Weak localization and negative magnetoresistance in semiconductor two-dimensional systems

682

Surface Science 170 (1986) 682-700 North-Holland, Amsterdam

WEAK L O C A L I Z A T I O N AND NEGATIVE M A G N E T O R E S I S T A N C E IN SEMICONDUCTOR TWO-DIMENSIONAL SYSTEMS S. KAWAJI Department of Physics, Gakushuin University, Mejiro, Toshima-ku, Tokyo 171, Japan Received 8 August 1985; accepted for publication 20 September 1985

Progress of experimental studies on weak localization in two-dimensional (2D) systems in semiconductor inversion layers by means of magnetoresistance measurements is reviewed. The inversion layers are those in Si MOS and GaAs/AI~,Ga 1_sAs heterojunction interfaces. Problems that remain to be solved are briefly presented.

1. Introduction

A theoretical framework for weak localization with a magnetic field effect on the weak localization was established in 1979 [1-3]. The characteristic properties of the weak localization in normal two-dimensional (2D) systems are the following: (1) the conductivity decreases with decreasing temperatures T with a correction term which is proportional to log T and (2) an external magnetic field B perpendicularly applied to 2D systems destroys the correction term or increases the conductivity. Before the development of the theoretical framework of the weak localization, observations of two characteristic properties of the weak localization in

'

•

~

. . . . .

1

5

_i]

10

T|KI Fig. 1. Temperature dependences of resistivity of n-channel inversion layers in Cs-Si (111) [4].

0039-6028/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) and Yamada Science Foundation

S. Kawaji / Weak localization in semiconductor 2D systems

683

HITI 0.5

1.0

1.5

-0.01

o._.~.- 0.0 2

- 0.03

5i O/Cs/p-Si ( 111 ) C-33

~.

~

a28

~.~..~.. ~ ~.7~.~

-0.04'

Fig. 2. Magnetoresistance of an n-channel inversion layer in Cs-Si (111) [4].

2D systems were reported in the 2nd EP2DS in 1977 by Kawaguchi, Kitahara and Kawaji [4] as shown in figs. 1 and 2. The 2D system in their work was inversion layer on Cs covered p-type Si(lll) surfaces. After the development of the scaling theory of localization by Abrahams, Anderson, Licciardello and Ramakrishnan [5] and the perturbation calculation of the weak localization by Anderson, Abrahams and Ramakrishnan [6], and Gor'kov, Larkin and Khmel'nitskii [7], a logarithmic temperature dependence or the conductivity was predicted. Then, the results of observations of a log T dependence of the resistivity in Si MOSFETs were reported in the 3rd EP2DS in 1979 by Bishop, Tsui and Dynes [8]. In December of this year, Kawaguchi and Kawaji [9] made the first analysis of negative magnetoresistance in Si MOSFETs based on the theory of magnetic field effect on weak localization derived by Hikami, Larkin and Nagaoka [10]. Hikami et al.'s calculation included the cases of the spin-orbit scattering and the spin-flip scattering due to impurities in addition to the case of ordinary impurity scattering which was studied by Altshuler, Khmel'niskii, Larkin, and Lee [11].

2. Weak localization

In 2D systems, the scaling theory of localization [5] predicted in a weakly localized regime that the sample conductance G(L) decreases with a correction term proportional to In L, L being the sample size, with increasing L. The correction term was calculated by means of the Green function diagramatic

684

S. Kawaji / Weak localization in semiconductor 2D systems

technique taking the r a n d o m potential into account as a perturbation for the electric field with frequency w as follows [6]:

e2D f , / , q dq rrZh ~o Dq 2 - io0'

0~(60)

(1)

where l = VFr, D = v 2 r / 2 and q = k + k ' for k and k ' are the incident and the scattered electron's wave n u m b e r vector, respectively. In the D C conductivity, o [ ( 0 ) diverges for q = 0. In actual cases, the wave n u m b e r vector is quantized by 2rr/L for the sample size L, one can replace the lower limit of the integral in eq. (1) by 1 / L and obtain ' -

