- Email: [email protected]
Role of the local density of states fluctuations in metallic point contacts
PHYSICAi
N
ELSEVIER
Physica B 218 (1996) 68 72
Role of the local density of states fluctuations in metallic point contacts Gergely Zar/md, L/tszl6 Udvardi* hlstitute o[Physies, Technical University of Budapest, H 1521 Buda[bki ut 8., Budapest. Hungar3,
Abstract The spatial and energy-dependent fluctuations of the local density of states (LDOS) are calculated by means of a transfer matrix formalism for a model point contact (PC). It is shown that decreasing the size of the orifice the L D O S fluctuations become more and more pronounced in the region near to the orifice. As the Kondo temperature of a magnetic impurity is determined by the L D O S at the Fermi energy a broadening of the K o n d o resonance is predicted for small metallic PC's in agreement with the results of recent measurements.
Distribution of energy levels in zero- and quasi-onedimensional mesoscopic systems and disordered metals has been the subject of intensive studies in the past few years [1 3]. It is well-known that in these systems at low temperatures where the phase breaking mechanisms are absent the interference of the conduction electrons may lead to strong fluctuations in the local density of states (I_DOS), ~ (c, r). The presence of these fluctuations can be manifested in an anomalous distortion of the N M R line shape [2] or an increase in the average K o n d o temperature [4]. In the present paper we report on the calculations of the L D O S fluctuations in a small metallic point contact (PC) in the ballistic region. As we shall see, in contrast to disordered and mesoseopic systems, for a PC even in the absence of random scattering the geometrical boundary conditions generate enormous L D O S fluctuations in the region near to the orifice. Naturally, these fluctuations are getting smaller far away from the boundary of the sample. However, measuring the I - V characteristics of a small PC of diameter d one is testing the physical
* Corresponding author.
processes in the region ~ d 3 where the L D O S fluctuations are large. Therefore, for small PC's these fluctuation effects become very important. In order to calculate the L D O S fluctuations we use a scattering matrix formalism [5], and we consider a simplified model, where the conduction electrons are represented by free electrons and a cylindrical geometry is assumed. The PC is modeled by two infinite half-spaces connected by a tube of radius R and length L (Fig. 1), and the vanishing of the wave functions is required at the boundaries of the PC. The advantage of this geometry is that the angular m o m e n t u m of the conduction electrons around the axis z of the PC is conserved during the different scattering processes. We use in all the three regions of Fig. 1 angular m o m e n t u m eigenstates propagating (decaying) along the z-axis. Then, for a fixed energy, c, and angular momentum m, our basis functions in region (b) can be written using cylindrical coordinates as (b) q)± .......[! " , z , q ~ ) = e
pn
~ln) N,(ml J,,(/,, r)e
+ ik (,;~"'~)~
.... ,
(1)
where the signs _+ refer to right- and left-going electrons, respectively, Jm denotes the m's Bessel function and the N~m~'s are appropriately chosen normalization constants.
0921-4526/96/$15.00 < 1996 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 5 ) 0 0 5 6 1 - 7
69
G. Zarand, L. Udvardi Phvsica B 218 (1996) 68 72
]j
L
~]
F
1
IR
rl
z
I
I a
b
Fig. 2. The simplified geometries defining the different scattering and reflection coefficients in Eqs. (3) and (41. (a)
(b)
(c)
Fig. 1. Cross section of the simple model PC used for the calculations.
