu et al. / Effect of finite electric field on QHE

207

As shown in figs. 3a-3c, the rapid increase of axx is reproduced by the calculations for every Hall step at the field positions close to respective E~r. Furthermore, the S-shaped characteristic of the calculated curves of ox~ for low Te's in the vicinity of the critical fields accounts for the experimentally observed inhomogeneity in the distribution of oxx [10]. For i = 6 and 4 the calculations well reproduce all the features of the ox~ versus E), curves at different TL'S. For i = 2, however, the agreement between the calculations and the experiments is not so excellent as for the other steps. To see how the S-shaped characteristic occurs in the o~x versus Ey curves, the T~-dependence of the energy difference Z ( T ~ ) - . ~ ( T [ ) with T~ = 2 K is compared with the TL-dependence (or the T~-dependence) of o~x for each Hall step in fig. 1. If the Te-dependence of rF is not too strong, it is evident from the figure that the rate of the energy gain balances with that of the energy loss at more than one values of T~ in a narrow interval of Ev, leading to the S-shaped negative differential conductivity. Hence we wish to emphasize that the breakdown of the QH effect of the type described here is almost an unavoidable consequence of the characteristic temperature dependence of o~x in the QH regime independently of the details of the calculation. Changing the values of the parameters in the presence calculation only affects the position and the magnitude of the interval of Ey where the breakdown occurs. We have confirmed the results of calculation to be quite insensitive to varying the value of parameter F from 0.5 meV. Further, for i = 6 and 4, the parameter C is uniquely determined as stated above and therefore no arbitrariness is present in the calculation to manipulate the position of the critical fields. Therefore the excellent agreement of the critical fields in the calculations with those in the experiments for i = 6 and 4 is definite evidence for the correctness of the present picture. We should also note that the value of C for i = 4 is smaller than that for i = 6, which is reasonable because the density of states near the Fermi level must be smaller for i = 4 than for i = 6. The results of calculation for i = 2, on the other hand, are not so rigorous as those for i = 6 and 4, since arbitrariness remains in the determination of C. However, the value of C used in the calculation for i = 2 may be also reasonable in that it is smaller than (but of the same order of) those for i = 6 and 4. Hence the results of calculation for i = 2 (fig. 3c) can also be viewed as a reasonable explanation for the observed breakdown. Other detailed features peculiar to the i = 2 step, such as the decrease of ox~ with Ey at high T L and the presence of hysteresis in the higher Ev range well above Ec~, however, cannot be explained by the calculations and suggest that an additional mechanism, such as argued in refs. [11,12], is incorporated with the electron heating effect to introduce the current instability in the wide range of both T L and Ev. In summary, we have studied the Ey-dependence of o~x for i - - - 6 - 2 Hall steps and compared the data with the calculations of the electron heating effect. The breakdown of the QH effect for every Hall step is consistently

208

72 Takamasu et a L / Effect of finite electric field on QHE

e x p l a i n e d in t e r m s of the S - s h a p e d i n s t a b i l i t y c a u s e d b y the e l e c t r o n h e a t i n g . E s p e c i a l l y for i = 6 a n d 4 H a l l steps, the c a l c u l a t i o n s a g r e e e x c e l l e n t l y w i t h the e x p e r i m e n t a l results, a s s u r i n g the v a l i d i t y a n d the a d e q u a t e n e s s o f the p r e s e n t t r e a t m e n t . F o r an i = 2 H a l l step, the c a l c u l a t i o n s a c c o u n t for the b r e a k d o w n b u t fail to e x p l a i n m o r e d e t a i l e d f e a t u r e s in the o b s e r v a t i o n s , a n d suggest i n c o r p o r a t i o n of an a d d i t i o n a l m e c h a n i s m o t h e r t h a n e l e c t r o n h e a t i n g . T h i s w o r k was s u p p o r t e d by the G r a n t s - i n - A i d for S c i e n t i f i c R e s e a r c h a n d for S p e c i a l P r o j e c t R e s e a r c h , b o t h f r o m the M i n i s t r y of E d u c a t i o n , S c i e n c e a n d Culture.

References [1] G. Ebert, K. von Klitzing, K. Ploog and G. Weimann, J. Phys. C (Solid State Phys.) 16 (1983) 5441. [2] M.E. Cag, R.F. Dziuba, B.F. Field, E.R. Williams, S.M. Girvin, A.C. Gossard, D.C. Tsui and R.J. Wagner, Phys. Rev. Letters 51 (1983) 1374. [3] P. StFeda and K. von Klitzing, J. Phys. C (Solid State Phys.) 17 (1984) L483. [4] H.L. Sti3rmer, A.M. Chang, D.C. Tsui and J.C.M. Hwang, in: Proc. 17th Intern. Conf. on Physics of Semiconductors, San Francisco, 1984, Eds. D.J. Chadi and W.A. Harrison (Springer, Berlin, 1985). [5] O. Heinonen, P.L. Taylor and S.M. Girvin, Phys. Rev. B30 (1984) 3016. [6] L. Smr~ka, J. Phys. C18 (1985) 2897. [7] F.F. Fang and A.B. Fowler, J. Appl. Phys. 41 (1970) 1825. [8] Y. Kawaguchi and S. Kawaji, Japan. J. Appl. Phys. 21 (1982) L709. [9] H. Sasaki, H. Hirakawa, J. Yoshino, S.P. Sensson, Y. Sekiguchi, T. Hotta and S. Nishii, Surface Sci. 142 (1984) 306. [10] S. Komiyama, T. Takamasu, S. Hiyamizu and S. Sasa, Solid State Commun. 54 (1985) 479. [11] T. Kurosawa, H. Maeda and H. Sugimoto, J. Phys. Soc. Japan 36 (1974) 491. [12] T. Kurosawa and S. Nagahashi, J. Phys. Soc. Japan 45 (1978) 707.

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