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Suppression of current noise due to the Coulomb correlation
Physica B 227 (1996) 116 118
ELSEVIER
Suppression of current noise due to the Coulomb correlation Fumiko Yamaguchi*, Kiyoshi Kawamura Department of Physics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kouhoku-ku, Yokohama 223, Japan
Abstract Effect of the Coulomb interaction on current noise in a quantum dot is studied from a theoretical point of view by using the Anderson model. Based on the perturbation expansion in the coupling between the dot and reservoirs in terms of the Keldysh Green function, current and current noise are computed including infinite order of the Coulomb interaction. We find that the Coulomb correlation shifts the maxima of equilibrium noise and suppresses nonequilibrium noise as functions of the bias voltage across the quantum dot. In addition, we examine the case in which the Kondo effect is possibly observed in transport through a quantum dot, and show the zero-bias anomalies in I-V characteristics.
Keywords: Current noise; Coulomb correlation
1. Introduction
2. Model
Current noise gives us information a b o u t the correlation between traveling electrons in the presence of transport [ 1 - 4 ] , which c a n n o t be obtained from average current. At zero-temperature, equilibrium noise which obeys the Nyquist theorem vanishes and only nonequilibrium noise remains. N o n e q u i l i b r i u m noise increases with current and becomes the classical shot noise if transport of each electron is independent. To deal with the correlation between the electrons which contribute to the current precisely, the effect of the C o u l o m b interaction on transport t h r o u g h a q u a n t u m dot connected to two electrodes on both the sides is studied.
O u r w o r k begins with modeling the system by the Anderson H a m i l t o n i a n [5-7],
* Corresponding author.
H = Ho + HT e-altl,
Ho = Hc + Ha, H¢ = ~ eka~aakff, k, a
Ha = ~ e,c~c~ + Un~n l, 6
* t HT = ~ (V,~at,~c. + V,.c~a,,~),
(1)
k,,r
where ak~ (a~) is the annihilation (creation) operator of an electron with m o m e n t u m k, spin a and energy eT, in the reservoirs t/ ( t / = L, R), with the vector k being (t/, k). The o p e r a t o r c~ (c~) destroys (creates) an electron with spin a (1 or J,) on the site,
0921-4526/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved PII S092 1 - 4 5 2 6 ( 9 6 ) 0 0 3 7 7 - 8
F. YamaguchL K. Kawamura / Physica B 227 (1996) 116-118
and n~ = c*~c~ is the number operator. Current is driven from one reservoir to the other by applied bias voltage, expressed by e V = PL -- #R.
117
( I ~ ) = 2er" [-L(~o)(1 - ( n ) o )
- {1 - L ( e o
+ U)}o-I,
and 3. Current and current noise
S,.,(0) = 4eEF"[f.(eo)(1 - ( n ) o )
Current operator between the dot and electric lead q is obtained from the continuity of current as
+ {1 -Lt~.o + c)}{n>o - 2{f,(eo) - f d e o
ie I,(t) =- - eN,(t) = -~ INn(t), H ] ,
+ u)}((n)o
-- (nTn+)o) ]. in terms of the creation and annihilation operators of electrons in the Heisenberg representation, where N, = L.~a~*~a~ is the number of electrons in electric lead q. The density matrix Po governed by Ho can be obtained as PO = Z - - l e
fl(H° '~/I'NE--'~/RNR--'~/cl~/c),
l = Yre fi(tt,./q. Ni.--ItaNR /*cNc) where N~ = ~£~c*~c, is the number of electrons in the quantum dot region, and fl is inverse temperature. The electrochemical potential which accounts for the state in the quantum dot is written by /t~ here, Obviously the equilibrium average of current vanishes ( l , ( t ) ) o = 0. Current becomes (I,(t)) = ~
dt' ([Hx(t'), l,(t)] )o,
(2)
v(:
up to the second-order term of Hx. The current noise spectrum is defined by S,~,,,,(o)) -
4. Calculated results
d(t - t ' ) { ( A I . ( t ) A I , , ( t ' ) ) + (AI.,(t')AI,(t))}ei°'"-"(
Then the condition of stationary current ( I L ) = determines the unknown quantity (n)o. Once the number of electrons in the dot is determined, the chemical potential/2c of the dot can be determined through the relation, (n)o = T r p o n T = Trpon~. The second unknown quantity (n~n+)o is calculated in terms of the density matrix, (n~n~)o -- Tr ponTn r The quantities (n)o and (nrn~) o are uniquely determined for given chemical potentials of the reservoirs. In particular, we discuss symmetrized zero-frequency current noise S o ( 0 ) I(SL, L -{- SR, R) , which has a simple analytical expression. Now we divide the zero-frequency current noise into the "Johnson-Nyquist noise" (equilibrium noise) S j N ( O ) = ½ ( 4 k B T G L + 4 k B T G R ) and nonequilibrium noise Sneq(0)= S ( 0 ) - Sin(0). The linear conductance G, of each potential barrier is defined by (~I,/O(ela)),~o where ~ = #, - ~c. --(IR) ,
(3)
The operator AI, - I. - ( I , ) represents deviation of the current operator from its average. For simplicity, we assume that the density of states p.(e) of electric lead r/is constant and the transfer V~(e) between the quantum dot and the electric lead is independent of energy and spin, so that the elastic couplings to the electric leads are constant, F~ = F ~ - - F ". Furthermore, we consider the nonmagnetic case, which corresponds to the case eT = el - eo and (n~)o = (n,)o - (n)o. Current (I~) and the zero-frequency current noise become
Examples of computed current and noise are presented in Fig. 1. The influence of the Coulomb interaction is as follows: the Coulomb interaction separates the single energy level eo in the dot into two levels, that is, e,o and eo + U. The equilibrium noise generated in the reservoirs is scanned by both the energy levels. Nonequilibrium noise shows a dip when the number of electrons (n)o is equal to ½. This suggests that the system becomes stable when the total number of electrons in the dot (Ndo,)O = 2 ( n ) o reaches unity because of the strong correlation, and subsequently the current fluctuates less.