The index n in Eq. (1) refers to the nth root of the ruth Bessel function determined by the condition: J,,(2~.")R) = 0. The z component of the momentum, k., can be expressed as
kz
= ~(2mCt: - ()~llm))2)1/2 [i(().lnm))2 -- 2meg) 1'2
if X / ~ if ~
> 21'''~' < )~,-1
(2)
where mo denotes the electron mass. Similarly to Eq. (1) the wave functions in regions (a) and (c) can be written as ,01+ a.c).... i~ v , z, ~o) = ei'P'J,,(),r)e + ik.(;._- where the discrete vertical wave vector 3o~,")has been replaced by the continuous variable, ), and k= is connected to ). by a formula similar to Eq. (2). In order to calculate the L D O S and the conductance of the PC one has to determine the scattering matrices connecting the wave functions in regions (a), (b), and (c). For this purpose it is useful to consider first the simple geometries shown in Figs. 2(a) and (b). Since the scattering is diagonal in m in order to make our formulas more transparent in the following we drop the index m in the different formulae. Then for the geometry in Fig. 2(a) one can look for the solution of the Schredinger equation in the form = T,,
'p+.... + 2",,,=~¢9 ......r,,., [j'o d).).O .... at()., n)
if z < 0 , if z > 0,
(3)
while for the geometry shown in Fig. 2(b) the following Ansatz is used 5fO+.~.a + ~g d/'.').'~P-.~.a,ra,.a 7Ja = ( }~,~=1 q) . . . . . t(n, 2)
if z < 0, if z > 0.
(4)
Requiring the continuity of the wave functions and their derivatives at the boundary, one obtains a set of equa-
tions for the different transmission and reflection coefficients t(),, n), t(n, 2), r~,a, and r,c .... which can be easily solved by using a Bessel Fourier transform in the radial variable r. Using the notation 0,,().) for the Bessel Fourier transform, 0,,()-) = joR d2).Nl',")J,,,(21,,")r)J,,,(),r), end defining the matrices F..,,, and K,,.,c by F,,.,,, = 6,.,,,).,, and K,,.,,, = ~'~ d2)00,(2)k:(),)0,,4).) one can express the different reflection and transition amplitudes as r =(r
+ g)- l(r -K),
t()o) = 2k:().)(F + K )
(5) (6)
1Q().),
t + (,:,) = Q(),)(I + r),
r().',2)=
k:(2) Q ( 2 ) ( k - ( 2 J I - F ) ( F k:()~') -
(7)
+K)
~Q(2),
(8)
where a vector notation 0,,(2)~ #().), t(n, ) . ) ~ t(£), and t(2, n) --, t + (2) was introduced. Having obtained the coefficients (5 8) we can proceed by calculating the scattering matrices for the complex geometry shown in Fig. 1, which describe the wave functions corresponding to the incoming waves s, ,~(a) +.,:.a and q~e~.~.a. The scattering matrix Sb+.~+(n, 2) corresponds, for example, to the scattering of an incoming wave in region (a) to a right-going state in region (b), and it can be calculated by taking into account the multiple scattering processes inside the tube: Sb+.~+ (2) = (1 -- (re~rL)-') If(2). Having derived the different scattering matrices, one can easily construct the eigenstates of the scattering problem and calculate different physical quantities. In the inside region of the constriction, for example, the wave functions scattered into the tube from region (a), ~ , - b , can be written as ~p~.~u= ~ + . , : S b + . a + ( j ~ ) __ ~O_ . , : S b _ . a + ( ) ) ' where the vector notation q~ + ......(r, z, ~p) -* ~o+ .,: has been introduced for the wave functions in Eq. (1). The wave function scattered in from region (c), q~.~b, can be given by an analogous expression.
70
G. Zardmd, L. Udvardi / Ptb'sica B 218 (1996) 68 72
Then the L D O S in the tube can be simply o b t a i n e d by weighting the free electron L D O S corresponding to the region outside the constriction by the scattered wave functions constructed. In the tube, for instance, we arrive at the expression
).(e) O, ,~b I 2 + ~e f di, --_fi"
l~.sblZ),
able, 2, and the angular m o m e n t u m index m, and a summ a t i o n has to be carried out over the different angular m o m e n t u m channels. The c o n d u c t a n c e of the PC can be determined by integrating the expectation value of the current density o p e r a t o r over a surface in the tube:
(9) GO:) =
0
2e,2
\~
"
W ~ ! d),/'k_ [S~-+.,+£RSb+.,+
where the integration limit is determined by the energy considered, ),(e) = x/2m,e. Note t h a t the wave functions in Eq. (9) depend implicitly b o t h o n the integration vari-
-- S~
.a + F R S b
.a + q- 2
Im [S~-+.a+ F l S b - . a
+ ~~
(I0)
[~x...__.~
5
!
