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ON THE MOBILITY OF GaAs-AlGaAs
HETEROSTRUCTURES WITH AN IMPURITY LAYER IN THE GaAs
A. Gold
To cite this version:
A. Gold. ON THE MOBILITY OF GaAs-AlGaAs HETEROSTRUCTURES WITH AN IMPU- RITY LAYER IN THE GaAs. Journal de Physique Colloques, 1987, 48 (C5), pp.C5-255-C5-258.
�10.1051/jphyscol:1987554�. �jpa-00226758�
ON THE MOBILITY OF GaAs-AlGaAs HETEROSTRUCTURES WITH AN IMPURITY LAYER IN THE GaAs
A . GOLD
Physik-Department E 16, Technische Universitdt Miinchen, D-8046 Garching, F.R.G.
La mobilit6 des GaAs-AlGaAs h'eterostructures avec une couche dotee de 6 d'impurit'e ionis'ee dans 1 e GaAs d l a distance z i de 1 'i n t e r f a c e e s t calculBe. On montre que l a mobilit'e e s t une bonne image de l a fonction d'ondes qui d'ecrit l'extension du gaz Blectronique dans l e bulk. La mobilitB depend fort6ment de l a densit6 de diiplgtion.
La densite dlimpuritB c r i t i q u e , ou' une t r a n s i t i o n vers une l o c a l i s a t i o n f o r t e e s t a' supposer, e s t calculiie e t l a diipendance de z i e s t pr6dite.
The mobility of GaAs-AlGaAs heterostructures w i t h a 6-doping l a y e r of ionized
impurities i n t h e GaAs a t a distance zi from t h e i n t e r f a c e i s calculated. I t is shown t h a t the mobility is a good image of t h e wave function which describes the extension of t h e electron gas i n t o the bulk. The mobility strongly depends on the depletion density. The c r i t i c a l impurity density where a t r a n s i t i o n t o strong localization i s expected i s calculated and the dependence on zi is predicted.
The mobility of GaAs-AlGaAs heterostructures has been calculated in recent years /I-4/. The transport theory i s based on S t e r n ' s and Howard's work on s i l i c o n metal-oxide-semiconductor systems /5/. The impurities a r e assumed t o be outside the region of t h e electron gas. For "6-doping" layers /6/ and f o r quantum wells /7/, experiments have shown t h a t a two-dimensional impurity sheet can be located i n the region of t h e electron gas.
In t h i s paper t h e mobility of GaAs-AlGaAs heterostructures w i t h a &-doping layer of ionized impurities i n t h e GaAs a t a distance Zi from t h e i n t e r f a c e i s c a l - culated. According t o t h e theory of Ref. /5/ t h e zero-temperature momentum relaxa- tion time T of a two-dimensional electron gas in the presence of a random potential
< l ~ ( q ) l ~ > is given by 2k- a a
EF and k~ a r e t h e Fermi energy and t h e Fermi wave number, r e s p e c t i v e l y . ~ (q) is the d i e l e c t r i c function of t h e two-dimensional electron gas /8/, including local f i e l d corrections G(q) /4,9/.
For a two-dimensional impurity sheet, para1 1 el t o the two-dimensional electron gas and a t a distance z i from t h e AlGaAs/GaAs i n t e r f a c e , the random potential f o r wave numer q i s expressed a s
ni i s t h e impurity density a n d e L L i s t h e background d i e l e c t r i c constant. The form f a c t o r F(q,zi) accounts f o r t h e f i n i t e extension of t h e electron wave f u n c t i o n y ( z ) i n t o the bulk and f o r t h e distance between t h e electron gas and t h e impurity layer /8/ :
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987554
C5-256 JOURNAL DE PHYSIQUE We use the variational form /8/
b 2 -bz 3 I Y ( Z ) I ~ = T z e
and neglect band bending effects. The extension parameter b i s determined by the electron density n and by the depletion density ND: b3 oc ND + 11 1-1/32. The mean distance between the charge and the interface i s given by zo = z = 3/b.
For an impurity layer in the GaAs (zi > 0) with the variational form of the wave function, the form factor can be evaluated analytically /8/
with ap = (1 + q/b), a1 = Zq(3 + qZ/b2)/ba$, a2 = 4q(l
-
q/b)/bao2, and a3 = q 1-
q/b)/bao.The q integral in Eq. ( 1 ) can be evaluated in an approximation, taking advan- tage of the q = 2 k ~ singularity in that case /lo/. This approximation works very well f o r 1 ow el ectron concentrations, resul t i n g in
1
= ~ ~ r - 1 n. F (2kFZi)''r n ((1 - G ( 2 k F ) ) Fc(2kF) + 2kFlqs) (6)
Fc(q) i s the form factor f o r the electron electron interaction potential. qs i s the Thomos Fermi screening wave number.
I calculated 1 / ~ according to Eqs. (1,2,5) for GaAs-AlGaAs heterostructures. The mobility p i s connected with T via y = es/m, and m i s the
lo6 electron mass. The density dependence of p i s shown
I I I 1 1 1 1 1 in Fig. 1 f o r zi = 0, 100
A
and 2001.