F. Yamaguchi,K. Kawamura/ PhysicaB 227(1996)116-118
118 (a) ,~
0 . 0 O.05eo ........... 0.1e0 ~
0.1
follows:
~
2he
i~'-
( I ) =--~-- 2S(S +
¢',1
(
el V I "~2
1)eV_l°g k.rK j ,
"-- 0.05
S(0)Ao) = 0
0.5
1
1.5 2 2.5 (gL-gR) / E0
3
3.5
4 for
0.1
2e(I)A~o,
{
1{ eV "]2~ 1-~\kBT~ j ~,
S(O)Aa~=[~( eV "~2{1 3( eV "~3; \kB TKJ - -5\kB TK] J
~, 0.05
g 0 0.5
1
1.5 2 2.5 (gL-gR) /
3
3.5
1( er "~2;-11
3 \k~KJ
J
2e A~o,
for el V[ < kBTK, at zero-temperature limit, when the density of states are the same in both the electrodes. In both the cases, voltage is scaled by the K o n d o temperature kBTK of the system. Current noise becomes the sub-shot noise.
6. Conclusions
5. Kondo effect If eo + U >/AL,~AR and eo < ]-/L,/AR, the g r o u n d state is given by the filled Fermi sea (the reservoirs are filled with electrons up to the chemical potentials) and a single electron occupying the q u a n t u m dot for the limit where the coupling V is zero. This system can be expressed by using the K o n d o H a m i l t o n i a n [-8], H = H~ + H~x,
x{1
4
Fig. 1. (a) Current I through a quantum dot as a function of the chemical potential #L of the left reservoir at various temperatures, in units of 2eFL, FR=0.1F L, U=2eo, #R=0. (b) Nonequilibrium noise in units of 4e2FL.
Based on the analytical studies, we have shown that the C o u l o m b interaction suppresses the nonequilibrium noise when there is a total of only one electron in the dot. This feature is seen as a dip in nonequilibrium noise as a function of bias voltage. In addition, when the system shows the K o n d o effect due to localized magnetic m o m e n t in the q u a n t u m dot, nonequilibrium noise is suppressed.
(4)
References
where Hox=
kB TK, and
2rte (I)=2x~-eV
~) %
e IVI >
1 -eA-~
t -- __J_J 2N ykk'' a~wak~(S" o)~,~,
(5)
where a is the Pauli matrix and S is the spin o p e r a t o r of the electron in the q u a n t u m dot. We have obtained expressions of average current ( I ) t h r o u g h a q u a n t u m dot and the current noise S(0) Aeo as functions of applied bias voltage V as
[1] R. Landauer, Physica D 38 (1989) 226. [2] M. Biittiker, Phys. Rev. Lett. 65 (1990) 2901. [3] A. Shimizu and M. Ueda, Phys. Rev. Lett. 69 (1992) 1403. I-4] L.Y. Chen, G. Levine, J. Yang and C.S. Ting, Mod. Phys. Lett. B 5 (1991) 315. 1-5] S. Hershfield, Phys. Rev. B 46 (1992) 7061. 1-6] Y. Meir, N.S. Wingreen and P.A. Lee, Phys. Rev. Lett. 70 (1993) 2601. 1-7] F. Yamaguchi and K. Kawamura, J. Phys. Soc. Japan 63 (1994) 1258. [8] J. Kondo, Progr. Theor. Phys. 32 (1964) 37.