,/
4
?_.--
3
/
t
/
(5 2
, /
1
0 0
i
!
I
|
I
i
2
4
6
8
10
12
14
[eV]
Fig. 3. The m = 0 channel contribution to the conductivity of a PC with R = 3,~ and L = 6A as a function of the Fermi energy.
30
/
25
_/
2O c~ C ©
' ,~_/,t/
15
/
10 5 0 0
I
)
i
!
I
i
2
4
6
8
10
12
14
E [eV]
Fig. 4. The total conductivity of a PC with R = 3 ,~ and L = 6 A as a function of the Fermi energy.
71
G. Zargmd, L. Udvardi /'Physica B 218 (1996) 68 72
only produce PC's with a diameter of a b o u t ~ 20 A the step-like plateaus in the small energy (or, equivalently, very small constriction size) regions are difficult to observe in a metallic PC. However, we think that these discrete c o n d u c t a n c e steps could be seen in PC's m a d e of n a r r o w gap semiconducting materials like PbTe alloys [7], where a Fermi surface exists, a n d the Fermi wavelength, which falls into the region ~ 10,~, can be m u c h smaller t h a n m e a n free path. Fig. 5 shows the calculated L D O S fluctuations inside the tube for a P C with R = 15A a n d L = 15A at the point r = 7 A and z = 7,~, where the coordinate z is
where we have restored the q u a n t u m unit of the conductance, e2/h, and FR a n d F, denote the real and the imaginary parts of the matrix F, respectively. The calculated c o n d u c t a n c e as a function of the Fermi energyoiS shown in Figs. 3 a n d 4 for a PC with radius R = 3 A and length L = 6,~. Each channel m gives a staircase-like 6 o n t r i b u t i o n shown in Fig. 3, however, in the total c o n d u c t a n c e of the PC in Fig. 4 these steps become less p r o n o u n c e d a n d the total c o n d u c t a n c e increases approximately linearly with the energy according to Sharvin's formula I-6]. Since the Fermi energy of an ordinary metal lies in the region ~v ~ 5 eV a n d one can
0.018 i 0016 I
. . . . .
0:014 I"
t
0.012 0.01
T
& o,_
0.008 0.006 0.004 0.002 0 0
i
i
i
i
l
i
2
4
6
8
10
12
4
E leVI
Fig. 5. LDOS fluctuations inside the tube for a PC with R = 15 ,~ and L = 15 ,~ at the point r = 7 A and - = 7 A. The dashed line is the free electron value corresponding to the bulk.
0.018 0016 0.014 0.012 'T d~
It //"
0.01
\,/
0.008 0.006 O.004
i
0.002 \
0 0
i
!
i
i
i
i
1
2
4
6
8
10
12
14
R
[#,]
~,
Fig. 6. The LDOS function ~o(~:,z, r) of a PC with R = 15,~ and L = 15A at z = 7.A and energy ~: = 7 eV as a function of the radius r.