OL

e2 rr2h

- -

L In

(2)

l '

which is in accordance with the result of the scaling theory. In real systems, we have not observed such a sample size dependence. Thus one considers that a characteristic length L c smaller than L exists and the integral in eq. (1) is cut off at L c. As Thouless [12] discussed, the inelastic diffusion length L, = D ~ , is a possible characteristic length. Then we have o~_

e2

q'c

--In 2rrZh

r

(3)

If r, depends on an T as r, o~ T -p and r is independent of T, we have a In T dependent term as given by ,

oL=

oQoe 2

2~rzh

In T,

(4)

where Anderson et al. [6] showed c~ = 1 for non-interating gases of electron with up or down spin. Bishop et al. [8] found that a p / 2 = 0 . 5 2 + 0 . 0 5 in Si (100) and (111) M O S F E T s . We remark here that the results in fig. 1 show a p = 0.9 _+ 0.3. Besides the localization effect, Altshuler, A r o n o v and Lee [13] and F u k u y a m a [14] independently pointed out that the mutual interaction effect also introduces a correction term o~ to the Drude conductivity. Then, we have to describe a In T dependent correction term to the D r u d e conductivity as follows: O ' = O Ll + 'O ' - -

aT e2

2~2h

In T,

a T=ap+g,

(5)

where g is an interaction constant [14]. We note here that, in spite of all these theoretical efforts, the experimental temperature dependence of the conductivity in Si M O S F E T s and other semiconductor 2D systems, in particular in samples with high electron mobility, is

S. Kawaji / Weak localization in semiconductor 2D systems

685

not simple and cannot be explained only by the localization and interaction effects.

3. Negative magnetoresistance experiments in Si MOSFETs As described in eq. (1), constructive interference causes electron localization. An external magnetic field affects the phase of electron wave function as ~b(B) = ~ ( 0 ) e x p ( i

efA . d r / h ) ,

where A is a vector potential which gives the magnetic field; B = rot .4. When an electron makes one quarter of a circular trajectory with radius L m , the change in phase becomes approximately one radian which is enough to destroy the constructive interference. Then, in a magnetic field with flux density, B, we can consider L m = v/h/eB as a characteristic length. When L m is shorter than L,, we have to u s e L m in the place of L in eq. (2). Therefore, the conductivity recovers with a term proportional to In B up to Drude conductivity and the magnetic field effect is stronger at lower temperature because L, c~ T-P~ 2. Experimental results in fig. 2 shows qualitatively these features. Hikami, Larkin and Nagaoka's expression for the conductivity in a magnetic field for 2D systems where there is no spin-orbit scattering nor spin-flip scattering is given by [10] o=%

e2[(

111

2rr2 h ~ 1 + ~

a~,

'

where ~b is the di-gamma function and a =4DeB/h; diffusion constant. Here, we note that a~" is given by

(6)

D = v~r/2 is the

a,c = 4DeB'r/h = 4 , v r ~0cr/h = 212/l~ = %(mho)/LB/(1.93 x 10 -5 nv), where l 0 = ¢ceh-/B is the magnetic length and /~ the electron mobility. In the derivation of eq. (6) by Hikami, Larkin and Nagaoka [10] and in the derivation of a similar expression to eq. (6) by Altshuler, Khmel'nitskii, Larkin and Lee [11], they used a quasiclassical approximation for the Green function in a magnetic field G(r, r') deriving from that in the absence of the field G o ( r - r'). The approximate expression is given as follows:

G(r, r') = exp

rA(l).dl

]

Go( r - r'),

(7)

where the integration path is along a straight line connecting r and r' under the assumption 10 >> l. Therefore, eq. (6) is valid only in weak fields where aT << 1. Eq. (6) is also correct at the high field limit where eq. (6) gives the Drude conductivity %.

686

S. Kawaji / Weak localization in semiconductor 2D systems

The change in the conductivity due to magnetic field is derived from eq. (6) as follows: 21r2h

~

+a~-~

- ~ -2 + ~

+In

.

(8)

For extreme cases, eq. (8) gives

AOL(B) =

I

48~r2h

ez

4DeB'q

2

(9)

(a~, >> 1).