ELSEVIER
Suppression of current noise due to the Coulomb correlation Fumiko Yamaguchi*, Kiyoshi Kawamura Department of Physics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kouhoku-ku, Yokohama 223, Japan
Abstract Effect of the Coulomb interaction on current noise in a quantum dot is studied from a theoretical point of view by using the Anderson model. Based on the perturbation expansion in the coupling between the dot and reservoirs in terms of the Keldysh Green function, current and current noise are computed including infinite order of the Coulomb interaction. We find that the Coulomb correlation shifts the maxima of equilibrium noise and suppresses nonequilibrium noise as functions of the bias voltage across the quantum dot. In addition, we examine the case in which the Kondo effect is possibly observed in transport through a quantum dot, and show the zero-bias anomalies in I-V characteristics.
Keywords: Current noise; Coulomb correlation
1. Introduction
2. Model
Current noise gives us information a b o u t the correlation between traveling electrons in the presence of transport [ 1 - 4 ] , which c a n n o t be obtained from average current. At zero-temperature, equilibrium noise which obeys the Nyquist theorem vanishes and only nonequilibrium noise remains. N o n e q u i l i b r i u m noise increases with current and becomes the classical shot noise if transport of each electron is independent. To deal with the correlation between the electrons which contribute to the current precisely, the effect of the C o u l o m b interaction on transport t h r o u g h a q u a n t u m dot connected to two electrodes on both the sides is studied.
O u r w o r k begins with modeling the system by the Anderson H a m i l t o n i a n [5-7],
* Corresponding author.
H = Ho + HT e-altl,
Ho = Hc + Ha, H¢ = ~ eka~aakff, k, a
Ha = ~ e,c~c~ + Un~n l, 6
* t HT = ~ (V,~at,~c. + V,.c~a,,~),
(1)
k,,r
where ak~ (a~) is the annihilation (creation) operator of an electron with m o m e n t u m k, spin a and energy eT, in the reservoirs t/ ( t / = L, R), with the vector k being (t/, k). The o p e r a t o r c~ (c~) destroys (creates) an electron with spin a (1 or J,) on the site,
0921-4526/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved PII S092 1 - 4 5 2 6 ( 9 6 ) 0 0 3 7 7 - 8
F. YamaguchL K. Kawamura / Physica B 227 (1996) 116-118
and n~ = c*~c~ is the number operator. Current is driven from one reservoir to the other by applied bias voltage, expressed by e V = PL -- #R.
117
( I ~ ) = 2er" [-L(~o)(1 - ( n ) o )
- {1 - L ( e o
+ U)}
and 3. Current and current noise
S,.,(0) = 4eEF"[f.(eo)(1 - ( n ) o )
Current operator between the dot and electric lead q is obtained from the continuity of current as
+ {1 -Lt~.o + c)}{n>o - 2{f,(eo) - f d e o
ie I,(t) =- - eN,(t) = -~ INn(t), H ] ,
+ u)}((n)o
-- (nTn+)o) ]. in terms of the creation and annihilation operators of electrons in the Heisenberg representation, where N, = L.~a~*~a~ is the number of electrons in electric lead q. The density matrix Po governed by Ho can be obtained as PO = Z - - l e
fl(H° '~/I'NE--'~/RNR--'~/cl~/c),
l = Yre fi(tt,./q. Ni.--ItaNR /*cNc) where N~ = ~£~c*~c, is the number of electrons in the quantum dot region, and fl is inverse temperature. The electrochemical potential which accounts for the state in the quantum dot is written by /t~ here, Obviously the equilibrium average of current vanishes ( l , ( t ) ) o = 0. Current becomes (I,(t)) = ~
dt' ([Hx(t'), l,(t)] )o,
(2)
v(:
up to the second-order term of Hx. The current noise spectrum is defined by S,~,,,,(o)) -
4. Calculated results
d(t - t ' ) { ( A I . ( t ) A I , , ( t ' ) ) + (AI.,(t')AI,(t))}ei°'"-"(
Then the condition of stationary current ( I L ) = determines the unknown quantity (n)o. Once the number of electrons in the dot is determined, the chemical potential/2c of the dot can be determined through the relation, (n)o = T r p o n T = Trpon~. The second unknown quantity (n~n+)o is calculated in terms of the density matrix, (n~n~)o -- Tr ponTn r The quantities (n)o and (nrn~) o are uniquely determined for given chemical potentials of the reservoirs. In particular, we discuss symmetrized zero-frequency current noise S o ( 0 ) I(SL, L -{- SR, R) , which has a simple analytical expression. Now we divide the zero-frequency current noise into the "Johnson-Nyquist noise" (equilibrium noise) S j N ( O ) = ½ ( 4 k B T G L + 4 k B T G R ) and nonequilibrium noise Sneq(0)= S ( 0 ) - Sin(0). The linear conductance G, of each potential barrier is defined by (~I,/O(ela)),~o where ~ = #, - ~c. --(IR) ,
(3)
The operator AI, - I. - ( I , ) represents deviation of the current operator from its average. For simplicity, we assume that the density of states p.(e) of electric lead r/is constant and the transfer V~(e) between the quantum dot and the electric lead is independent of energy and spin, so that the elastic couplings to the electric leads are constant, F~ = F ~ - - F ". Furthermore, we consider the nonmagnetic case, which corresponds to the case eT = el - eo and (n~)o = (n,)o - (n)o. Current (I~) and the zero-frequency current noise become
Examples of computed current and noise are presented in Fig. 1. The influence of the Coulomb interaction is as follows: the Coulomb interaction separates the single energy level eo in the dot into two levels, that is, e,o and eo + U. The equilibrium noise generated in the reservoirs is scanned by both the energy levels. Nonequilibrium noise shows a dip when the number of electrons (n)o is equal to ½. This suggests that the system becomes stable when the total number of electrons in the dot (Ndo,)O = 2 ( n ) o reaches unity because of the strong correlation, and subsequently the current fluctuates less.
