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Transient transport through a quantum dot: The impurity Anderson model
. iI2
s
9 December
1996
mm
PHYSICS
j__
ELSEYIER
LETTERS
A
Physics Letters A 223 (1996) 285-288
Transient transport through a quantum dot: The impurity Anderson model Dafang Zheng alb,c, Wai-Sang Li ‘, Youyan Liu a*b a Department of Physics, South China University of Technology, Guangzhou 510641. China b International Center for Materials Physics, Academia Sinica, Shenyang 110015, China c Department of Electronic Engineering, The Hong Kong Polytechnic University, Hung Horn, Kowloon, Hong Kong Received 21 June 1996; accepted for publication 25 September Communicated by J. Plouquet
1996
Abstract
The transient process through a quantum dot is studied via the impurity Anderson model. The nonequilibrium Green’s function technique and the Feynman path-integral method are used. The dependence of the renormalized transition time r/r0 for electrons tunneling through the dot with spin cr is found depending on the occupation probability (Q) of electrons with the opposite spin 0 in the dot. We conclude that the electronic correlation effect will shorten the time required for a system to arrive at a steady-state. PACS: 73.40.Gk;
73.20.D~
In addition to the steady-state properties of mesoscopic systems, the transient process is also of crucial importance both for physical understanding and future device applications. The latter is not yet thoroughly understood. In a recent paper [ 11, Chen and Ting have presented a path-integral approach to investigate the transient transport of a carrier in a double-barrier resonant-tunneling structure. They have demonstrated that the transient process can be characterized by the lifetime of a resonant level or the time dissipation of electrons in the leads. However, the electronic correlation effect, which often leads to a variety of interesting phenomena in mesoscopic systems, has not been taken into account in their work. We have included this effect in the present work. We are going to investigate the transient transport of electrons through a quantum dot. Here we adopt the following one-dimensional Anderson impurity Hamiltonian [ 2-51,
+~&C,+bk,+ T,;&&).
(1)
ku a& (ako), bzr ( bka) are the creation (annihilation) operators of electrons with momentum k and spin (+ in, respectively, the left and the right leads. c,’ (c,) is the operator which creates (annihilates) a spin-a electron in the quantum dot. &, and $ are the corresponding single-particle energy of conduction electrons in the two 0375-9601/96/.$12.00 Copyright PII SO375-9601(96)00737-2
0 1996 Elsevier Science B.V. All rights reserved.
D. Zheng et al. /Physics
286
Letfers A 223 (1996) 285-288
leads. For simplicity we have assumed only two resonant levels E,, ( IJ =t, I>, which may be degenerated in the dot. The fourth term is the on-site Coulomb interaction on the dot, and the Coulomb interaction energy between electrons U is approximately equal to the capacitive charging energy of the quantum dot e2/c N 0.5 meV[2]. The fifth and sixth terms in (1) describe the coupling between electrons in the dot and those in the two leads with the tunneling matrices Tt_p and Tnk. Since the electron distribution in the dot is highly non-equilibrium, we shall, following Chen and Ting [ 161, utilize the Schwinger-Keldysh nonequilibrium Green’s function formalism [7-IO] and the Feynman pathintegral method [ I I-121. We introduce the following retarded, advanced and distribution Green’s functions to characterize the nonequilibrium state of electrons in the dot, iGL(fl,f2)
=@([I
iGz(rt,rz)
=-O(t2
iG:(rt , l2)
= -(CS(~~)C,(~I)).
A more symmetrical iGZP(rt,rz)
=
-
t2)({co(tl),c~(t2)}), -
(2)
tl>({c,,(rl),c~(f2)}),
(3) (4)
form of the nonequilibrium
closed time-path
Green’s function
defined as (5)
(T,,[c,,,(fl)c~~(t2)1)
would be much easier for mathematics. T,) is the generalized chronological ordering operator along the closed time-path, and a(P) = + or - corresponds to tt (t2) located on the “ + ” or “ - ” branch. The Green’s function in Keldysh representation (5) can be easily transformed to the physical representation by G; = ;Sn@$?