72
G. Zarimd, L. Udvardi / Physica B 218 (1996) 68-72
measured from the left wall of the PC. Each time a new conduction channel is opening a peak appears in the function 0(~:, r, z). As one can see in Fig. 6 the strong interference effects cause strong fluctuations even for a fixed energy if the spatial coordinate is varied, Therefore, any physical quantity of an impurity which depends on the L D O S at the Fermi energy (Knight shift, Korringa relaxation time, K o n d o energy, etc.) will depend on the spatial position of the impurity, and if the scatterers are distributed randomly in the PC then one is always measuring a di,stribution of the physical quantities. It is also evident that the fluctuations found must be present in small diffusive PC's as well. A thorough analysis shows [8] that these L D O S fluctuations can quantitatively explain both the size dependence of the Kondo temperature measured in C u M n PC's [9] and the broadening of the two level system's K o n d o peak in high-resistivity disordered PC's [10, 11]. The authors would like to thank A. Zawadowski for helpful discussions. The present research was supported by the Hungarian Grants O T K A 7283/93 and O T K A F016604.
References [1] P.A. Mello, P. Pereyra and A. Kumar, Ann. Phys. (NY) 181 (1988) 290. [2] K.B. Yefetov and V.N. Prigodin, Phys. Rev. Lett. 70(1993) 1315. [3] Y.V. Fyodorov and A.D. Mirilin, Phys. Rev. Lett. 71 (1993) 412~ A.D. Mirilin and Y.V. Fyodorov, Phys. Rev. Lett. 72 (1994) 526. [4] V. Dobroavljevi& T.R. Kirkpatrick and G. Kotliar, Phys. Rev. Lett. 69 {1992) 1113. [5] E. Tekman and S. Ciraei, Phys. Rev. B 39 (1989) 8772. [6] Yu.V. Sharvin, Zh. Eksp. Teor. Fiz. 48 (1965) 984 (Soy. Phys.- JETP 21 (19651 655). [7] Katayama, S. Maekawa and H. Fukuyama, J. Phys. Soc. Japan 50 (1987) 694. [8] G. ZarS_nd and L. Udvardi, to be published. [9] I.K. Yanson et al., Phys. Rev. Lett. 74 (1995) 302. [10] R.J.P. Keijsers, O.I. Sbklyarevskii and H. van Kempen, Phys. Rev. B 51 (1995) 5628. [11] K. Vladfir and A. Zawadowski, Phys. Rev. B 28 (1983) 1564, 1582, 1596.
N
ELSEVIER
Physica B 218 (1996) 68 72
Role of the local density of states fluctuations in metallic point contacts Gergely Zar/md, L/tszl6 Udvardi* hlstitute o[Physies, Technical University of Budapest, H 1521 Buda[bki ut 8., Budapest. Hungar3,
Abstract The spatial and energy-dependent fluctuations of the local density of states (LDOS) are calculated by means of a transfer matrix formalism for a model point contact (PC). It is shown that decreasing the size of the orifice the L D O S fluctuations become more and more pronounced in the region near to the orifice. As the Kondo temperature of a magnetic impurity is determined by the L D O S at the Fermi energy a broadening of the K o n d o resonance is predicted for small metallic PC's in agreement with the results of recent measurements.
Distribution of energy levels in zero- and quasi-onedimensional mesoscopic systems and disordered metals has been the subject of intensive studies in the past few years [1 3]. It is well-known that in these systems at low temperatures where the phase breaking mechanisms are absent the interference of the conduction electrons may lead to strong fluctuations in the local density of states (I_DOS), ~ (c, r). The presence of these fluctuations can be manifested in an anomalous distortion of the N M R line shape [2] or an increase in the average K o n d o temperature [4]. In the present paper we report on the calculations of the L D O S fluctuations in a small metallic point contact (PC) in the ballistic region. As we shall see, in contrast to disordered and mesoseopic systems, for a PC even in the absence of random scattering the geometrical boundary conditions generate enormous L D O S fluctuations in the region near to the orifice. Naturally, these fluctuations are getting smaller far away from the boundary of the sample. However, measuring the I - V characteristics of a small PC of diameter d one is testing the physical
* Corresponding author.