2~r2~ In Here, we note again, the condition a~" << 1 must be satisfied in the latter extreme case. We note also AaL(B ~ ~ ) leads in the correct saturation value of Aa L which is the same value as eq. (3) with the opposite sign. The classical magnetoconductivity due to magnetic deflection of electron motion is given by A O c l ( B ) = -- o0(~c'r) 2 at low field. We have a ratio Aocl(B ) to A o c ( B ) as follows: AOL(B)-- -- 3'B" £F "r ~

.

(10)

In the usual cases we have %¢/h -- 10 and ¢,/~" > 10; therefore, the q u a n t u m mechanical magnetoconductivity due to the destruction of the localization is more than about 100 times the classical magnetoconductivity. F u k u y a m a [16] found that the contribution from the mutual interaction (g2 and g4 processes [15]) causes magnetoresistance due to an orbital effect. His result is given by [15]: 82

Alo.( B ) = o , ( B ) - o l ( O ) = 2 ~ ( g 2 -

2g4) q~(h, r),

(11)

where h = aT,, y = (h/2~r'r, kBT ) and the function q,(h, r ) is given in appendix A. In the field region h >> 1 and h7 >> 1, ~ ( h , y) depends on In B. In this field region, therefore, the total magnetoconductivity Ao(B)=AoL(B)+ AoI(B ) is given by eq. (8) multiplied by [1 + (g2 - 294)]- However, a higher order calculation [17] results in 1 g2 -- 2g4 = -- ½F 1 + ½F ln(1.13ho~o/ksT ) '

(12)

where he00 = ~F for the case of repulsive C o u l o m b interaction and F is a function introduced by Altshuler et al. [13] (see appendix B). At low temperatures, the contribution from mutual interactions to the magnetoconductivity is very small; I g2 - 294 I << 1.

S. Kawaji / Weak localization in semiconductor 2D systems

30

. . . . . . . . .

L

. . . . . . . . .

I

S~ M O S (100) N 5-10

. . . . . . . . .

687

]

T=4.2KI~("

. . y - - - ¢ ~ - "" .'-

Ns= 50xlO12cm2

~

Ct=0.29 '~= 1.Bxl6Ns

c-

~oE 20 I0

/

12,1K

b I "-c

~1o II

b

,

,

,

,

,

,

,

0

,

i

,

,

,

,

,

1000

,

,

I

t

[

. . . .

2000

,

. . . .

3000

MAGNETIC F I E L D ( O e )

Fig. 3. Field dependences of the magnetoconductivityin an n-channel Si (100) MOSFET [9]. Besides the orbital effect, Kawabata [18,19] and Lee and Ramakrishnan [20] noted that the s p i n - Z e e m a n splitting also affects the quantum correction due to the interaction o I. However, the characteristic field strength B o to observe the orbital effect and B S for the spin effect are very different from each other: for the orbital effect Bo - (h/4DeT,)=

(h/4,FT)(m/eT,),

and for the spin effect Bs - kaT/21xB (/~s: Bohr magneton) = m k a T f eh.

Since Bs/B o- (4,FT/h)(kaTT,/h)

>> 1

if

,FT/h

>>

1,

the orbital effect dominates the spin effect in the field region B - B o. Right after the derivation of eq. (8) by Hikami et al. [10], Kawaguchi and Kawaji [9] fitted their experimental magnetoconductivity in Si (100) M O S F E T inversion layers with /~peak(4.2 K) = 13,000 cm2/V • s to eq. (8). They used the approximantion ~b(1/2 + 1 ~ a T ) - ln(1/aT) in eq. (8) and multiplied by nva, where n v = 2 is the valley degeneracy factor in the Si (100) inversion layers and a is a phenomenological factor. In order to reproduce the experimental magnetoconductivity, a and T, are determined as shown in fig. 3. In fig. 3, aT is about 3 at a field of 3 kOe, although aT << 1 is the region where the original equation (8) can be applied. The values of Tc and its temperature dependence

688

S. Kawaji / Weak localization in semiconductor 2D systems ---2 Ns= I .9 x ld 2 crn ~,~,~,~,~-~

10.0

/

~,~,.~-~-~°~°) ° T=I.9K

~~.~*~

~.