(6)
G; = ;~&JG;~,
(7)
G& =G,+-,
(8)
with ,$+ = & = r]+ = -v_ = 1. In the steady-state transport, the Green’s functions G$@ are invariant under time translation. In the transient process, G$? will depend on both tt and t2. The dependence of G;( tt ,t2) on the relative time t = tl - t2 describes the spectrum of electrons in the quantum dot, and its dependence on T = ( tl + t2) /2 characterizes the transient evolution of their state [ 1,6]. So we use (1, T) variables for G: and investigate the Fourier coefficient G,<( w, T) in the transformation of G,<( t, T) with respect to t. To do this we differentiate Eq. (5) with respect to tl and t2 respectively, and obtain a set of equations as follows, ia,,Gz’(tt,t2) -
= ;(v,
i~TRk(~,bkrro(tI
+ vp)&tt )&(t2))
- 12) - iEc,,(T,,c,c,,(tl)c~~(f:!)) - iU(T,,c&,(tl
)cca(tI)c,(tl
- iCTLk(T~akon(tl)c~~(tZ)) )c&(t2)),
(9)
k
(10) There are two new two-particle iGzt(tl,tz)
Green’s functions,
= (T,,c~*(tl)c~~(tl)c,,(tl)c~~(t:!)),
(11)
and iGz$(tl,
12) = (T,c,,(t,)c~~(f2)c~~(t2)c,p(f2))
(12)
D. Zheng et al./Physics
Letters A 223 (19%) 285-288
in Eqs. (9) and ( lo), which are due to the Coulomb Ref. [ 21, we can find the equations
term Untnl.
of motion for Gi:(
- i(nb)CF_k(T,,ak,(tl )c&(t2))
tt ,
Following
287
the decoupling
procedure
t2) and GzE( tl , t2) as
- i(n0)C&k(Tpbkon(tt )c&(t2)),
k
used in
(13)
k
and
respectively. In what follows, we will express the nonequilibrium
(C...))=
s
[da~vl[dak,l
s
[dbkfll[dbkrl
ensemble
average in the path-integral
J
[dc,+l[dc,l(...)exp
form [ 1,6,10].
(iJL(t)dt),
(15)
I’
where L(t)
=~U&(t)(iat)Ukg(f)
-I- ~b~C(t)(i4)bk,(t)
is the Lagrangian of the total system. J, dt = /-‘,” We carry out the Gaussian functional integrations (tl , t2), and performing the Fourier transformation results G,<(w,T)
+ ~C~(t)(i&)G(t)
= -{F(w)
with a Fermi-Dirac-type
- [F(o)
- H(t)
(16)
0
ku
ku
dt+ +Jiz dt_ is the integration along the closed time-path. over Uka and bko variables in Eq. (15). Putting (t, T) for with respect to t, we obtain, after lengthy algebra, the final
- Fo(w>]e-2Ti’}[G~(w>
-GE(w)],
(17)
distribution,
for electrons in the quantum dot. A,.(,, (k, o) and B,,,) (k, w) are, respectively, the retarded and advanced Green’s functions for conduction electrons in the left and the right leads. They adopt the equilibrium forms because of the very fast response of the electrons to the external field. Fn(w) is the initial distribution. In Eq. (17), 7 is the time scale relevant to the transient process and takes the form
70 is the transient time in the absence of Coulomb interaction. A’ (G I$,, - E,,) accounts for the overall modification of the renormalized resonant level due to tunneling and the Coulomb correlation effect. It is given by ” =
(
u(ni,) ’ + A’ _ u+-)
>
(20)
‘.
A E Re[ xk 17’Lk12A,(k, w) + zk l&k12Br( k, w) ] is the modification of the renormalized tunneling, which is assumed constant in the wide-band limit [ 14,151.
resonant
level due to
288
D. Zheng et al. /Physics
Letters A 223 (1996) 285-288
0.6
0.6
0.2
“0
-0.0
0.2
0.4
0.6
0.6
1.0
Fig. 1. Renormalized respectively.
transient time r/q
versus (ire) with parameters
U/A = 2,10,20
and 100, which correspond
to curves a, b, c and d,
Apart from a trivial solution for A’ (A’ = 0) obtained from Eq. (20), we have A’ = A+ V(n*), is reduced to
q1+!$q.