processes in the region ~ d 3 where the L D O S fluctuations are large. Therefore, for small PC's these fluctuation effects become very important. In order to calculate the L D O S fluctuations we use a scattering matrix formalism [5], and we consider a simplified model, where the conduction electrons are represented by free electrons and a cylindrical geometry is assumed. The PC is modeled by two infinite half-spaces connected by a tube of radius R and length L (Fig. 1), and the vanishing of the wave functions is required at the boundaries of the PC. The advantage of this geometry is that the angular m o m e n t u m of the conduction electrons around the axis z of the PC is conserved during the different scattering processes. We use in all the three regions of Fig. 1 angular m o m e n t u m eigenstates propagating (decaying) along the z-axis. Then, for a fixed energy, c, and angular momentum m, our basis functions in region (b) can be written using cylindrical coordinates as (b) q)± .......[! " , z , q ~ ) = e
pn
~ln) N,(ml J,,(/,, r)e
+ ik (,;~"'~)~
.... ,
(1)
where the signs _+ refer to right- and left-going electrons, respectively, Jm denotes the m's Bessel function and the N~m~'s are appropriately chosen normalization constants.
0921-4526/96/$15.00 < 1996 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 5 ) 0 0 5 6 1 - 7
69
G. Zarand, L. Udvardi Phvsica B 218 (1996) 68 72
]j
L
~]
F
1
IR
rl
z
I
I a
b
Fig. 2. The simplified geometries defining the different scattering and reflection coefficients in Eqs. (3) and (41. (a)
(b)
(c)
Fig. 1. Cross section of the simple model PC used for the calculations.
The index n in Eq. (1) refers to the nth root of the ruth Bessel function determined by the condition: J,,(2~.")R) = 0. The z component of the momentum, k., can be expressed as
kz
= ~(2mCt: - ()~llm))2)1/2 [i(().lnm))2 -- 2meg) 1'2
if X / ~ if ~
> 21'''~' < )~,-1
(2)
where mo denotes the electron mass. Similarly to Eq. (1) the wave functions in regions (a) and (c) can be written as ,01+ a.c).... i~ v , z, ~o) = ei'P'J,,(),r)e + ik.(;._- where the discrete vertical wave vector 3o~,")has been replaced by the continuous variable, ), and k= is connected to ). by a formula similar to Eq. (2). In order to calculate the L D O S and the conductance of the PC one has to determine the scattering matrices connecting the wave functions in regions (a), (b), and (c). For this purpose it is useful to consider first the simple geometries shown in Figs. 2(a) and (b). Since the scattering is diagonal in m in order to make our formulas more transparent in the following we drop the index m in the different formulae. Then for the geometry in Fig. 2(a) one can look for the solution of the Schredinger equation in the form = T,,
'p+.... + 2",,,=~¢9 ......r,,., [j'o d).).O .... at()., n)
if z < 0 , if z > 0,
(3)
while for the geometry shown in Fig. 2(b) the following Ansatz is used 5fO+.~.a + ~g d/'.').'~P-.~.a,ra,.a 7Ja = ( }~,~=1 q) . . . . . t(n, 2)
if z < 0, if z > 0.