3

0 r-

B ? 0 v

9.5 I

9.1,

o

5

I0

H (kOe) Fig. 4. Field dependence of conductivity in an n-channel Si (100) M O S F E T [21].

oc T 1.8 are not u n r e a s o n a b l e from the fact that z, >> ~" a n d that a relation ~- cc T - 2 is expected for the e l e c t r o n - e l e c t r o n process in p u r e metals [6]. Fig. 4 shows the field d e p e n d e n c e o f the m a g n e t o c o n d u c t i v i t y of a Si (100) M O S F E T with a low electron m o b i l i t y (~peak(4.2 K) = 3500 c m 2 / V • s). The e x p e r i m e n t a l d a t a are also fitted to eq. (8) with a m u l t i p l i c a t i o n factor n va (n v = 2) a n d an a p p r o x i m a t e d form of ~ ( 1 / 2 + 1 / a ' r ) g ln(1/a~-). The temp e r a t u r e d e p e n d e n c e of ~-, is shown in fig. 5. T h e t e m p e r a t u r e d e p e n d e n c e of ~', is described b y "r oz T-P at t e m p e r a t u r e s higher than 2 K. The e x p o n e n t p increases with increasing Ns: p = 1.33 at N 2 = 0.8 x 1012 cm 2 a n d p = 1.75 at N s = 1.9 X 1012 c m -2. The value of ~ is 0.5 at N s = 0.8 X 1012 c m - 2 a n d g r a d u a l l y decreases with decreasing Ns to 0.3 at N s = 6 x 1012 cm -2 [21]. By the use of AOL(B ) = nva X eq. (8) without the a p p r o x i m a t i o n for ~b(1/2 + 1 / a z ) , a similar Ns-dependence of a was o b s e r v e d [22]. A T - d e p e n d e n c e of c~, which has a similar trend as a p p e a r e d in fig. 3, a n d a relation ~-, cc Ns, were also o b s e r v e d [22]. I n H i k a m i et al.'s theory, a is unity for o r d i n a r y scattering. F u k u y a m a [23-25] solved the puzzle of the e x p e r i m e n t a l a which is smaller than 1 a n d d e p e n d s on N s a n d T, b y taking into account an intervalley scattering effect. T h e calculated result of a as a function of %/'q, where ~i is the intervalley scattering time, is shown in fig. 6 [2]. T h e use of H i k a m i et al.'s theory [10] assuming nva = 1 to extract z, from m a g n e t o c o n d u c t i v i t y d a t a in Si M O S F E T s was m a d e b y W h e e l e r [26] at

S. Kawaji / Weak localization in semiconductor 2D systems

1 30

689

Si MOS (O01)N 55-8

NS=I.9xlO16 m 2

.

- -~.,.

LL',

"q

ul

~s

v

3 A O

(th) Kawabata

HLN 5

7

10

T(K) Fig. 5. Temperature dependence of inelastic scattering time in the sample in fig. 4 [35]. A numerical error in the calculation of ¢o (eq. (20)) in fig. 2 in ref. [35] has been corrected in ~,(th) in this figure. T = 1 - 4 K a n d b y Bishop, D y n e s a n d T s u i [27] at T = 0 . 0 5 - 4 K. T h e y o b t a i n e d p = 1 i n the e x p o n e n t of ~',. D a v i e s a n d P e p p e r [28] a p p l i e d H i k a m i et al.'s t h e o r y b y u s i n g n va :~ 1 to extract ~', f r o m their m a g n e t o c o n d u c t i v i t y

1.0

i

i

!

i

i

i

i

,

J

i

,

0.9 0.8 0.7 0.6 0.5

I

I

|

i

1

2

3

4

6

i

I

810

r~/~-~ Fig. 6. Prefactor a against "r,/~"i for the case where q'i/% = 10 calculated by Fukuyama [2]. Here ¢0, ~'i and ¢, are lifetimes due to intravalley impurity, intervalley impurity and inelastic scattering, respectively.

690

s. Kawaji / Weak localization in semiconductor 2D systems

data at T = 0 . 5 to 7 K. The fitted a form I / , r ( = A a T + A 2 T 2 to their experimental T, data and determined A a and A 2. Choi [29] extracted ~-( from his magnetoconductivity data at T = 1 - 4 K by fitting Hikami et al.'s theory assuming n v a --- 1. He tried also to separate 1/,( into two terms as was done by Davies and Pepper [28].