F. Yamaguchi,K. Kawamura/ PhysicaB 227(1996)116-118
118 (a) ,~
0 . 0 O.05eo ........... 0.1e0 ~
0.1
follows:
~
2he
i~'-
( I ) =--~-- 2S(S +
¢',1
(
el V I "~2
1)eV_l°g k.rK j ,
"-- 0.05
S(0)Ao) = 0
0.5
1
1.5 2 2.5 (gL-gR) / E0
3
3.5
4 for
0.1
2e(I)A~o,
{
1{ eV "]2~ 1-~\kBT~ j ~,
S(O)Aa~=[~( eV "~2{1 3( eV "~3; \kB TKJ - -5\kB TK] J
~, 0.05
g 0 0.5
1
1.5 2 2.5 (gL-gR) /
3
3.5
1( er "~2;-11
3 \k~KJ
J
2e A~o,
for el V[ < kBTK, at zero-temperature limit, when the density of states are the same in both the electrodes. In both the cases, voltage is scaled by the K o n d o temperature kBTK of the system. Current noise becomes the sub-shot noise.
6. Conclusions
5. Kondo effect If eo + U >/AL,~AR and eo < ]-/L,/AR, the g r o u n d state is given by the filled Fermi sea (the reservoirs are filled with electrons up to the chemical potentials) and a single electron occupying the q u a n t u m dot for the limit where the coupling V is zero. This system can be expressed by using the K o n d o H a m i l t o n i a n [-8], H = H~ + H~x,
x{1
4
Fig. 1. (a) Current I through a quantum dot as a function of the chemical potential #L of the left reservoir at various temperatures, in units of 2eFL, FR=0.1F L, U=2eo, #R=0. (b) Nonequilibrium noise in units of 4e2FL.
Based on the analytical studies, we have shown that the C o u l o m b interaction suppresses the nonequilibrium noise when there is a total of only one electron in the dot. This feature is seen as a dip in nonequilibrium noise as a function of bias voltage. In addition, when the system shows the K o n d o effect due to localized magnetic m o m e n t in the q u a n t u m dot, nonequilibrium noise is suppressed.
(4)
References
where Hox=
kB TK, and
2rte (I)=2x~-eV
~) %
e IVI >
1 -eA-~
t -- __J_J 2N ykk'' a~wak~(S" o)~,~,
(5)
where a is the Pauli matrix and S is the spin o p e r a t o r of the electron in the q u a n t u m dot. We have obtained expressions of average current ( I ) t h r o u g h a q u a n t u m dot and the current noise S(0) Aeo as functions of applied bias voltage V as
[1] R. Landauer, Physica D 38 (1989) 226. [2] M. Biittiker, Phys. Rev. Lett. 65 (1990) 2901. [3] A. Shimizu and M. Ueda, Phys. Rev. Lett. 69 (1992) 1403. I-4] L.Y. Chen, G. Levine, J. Yang and C.S. Ting, Mod. Phys. Lett. B 5 (1991) 315. 1-5] S. Hershfield, Phys. Rev. B 46 (1992) 7061. 1-6] Y. Meir, N.S. Wingreen and P.A. Lee, Phys. Rev. Lett. 70 (1993) 2601. 1-7] F. Yamaguchi and K. Kawamura, J. Phys. Soc. Japan 63 (1994) 1258. [8] J. Kondo, Progr. Theor. Phys. 32 (1964) 37.