(4)
Requiring the continuity of the wave functions and their derivatives at the boundary, one obtains a set of equa-
tions for the different transmission and reflection coefficients t(),, n), t(n, 2), r~,a, and r,c .... which can be easily solved by using a Bessel Fourier transform in the radial variable r. Using the notation 0,,().) for the Bessel Fourier transform, 0,,()-) = joR d2).Nl',")J,,,(21,,")r)J,,,(),r), end defining the matrices F..,,, and K,,.,c by F,,.,,, = 6,.,,,).,, and K,,.,,, = ~'~ d2)00,(2)k:(),)0,,4).) one can express the different reflection and transition amplitudes as r =(r
+ g)- l(r -K),
t()o) = 2k:().)(F + K )
(5) (6)
1Q().),
t + (,:,) = Q(),)(I + r),
r().',2)=
k:(2) Q ( 2 ) ( k - ( 2 J I - F ) ( F k:()~') -
(7)
+K)
~Q(2),
(8)
where a vector notation 0,,(2)~ #().), t(n, ) . ) ~ t(£), and t(2, n) --, t + (2) was introduced. Having obtained the coefficients (5 8) we can proceed by calculating the scattering matrices for the complex geometry shown in Fig. 1, which describe the wave functions corresponding to the incoming waves s, ,~(a) +.,:.a and q~e~.~.a. The scattering matrix Sb+.~+(n, 2) corresponds, for example, to the scattering of an incoming wave in region (a) to a right-going state in region (b), and it can be calculated by taking into account the multiple scattering processes inside the tube: Sb+.~+ (2) = (1 -- (re~rL)-') If(2). Having derived the different scattering matrices, one can easily construct the eigenstates of the scattering problem and calculate different physical quantities. In the inside region of the constriction, for example, the wave functions scattered into the tube from region (a), ~ , - b , can be written as ~p~.~u= ~ + . , : S b + . a + ( j ~ ) __ ~O_ . , : S b _ . a + ( ) ) ' where the vector notation q~ + ......(r, z, ~p) -* ~o+ .,: has been introduced for the wave functions in Eq. (1). The wave function scattered in from region (c), q~.~b, can be given by an analogous expression.
70
G. Zardmd, L. Udvardi / Ptb'sica B 218 (1996) 68 72
Then the L D O S in the tube can be simply o b t a i n e d by weighting the free electron L D O S corresponding to the region outside the constriction by the scattered wave functions constructed. In the tube, for instance, we arrive at the expression
).(e) O
l~.sblZ),
able, 2, and the angular m o m e n t u m index m, and a summ a t i o n has to be carried out over the different angular m o m e n t u m channels. The c o n d u c t a n c e of the PC can be determined by integrating the expectation value of the current density o p e r a t o r over a surface in the tube:
(9) GO:) =
0
2e,2
\~
"
W ~ ! d),/'k_ [S~-+.,+£RSb+.,+
where the integration limit is determined by the energy considered, ),(e) = x/2m,e. Note t h a t the wave functions in Eq. (9) depend implicitly b o t h o n the integration vari-
-- S~
.a + F R S b
.a + q- 2
Im [S~-+.a+ F l S b - . a
+ ~~
(I0)
[~x...__.~
5
!
,/
4
?_.--
3
/
t
/
(5 2
, /
1
0 0
i
!
I
|
I
i
2
4
6
8
10
12
14
[eV]
Fig. 3. The m = 0 channel contribution to the conductivity of a PC with R = 3,~ and L = 6A as a function of the Fermi energy.
30
/
25
_/
2O c~ C ©
' ,~_/,t/
15
/
10 5 0 0
I
)
i
!
I
i
2
4
6
8
10
12
14
E [eV]
Fig. 4. The total conductivity of a PC with R = 3 ,~ and L = 6 A as a function of the Fermi energy.
71
G. Zargmd, L. Udvardi /'Physica B 218 (1996) 68 72
only produce PC's with a diameter of a b o u t ~ 20 A the step-like plateaus in the small energy (or, equivalently, very small constriction size) regions are difficult to observe in a metallic PC. However, we think that these discrete c o n d u c t a n c e steps could be seen in PC's m a d e of n a r r o w gap semiconducting materials like PbTe alloys [7], where a Fermi surface exists, a n d the Fermi wavelength, which falls into the region ~ 10,~, can be m u c h smaller t h a n m e a n free path. Fig. 5 shows the calculated L D O S fluctuations inside the tube for a P C with R = 15A a n d L = 15A at the point r = 7 A and z = 7,~, where the coordinate z is
where we have restored the q u a n t u m unit of the conductance, e2/h, and FR a n d F, denote the real and the imaginary parts of the matrix F, respectively. The calculated c o n d u c t a n c e as a function of the Fermi energyoiS shown in Figs. 3 a n d 4 for a PC with radius R = 3 A and length L = 6,~. Each channel m gives a staircase-like 6 o n t r i b u t i o n shown in Fig. 3, however, in the total c o n d u c t a n c e of the PC in Fig. 4 these steps become less p r o n o u n c e d a n d the total c o n d u c t a n c e increases approximately linearly with the energy according to Sharvin's formula I-6]. Since the Fermi energy of an ordinary metal lies in the region ~v ~ 5 eV a n d one can
0.018 i 0016 I
. . . . .