4. Negative magnetoresistance experiments in G a A s / A i x G a , _ x As

The electrons in G a A s / A l x G a ] _ x A s heterojunction interfaces constitute the simplest 2D system if N s is sufficiently small to maintain a single subband system ( N s < 6 × 1011/cm2). Lin, Paalanen, Gossard and Tsui [30] made systematic experiments on samples with low electron mobility of (0.55-3) × 10 4 cm2/V • s at T = 0.04-1 K. By fitting their magnetoconductivity data to eq. (8) with the prefactor a, they obtained a = 0.75-0.85 and p = 1 in r cc T -p. The slopes in the In T dependence of the conductivity was larger than the theoretical prediction of the localization and interaction effects. N a m b u et al. [31] made careful experiments of magnetoconductivity in low mobility samples. The angular dependence of the magnetoconductivity shown in fig. 7 shows that the present system is an ideal 2D system for the study of weak localization. However, as shown in fig. 8, their magnetoconductivity data could not be fitted to eq. (8) except in an extremely low field region. Recently, Kawabata [32] developed a theory of negative magnetoresistance I

I

I

GaAs/Alo.3Gaa7 As 3 1 2 ( I ) T=I.IK Eso=4Vlm 0"=1,54.10 -3 mho/u AA A A A A A

1.0

A

A

A

A

ti

A a

A

A

v

b b

AAAA6

A A

~

A A ~

A •

.5

A • A •

• •

"2

16G&u

0

•

"-:"

•

A A

A •

•

• •

~, •

o•°ee

• • A ° e e e S G a u . 5S eeAo

•

b A

•e ~ A •

••o•

• •

• eAA• • • • ee lea • •

tie

"U

"'"

I

,

I

,

0

90

180

270

I 360

O ( Degree ) Fig. 7. Angular dependence of magnetoconductivitywhere 8 = 0 at BII interface [31].

S. Kawaji / Weak localization in semiconductor 2D systems

c"

691

Q

E

%

o t-" tt~

<3

E

'O v

b<~

0

| 2 3 B (10 G a u s s

4 )

5

0

2

4 6 8 B (10 G a u s s )

10

Fig. 8. Experimental magnetoconductivity data (points) are fitted to eq. (8) in the region B _<10 G [31]. Numerical errors in the ordinate in fig. 4 in ref. [31] have been corrected in this figure. Fig. 9. Experimental data in fig. 8 are fitted to eq. (14) in the region B < 100 G.

which is a 2D version of his earlier theory for three-dimensional systems [33]. He did not use the quasiclassical approximation for the Green function given by eq. (7). Kawabata [32] derived a Green function in a magnetic field given by

G ( r , r', , ) = exp[i(x + x ' ) ( y ' - y ) / 2 1 g] Go(r - r', , ) ,

(13)

where Go(r - r', c) is the Green function in the absence of a magnetic field. Eq. (13) is valid under the conditions toc = e B / m << 1/~-, CF/h. Therefore, Kawabata's theory can be applicable in the fields where a r < 1 under the condition ~oc~', h/~FZ << 1. Kawabata's expression for the magnetoconductivity in a simple 2D system is given by 2e 2 AOL(B ) - - S(1). (14) ~rZh The function S(fl) is given in appendix C. Nambu et al.'s experimental data were fitted to Kawabata's theory as shown in fig. 9. Note that the range of the magnetic field is twice that in fig. 8. Near the maximum field in fig. 9, we have ar = 1.

S. Kawaji / Weak localization in semiconductor 2D systems

692

100

50

GaAs AlCoAs 399(I) Z',= 0.86ps

.\ \o\

"~30

\

o Kawabat~

10

eHLN A

w

2

T(I'O

3

4

5

Fig. 10. Temperaturedependenceof inelastic scattering time [34]. The temperature dependence of T, extracted from a fit of experimental fig. 10 [34]. In fig. 10, r, data extracted from a fit of the experimental data to eq. (8) as in fig. 8 are also shown. Theoretical calculation of ~-,(th) in fig. 10 will be described in section 6.