and Eq. ( 19)
(21)
70
In Fig. 1 we have plotted r/r0 against (n,) with V/A = 2,10,20 and 100, respectively. In the figure one can see that r/r0 is in all cases decreasing monotonically as (n,) increases, and clearly this behavior will be enhanced by large values of V/A. When a dc bias is applied to the system at T = 0, the occupation of an electron state in the dot with the opposite spin as the incoming electron would help the system to attain faster a steady-state distribution F(w). This supports our conclusion that electronic correlation plays an important role in transient transport through the quantum dot. The authors acknowledge tion of Hong Kong.
the support from Research Grant Committee
of HKPOLYU
References [ 1; L.Y. Chen and C.S. Ting, Phys. Rev. B 41 (1990) 8533. [ 21 Y. Meir, N.S. Wingreen and PA. Lee, Phys. Rev. Lett. 65 ( 1991) 3048. [ 31 Y. Meir, N.S. Wingreen and PA. Lee, Phys. Rev. L&t. 70 (1993) 2601. [4] [5] [6] [7] [ 81 [9] [ 101 [ 111 [ 121 [ 131 [ 141 [ 15 ]
E.V. Anda and A. LatgC, Phys. Rev. B 50 (1994) 8559. C. Niu, L. Liu and T. Lin, Phys. Rev. B 51 (1995) 5130. L.Y. Chen and C.S. Ting, Phys. Rev. B 43 (1991) 2097. J. Schwinger, J. Math. Phys. 2 (1961) 407. L.V. Keldysh, Zh. Eksp. Tear. Fiz. 47 ( 1964) 1551 [Sov. Phys. JETP 20 ( 1965) 1018]. L.P. Kadanoff and G. Baym, Quantum statistical mechanics (Benjamin, New York, 1962). KC. Chou, Z.B. Su, B.L. Hao and L. Yu, Phys. Rep. 118 (1985) 1. A.R. Hibbs and RI? Feynman, Quantum mechanics and path integrals (McGraw-Hill, New York, 1965). RI? Feynman and FL. Vernon, Ann. Phys. (N.Y.) 24 (1963) 118. L.Y. Chen and C.S. Ting, Phys. Rev. B 43 (1991) 4534. AI? Jauho, N.S. Wingreen and Y. Meir, Phys. Rev. B 50 (1994) 5528. M.P. Anantram and S. Datta, Phys. Rev. B 51 (1995) 7632.
and Croucher Founda-
s
9 December
1996
mm
PHYSICS
j__
ELSEYIER
LETTERS
A
Physics Letters A 223 (1996) 285-288
Transient transport through a quantum dot: The impurity Anderson model Dafang Zheng alb,c, Wai-Sang Li ‘, Youyan Liu a*b a Department of Physics, South China University of Technology, Guangzhou 510641. China b International Center for Materials Physics, Academia Sinica, Shenyang 110015, China c Department of Electronic Engineering, The Hong Kong Polytechnic University, Hung Horn, Kowloon, Hong Kong Received 21 June 1996; accepted for publication 25 September Communicated by J. Plouquet
1996
Abstract
The transient process through a quantum dot is studied via the impurity Anderson model. The nonequilibrium Green’s function technique and the Feynman path-integral method are used. The dependence of the renormalized transition time r/r0 for electrons tunneling through the dot with spin cr is found depending on the occupation probability (Q) of electrons with the opposite spin 0 in the dot. We conclude that the electronic correlation effect will shorten the time required for a system to arrive at a steady-state. PACS: 73.40.Gk;
73.20.D~
In addition to the steady-state properties of mesoscopic systems, the transient process is also of crucial importance both for physical understanding and future device applications. The latter is not yet thoroughly understood. In a recent paper [ 11, Chen and Ting have presented a path-integral approach to investigate the transient transport of a carrier in a double-barrier resonant-tunneling structure. They have demonstrated that the transient process can be characterized by the lifetime of a resonant level or the time dissipation of electrons in the leads. However, the electronic correlation effect, which often leads to a variety of interesting phenomena in mesoscopic systems, has not been taken into account in their work. We have included this effect in the present work. We are going to investigate the transient transport of electrons through a quantum dot. Here we adopt the following one-dimensional Anderson impurity Hamiltonian [ 2-51,
+~&C,+bk,+ T,;&&).