1
-
G ~ A S- A I G ~ A S ,{
-
ni = 1 x l ~ ~ l c r n - ~-
-
/ I1
-
~ ~ l1/ /0 l ~ ~ ~ c ~ ~The mobility increases with increasing electron density according t o
p oc nd (7)
For zi = 0, 100 Rand 200
A,
we find o < s 1.1, 1.3, and 2.0, respectively. With increasing ND the elec- tron layer i s pushed t o the interface. This i s why e(Np = 0,; v(Np > O J holds f o r zi c 100A,
while p ND = 0 c p N D > 0) holds f o r i!i > 100
A.
Mobility versus density f o r zi = 0 (solid
fM
zi = 100 (dotted l i n e s ) , and =I = 200R
'012
(dashed lines) f o r two values of ND.
m y versus Z i f o r t h r e e values of Nd a s s o l i d l i n e s . I / J Y ( z ~ ) ~ 2 according t o Eq. (4) a s dotted, dashed, and dashed dotted l i n e s .
In Fig. 2, t h e mobility versus Z i i s shown f o r fixed n and n i . W i t % increasing 21% t h e mobility f i r s t decreases and then increases.
The minimum i s reached when the impurity layer matches t h e maximum of the electron charge. For Z i
>
150 A t h e mobility strongly depends on t h e depletion density. This e f f e c t could be used t o determine t h e depl tion den-t
s i t y i n real samples. 1 / JY(z = z i )
1
accord- ing t o Eq. (4) is a l s o shown i n Fig. 2.I t i s obvio s t h a t 1 /
1
y 12 matches t h e Z i dependence of t h e mobility q u i t e well f o r z i 401.
I conclude t h a t (macroscopic) mobi 1 i t y measurements f o r 40A
e zi e 200 1( can determine the (microscopic) wave function of the electron gas.The above mentioned e f f e c t can be understood a s follows: if i n a f i r s t approxi- mation one uses 1 y ( z ) l 2 = & ( z
-
zO) i n Eq. ( 3 ) , one obtains F ~ ( q , z i ) =exp (-q(zo
-
z i1).
If i n a second approximation one use? exp ( - q ( z-
Z i J ) 2 6 ( z-
Zij/q i n Eq. ( 3 ) , one g e t s FR(q,ti) = Z ( Y / ( z i ) J /q. According t o Eq. ( 6 ) the mobility is given a s1
2kFIzo - zil
/~('i) FR(2kF,
4)
2 % I Y ~ Z ~ I I ~ (8) For 2 k ~ I zo-
Zi1
(< 1 one f i n d s the desired r e s u l t : # 1 / I Y ( Z i ) l 2. The same con- clusion holds f o r quantum wells / I t / , i f they a r e doped i n t h e region of t h e two- dimensional electron gas /7/.The transport theory of Ref. / 5 / is valid only f o r low impurity concentrations.
Multiple s c a t t e r i n g e f f e c t s give r i s e t o a metal i n s u l a t o r t r a n s i t i o n i f t h e impurity concentration exceeds a c r i t i c a l impurity concentration n i C /I 21. The density
Fig. 3: C r i t i c a l impurity concen- t r a t i o n versus electron concentration f o r various values of z i a s s o l i d 1 ines. The dotted 1 i n e corresponds t o ni = n. In the i n s e r t , n . versus z j is shown f o r nit = n anaC
N,, = 1 x ''01 crn -2
.
C5-25 8 JOURNAL DE PHYSIQUE
dependence of n i C is shown i n Fig. 3 f o r various values of z i . With increasing z i ( ~ i
<
100 A), t h e e l e c t r o n d e n s i t y range f o r m e t a l l i c conduction decreases and reaches a minimum f o r Z i w 100A.
This minimum r e f l e c t s t h e mobility minimum i n Fig. 2. For higher z i ( z i > 100 A), t h e m e t a l l i c phase i n c r e a s e s again.For n i & n, band bending e f f e c t s a r e expected t o be important. I have neglected t h e s e e f f e c t s i n my calculation. However, f o r quantum wells i t has been shown t h a t t h e mobility does n o t s t r o n g l y depend on t h e quantum well thickness i f t h e doping is i n t h e middle of t h e quantum w e l l , s e e Fig. 12 of Ref. /11/. Therefore, I believe t h a t f o r z i & 100 A my c a l c u l a t i o n i s q u i t e r e a l i s t i c , and t h a t Fig. 3 can be used t o e s t i m a t e t h e i m p u r ~ t y and e l e c t r o n d e n s i t y range where strong l o c a l i z a t i o n e f f e c t s a r e expected. A metal i n s u l a t o r t r a n s i t i o n f o r f i x e d impurity concentration has been found i n a density sweep of a "6-doped" GaAs sample /13/. The c r i t i c a l electron d e n s i t y was i n reasonable agreement with t h e theory.
In conclusion we have shown how m o b i l i t y measurements can be used t o determine t h e e l e c t r o n wave function. A low e l e c t r o n d e n s i t y and a doping of t h e region of t h e e l e c t r o n charge is necessary f o r t h i s prediction.
Acknowl edgement
I thank Prof. F. Koch f o r valuable discussions. This work was supported by t h e Ernst von Siemens Stipendium of t h e Siemens AG.
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