Ao(B) to eq. (14) is shown in

5. Analysis of negative magnetoresistance in Si MOSFETs by Kawabata's theory Kawabata [32] calculated also the magnetoconductivity including intervalley scattering effect in the case of the two-valley systems. The result is given by

Ao(B) =

- 4e----~2(S(fl3) + ½ I S ( l ) -

S(fl4)] )

(15)

,2.r2 ~

with f13 = 1 - ~'/zi and/34 = 1 - 2~'/~'i, Ti being the intervalley scattering time. The function S(fl) i~ given in appendix C. Kawaji and Kawaguchi [35] fitted their experimental data for Ns = 1.9 × 1012 cm -2 shown in fig. 4 to eq. (15) and extracted ~',. In their analysis, i-i/% = 4, % being the intravalley scattering time, gave the best fitting of the experimental data to the theory as shown in fig. 11. The temperature dependence of z, extracted by fitting Kawabata's theory to the experimental data is shown in fig.

S. Kawaji / Weak localization in semiconductor 2D systems

693

.=**** s j * * l

10.2

[] o

l'e= 2.1 x 10-115

10.0

.t:

E

b

9.8 / i

N = 1.9 x 1016n~z ,ri pro=/,

20= 4.5 x 10-13S

9.6~)

i

i

i

i

i

0.5

i

i

i

i

i

1.0

i

i

i

B(T) Fig. l l . Experimental data in fig. 4 are fitted to eq. (15) [35].

5 in addition to the former result obtained based on Hikami et al.'s theory. Results of a fit of Hikami et al.'s theory [10] with a phenomenological parameter a to the experimental magnetoconductivity data as shown in fig. 4, similar results of a fit of Kawabata's theory [32] shown in fig. 11, and the fact that ~',s extracted from the same data based on two theories are in fair agreement in fig. 5 show that a is a useful parameter to take the intervalley scattering effect into account. In the next section, theoretical calculation of ~-,(th) shown in fig. 5 will be described.

6. Inelastic scattering time The inelastic scattering time z, was first introduced phenomenologically to the weak localization theory by Anderson et al. [6]. Since then, several theories of % in 2D systems due to electron-electron interactions have been presented [36-40]. Abrahams, Anderson, Lee and Ramakrishnan [36] first calculated the

694

S. Kawaji / Weak localization in semiconductor 20 system

electron quasiparticle life time in the dirty 2D metal case, proposed that l/r, is given as follows: 1

k,T =GlnT,

+ALK)

h/7 % k,T,

Ti

and

(16)

F

with k,T, q,

= 4 %f ‘n&3, ( 1

being the inverse

T,(K)

screening

= 32.6 x [e&nho)/1.23

However, given by 1 _=%’

(17)

k,T

Altshuler,

Aronov

length.

The numerical

value of T, is given as

x 1O-s]3 x (7.7/K)2(m*,‘0.19m0)(2/n,). and

Khmel’nitskii

[37] claimed

that

l/r,

cFr

is

(18)

2E F 7%

by a direct evaluation of a[. Considering the orginal expression of eq. (47) in ref. [37], we note that eF7 in eq. (18) is multiplied by n, in the case n, # 1. Fukuyama and Abrahams [38] derived the same expression as eq. (16) by a diagramatical calculation of the particle-particle diffusion propagator. However, the experimentally extracted r( data from negative magnetoresistance by many authors were too large to be explained by eq. (16). Fukuyama [39] considered that the strong q-dependence of Fukuyama and Abrahams’ calculation which leads to eq. (16) must be treated carefully, and he derived a result similar to eq. (18) as follows:

(19) For a pure 2D metal following result: 1 = ,p=?go

2k;T2 he,

1 ins,

case, Fukuyama

and

Abrahams

[38] obtained

the

(20)

where go is replaced by F in appendix B and S is the cutoff parameter Max(ti/e,r, k,T/r,). In the case n, # 1, l/7,’ is given by (l/n,) X eq. (20). A similar result to eq. (20) but with a different numerical factor was obtained by Giuliani and Quinn [40]. The pure metal case corresponds to large momentum transfer processes and the dirty metal case corresponds to small momentum transfer processes. Therefore, the total inelastic scattering probability l/r, is given by the sum of eq. (20) and eq. (18) as L=‘+L 7, (th)

r,”

%’ .