(1)
ku a& (ako), bzr ( bka) are the creation (annihilation) operators of electrons with momentum k and spin (+ in, respectively, the left and the right leads. c,’ (c,) is the operator which creates (annihilates) a spin-a electron in the quantum dot. &, and $ are the corresponding single-particle energy of conduction electrons in the two 0375-9601/96/.$12.00 Copyright PII SO375-9601(96)00737-2
0 1996 Elsevier Science B.V. All rights reserved.
D. Zheng et al. /Physics
286
Letfers A 223 (1996) 285-288
leads. For simplicity we have assumed only two resonant levels E,, ( IJ =t, I>, which may be degenerated in the dot. The fourth term is the on-site Coulomb interaction on the dot, and the Coulomb interaction energy between electrons U is approximately equal to the capacitive charging energy of the quantum dot e2/c N 0.5 meV[2]. The fifth and sixth terms in (1) describe the coupling between electrons in the dot and those in the two leads with the tunneling matrices Tt_p and Tnk. Since the electron distribution in the dot is highly non-equilibrium, we shall, following Chen and Ting [ 161, utilize the Schwinger-Keldysh nonequilibrium Green’s function formalism [7-IO] and the Feynman pathintegral method [ I I-121. We introduce the following retarded, advanced and distribution Green’s functions to characterize the nonequilibrium state of electrons in the dot, iGL(fl,f2)
=@([I
iGz(rt,rz)
=-O(t2
iG:(rt , l2)
= -(CS(~~)C,(~I)).
A more symmetrical iGZP(rt,rz)
=
-
t2)({co(tl),c~(t2)}), -
(2)
tl>({c,,(rl),c~(f2)}),
(3) (4)
form of the nonequilibrium
closed time-path
Green’s function
defined as (5)
(T,,[c,,,(fl)c~~(t2)1)
would be much easier for mathematics. T,) is the generalized chronological ordering operator along the closed time-path, and a(P) = + or - corresponds to tt (t2) located on the “ + ” or “ - ” branch. The Green’s function in Keldysh representation (5) can be easily transformed to the physical representation by G; = ;Sn@$?
(6)
G; = ;~&JG;~,
(7)
G& =G,+-,
(8)
with ,$+ = & = r]+ = -v_ = 1. In the steady-state transport, the Green’s functions G$@ are invariant under time translation. In the transient process, G$? will depend on both tt and t2. The dependence of G;( tt ,t2) on the relative time t = tl - t2 describes the spectrum of electrons in the quantum dot, and its dependence on T = ( tl + t2) /2 characterizes the transient evolution of their state [ 1,6]. So we use (1, T) variables for G: and investigate the Fourier coefficient G,<( w, T) in the transformation of G,<( t, T) with respect to t. To do this we differentiate Eq. (5) with respect to tl and t2 respectively, and obtain a set of equations as follows, ia,,Gz’(tt,t2) -
= ;(v,
i~TRk(~,bkrro(tI
+ vp)&tt )&(t2))
- 12) - iEc,,(T,,c,c,,(tl)c~~(f:!)) - iU(T,,c&,(tl
)cca(tI)c,(tl
- iCTLk(T~akon(tl)c~~(tZ)) )c&(t2)),
(9)
k
(10) There are two new two-particle iGzt(tl,tz)
Green’s functions,
= (T,,c~*(tl)c~~(tl)c,,(tl)c~~(t:!)),
(11)
and iGz$(tl,
12) = (T,c,,(t,)c~~(f2)c~~(t2)c,p(f2))
(12)
D. Zheng et al./Physics
Letters A 223 (19%) 285-288
in Eqs. (9) and ( lo), which are due to the Coulomb Ref. [ 21, we can find the equations
term Untnl.
of motion for Gi:(
- i(nb)CF_k(T,,ak,(tl )c&(t2))
tt ,
Following
287
the decoupling
procedure
t2) and GzE( tl , t2) as
- i(n0)C&k(Tpbkon(tt )c&(t2)),
k
used in
(13)
k
and
respectively. In what follows, we will express the nonequilibrium
(C...))=
s
[da~vl[dak,l
s
[dbkfll[dbkrl
ensemble
average in the path-integral
J
[dc,+l[dc,l(...)exp
form [ 1,6,10].