0:014 I"
t
0.012 0.01
T
& o,_
0.008 0.006 0.004 0.002 0 0
i
i
i
i
l
i
2
4
6
8
10
12
4
E leVI
Fig. 5. LDOS fluctuations inside the tube for a PC with R = 15 ,~ and L = 15 ,~ at the point r = 7 A and - = 7 A. The dashed line is the free electron value corresponding to the bulk.
0.018 0016 0.014 0.012 'T d~
It //"
0.01
\,/
0.008 0.006 O.004
i
0.002 \
0 0
i
!
i
i
i
i
1
2
4
6
8
10
12
14
R
[#,]
~,
Fig. 6. The LDOS function ~o(~:,z, r) of a PC with R = 15,~ and L = 15A at z = 7.A and energy ~: = 7 eV as a function of the radius r.
72
G. Zarimd, L. Udvardi / Physica B 218 (1996) 68-72
measured from the left wall of the PC. Each time a new conduction channel is opening a peak appears in the function 0(~:, r, z). As one can see in Fig. 6 the strong interference effects cause strong fluctuations even for a fixed energy if the spatial coordinate is varied, Therefore, any physical quantity of an impurity which depends on the L D O S at the Fermi energy (Knight shift, Korringa relaxation time, K o n d o energy, etc.) will depend on the spatial position of the impurity, and if the scatterers are distributed randomly in the PC then one is always measuring a di,stribution of the physical quantities. It is also evident that the fluctuations found must be present in small diffusive PC's as well. A thorough analysis shows [8] that these L D O S fluctuations can quantitatively explain both the size dependence of the Kondo temperature measured in C u M n PC's [9] and the broadening of the two level system's K o n d o peak in high-resistivity disordered PC's [10, 11]. The authors would like to thank A. Zawadowski for helpful discussions. The present research was supported by the Hungarian Grants O T K A 7283/93 and O T K A F016604.
References [1] P.A. Mello, P. Pereyra and A. Kumar, Ann. Phys. (NY) 181 (1988) 290. [2] K.B. Yefetov and V.N. Prigodin, Phys. Rev. Lett. 70(1993) 1315. [3] Y.V. Fyodorov and A.D. Mirilin, Phys. Rev. Lett. 71 (1993) 412~ A.D. Mirilin and Y.V. Fyodorov, Phys. Rev. Lett. 72 (1994) 526. [4] V. Dobroavljevi& T.R. Kirkpatrick and G. Kotliar, Phys. Rev. Lett. 69 {1992) 1113. [5] E. Tekman and S. Ciraei, Phys. Rev. B 39 (1989) 8772. [6] Yu.V. Sharvin, Zh. Eksp. Teor. Fiz. 48 (1965) 984 (Soy. Phys.- JETP 21 (19651 655). [7] Katayama, S. Maekawa and H. Fukuyama, J. Phys. Soc. Japan 50 (1987) 694. [8] G. ZarS_nd and L. Udvardi, to be published. [9] I.K. Yanson et al., Phys. Rev. Lett. 74 (1995) 302. [10] R.J.P. Keijsers, O.I. Sbklyarevskii and H. van Kempen, Phys. Rev. B 51 (1995) 5628. [11] K. Vladfir and A. Zawadowski, Phys. Rev. B 28 (1983) 1564, 1582, 1596.