(21)

S. Kawaji / Weak localization in semiconductor 2D systems

695

The calculated results of ~',(th) are shown in fig. 5 for Si(100) inversion layers and in fig. 10 for G a A s / A l x G a I _xAs heterojunction interfaces. In these figures, agreement between experimental results and theoretical results is very good.

7. Other topics and future problems Kawaguchi and Kawaji [41] applied the negative magnetoresistance to the measurment of electron temperature of Si MOSFETs in high electric fields. As is discussed above, T, refers to electron-electron interactions at low temperatures. Its temperature dependence is so strong that we can use it as a good thermometer for the electron temperature. They obtained the result that the deformation potential constants in electron-phonon coupling in Si inversion layers are the same as the bulk values. Bishop, Dynes, Lin and Tsui [42] measured an anisotropic negative magnetoresistance in Si (011) MOSFETs. Their results are in agreement with the theoretical results by WiSlfle and Bhatt [43] on this system. We note here that the theory of weak localization was first extended to anisotropic systems by Kawabata [33] in his theory of negative magnetoresistance of three-dimensional systems. His theory explained an anisotropic negative magnetoresistance in Ge observed by Sasaki [44]. In 2D systems, the weak localization and magnetoresistance in anisotropic systems were also studied for Ge bicrystal interfaces by Landwehr's group [45]. Kawaji, Shigeno, Yoshino and Sakaki [46] found a negative magnetoresistance to be proportional to B z in G a A s / A l x G a l _ x A s heterojunction interfaces with /~(4.2 K ) - - 2 × 105 cmZ/V • s in the low field region of t%~-< 0.2. The parabolic negative magnetoresistance is added to a positive magnetoresistance which saturates at high field as shown in fig. 12. This parabolic negative magnetoresistance is a phenomenon different from Paalanen, Hwang and Tsui's finding [47] which appeared in high field regions up to ~c~-- 10. Kawaji and Maeda [48] found a positive magnetoresistance in Si (100) MOSFETs which was independent of the field direction. The magnetoresistance is described as a function of B / ( T - 0), where 0 is a negative characteristic temperature as shown in fig. 13. The magnitude [0[ decreases with decreasing conductivity and approaches zero near the transition from the weak to the strong localization as shown in fig. 14. No existing theory can explain the parabolic negative magnetoresistance and the isotropic positive magnetoresistance described above. Poole, Pepper and Hughes [49] observed a positive magnetoresistance at low magnetic fields B < 0.015 T and a negative magnetoresistance at higher magnetic fields in an InP MOSFET in the temperature range T--- 4.2-0.3 K. They attributed this behaviour to the presence of spin-orbit interaction which

696

S. Kawaji / Weak localization in semiconductor 2D systems

D3 o

.12

E2 ,.t q

o

vl

b

.~0 0 0

0 0 -1

0

0.5

1.0 1.5 B 2 (10-~ T 2 )

Fig. 12. Magnetoconductivity against 9 2 in a high mobility GaAs/AI~,Ga 1 xAs [46].

0 ~

Si l',4OS N(100) 62- H53-17-L1 Vg =1.60 V Ns=1.4.6 x 1015 m -2

%

%

-.05

Esd=0.10 Vim

(3 =-0.45 K

=t,~, zl J

O v

!1 ilu e¢l

A

N

m -.1 5

o

I:4.20K

O " ( 0 ) = 3 . 3 4 7 x 1 0 - s r'nt~/E] o

o : 2 , 0 0 K C,-(0) = 3 616 x l 0 - s n'xho/D ~ : 1 . 2 6 K CT'(0) = 3.922 x l 0 - 5 m h o / D

<3

o

o : 1 06 K O'(0) =4 0 5 9 x 1 0 - 5 r n h o / l J -.2

,

0

. . . .

5 B(T)2/(T(K)

z

. . . .

1

15

- G(K))2

Fig. 13. Isotropic magnetoresistance in Si (100) MOS inversion layers versus [ B / ( T -

0)] 2

[48].

S. Kawaji / W e a k localization in semiconductor 2 D systems

697

kFt 0 1.5

1

2

i

|

3

4

i

5

;

D

Si MOS N(100)

./

1.0 ,,( v

/

(3:) I

ii J

0.5 • 62-H53-17-L1 A N9-8-L1 I ~

00

I

I

|

i

I

l

I

I

l

5 10 O" ( 1 0 - S m h o l a )

Fig. 14. Characteristic temperature 0 changes as a function of conductivity[48].

at B = 0 reduces the magnitude of the weak localization. Kawaji et al. [34] also found a positive magnetoresistance at low magnetic fields B _< 0.004 T and a negative magnetoresistance at higher fields in a G a A s / A l x G a 1_xAs (x = 0.3) heterojuction interfaces in the temperatures range T = 1.0-0.04 K. These authors also attributed this behaviour to the presence of the spin-orbit interaction. Theoretically, effects of the spin-orbit interaction in magnetoresistance in the weak localization was calculated first by Hikami, Larkin and Nagaoka [10]. Maekawa and Fukuyama [50] extended the theory of Hikami et al. to take effects of the spin-Zeeman effect into account. Kawabata [32] calculated also magnetoresistance with the spin-orbit scattering in the weak localization. All these theoretical results show that the effect of the spin-orbit interaction cannot be observed in a system of strictly two-dimensions where the spin-orbit interaction has only the z-component (1/~'sx = 0, 1 / z ~ 4: 0). Electrons whose motion along the z-direction is quantized in semiconductor inversion latyers constitute strict 2D systems. In this respect, metallic thin films, where Bergmann [51] observed magnetoresistance which was explained by the spin-orbit interaction, are not strict 2D systems. Kawaji et al. [34] could explain the magnetic field dependence of the magnetoresistance by Maekawa and Fukuyama's theory and Kawabata's theory based on an assumption of ~'sx = ~'do-However, the temperature dependence of the inelastic scattering time became flat at lower temperatures where the positive magnetoresistance appears [34]. We consider that the existing theories which include the spin-orbit interaction in magnetic fields are incomplete.

698

S. Kawaji / Weak localization in semiconductor 2D systems

In conclusion, problems remain unsolved in the localization of high mobility 2D systems and 2D systems near the strong localization, although the properties of the normal weak localization appear being well understood at present. Interaction effects play probably some important roles in these remaining problems. The spin-orbit interaction in localization in semiconductor 2D systems under magnetic fields is another problem to be solved.

Acknowledgements The author thanks A. Kawabata and H. Fukuyama for helpful discussions and comments. Part of this work is supported by a Grant-in-Aid for Special Distinguished Research from the Ministry of Education, Science and Culture.

Appendix A

~ { 5 [(-~h)2 ~k" ( l-~+ h + ~(h, y ) = - t=,

tz(

h)"

/ ) + ~(1+7)2 / ~-~

5+h+

,]}

(z+y) 2

"

Here, h -- am,, y = (h/2~r'r, k s T ) and ~p(') (z) is the poly-gamma function.

Appendix B 1 fo'~

r=-~

dO 1 + (ZkF/qs) sin(0/Z)'

where qs is the reciprocal screening length.

Appendix C The function S(fl) is given by No A S(fl) = E l _ ~ p N F z ( A ( N o + I ) , f l ) + F z ( O , fl),

(c.1)

N~O

with .4 = ~jrZcv/h = a'r/4 and

[ dx L u ( x z) e . . . .

s P N - - 1 7 z Jo

x2/2

(C.2)

S. Kawafi / Weak localization in semiconductor 2D systems

699

w h e r e z = z / T , , L N ( x 2) is t h e L a g u e r r e p o l y n o m i a l a n d

s

=

(1

1/2.

+

(C.3)

T h e f u n c t i o n F2(y, fl) is g i v e n b y

F2( y , fl ) = [ t 2 / 2 + flt + fl 2 ln( t - fl ) ] / 4 , , = [ 8 y + (1

+ z)2] 1/2.

(C.4) (C.5)

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