(iJL(t)dt),
(15)
I’
where L(t)
=~U&(t)(iat)Ukg(f)
-I- ~b~C(t)(i4)bk,(t)
is the Lagrangian of the total system. J, dt = /-‘,” We carry out the Gaussian functional integrations (tl , t2), and performing the Fourier transformation results G,<(w,T)
+ ~C~(t)(i&)G(t)
= -{F(w)
with a Fermi-Dirac-type
- [F(o)
- H(t)
(16)
0
ku
ku
dt+ +Jiz dt_ is the integration along the closed time-path. over Uka and bko variables in Eq. (15). Putting (t, T) for with respect to t, we obtain, after lengthy algebra, the final
- Fo(w>]e-2Ti’}[G~(w>
-GE(w)],
(17)
distribution,
for electrons in the quantum dot. A,.(,, (k, o) and B,,,) (k, w) are, respectively, the retarded and advanced Green’s functions for conduction electrons in the left and the right leads. They adopt the equilibrium forms because of the very fast response of the electrons to the external field. Fn(w) is the initial distribution. In Eq. (17), 7 is the time scale relevant to the transient process and takes the form
70 is the transient time in the absence of Coulomb interaction. A’ (G I$,, - E,,) accounts for the overall modification of the renormalized resonant level due to tunneling and the Coulomb correlation effect. It is given by ” =
(
u(ni,) ’ + A’ _ u+-)
>
(20)
‘.
A E Re[ xk 17’Lk12A,(k, w) + zk l&k12Br( k, w) ] is the modification of the renormalized tunneling, which is assumed constant in the wide-band limit [ 14,151.
resonant
level due to
288
D. Zheng et al. /Physics
Letters A 223 (1996) 285-288
0.6
0.6
0.2
“0
-0.0
0.2
0.4
0.6
0.6
1.0
Fig. 1. Renormalized respectively.
transient time r/q
versus (ire) with parameters
U/A = 2,10,20
and 100, which correspond
to curves a, b, c and d,
Apart from a trivial solution for A’ (A’ = 0) obtained from Eq. (20), we have A’ = A+ V(n*), is reduced to
q1+!$q.
and Eq. ( 19)
(21)
70
In Fig. 1 we have plotted r/r0 against (n,) with V/A = 2,10,20 and 100, respectively. In the figure one can see that r/r0 is in all cases decreasing monotonically as (n,) increases, and clearly this behavior will be enhanced by large values of V/A. When a dc bias is applied to the system at T = 0, the occupation of an electron state in the dot with the opposite spin as the incoming electron would help the system to attain faster a steady-state distribution F(w). This supports our conclusion that electronic correlation plays an important role in transient transport through the quantum dot. The authors acknowledge tion of Hong Kong.
the support from Research Grant Committee
of HKPOLYU
References [ 1; L.Y. Chen and C.S. Ting, Phys. Rev. B 41 (1990) 8533. [ 21 Y. Meir, N.S. Wingreen and PA. Lee, Phys. Rev. Lett. 65 ( 1991) 3048. [ 31 Y. Meir, N.S. Wingreen and PA. Lee, Phys. Rev. L&t. 70 (1993) 2601. [4] [5] [6] [7] [ 81 [9] [ 101 [ 111 [ 121 [ 131 [ 141 [ 15 ]
E.V. Anda and A. LatgC, Phys. Rev. B 50 (1994) 8559. C. Niu, L. Liu and T. Lin, Phys. Rev. B 51 (1995) 5130. L.Y. Chen and C.S. Ting, Phys. Rev. B 43 (1991) 2097. J. Schwinger, J. Math. Phys. 2 (1961) 407. L.V. Keldysh, Zh. Eksp. Tear. Fiz. 47 ( 1964) 1551 [Sov. Phys. JETP 20 ( 1965) 1018]. L.P. Kadanoff and G. Baym, Quantum statistical mechanics (Benjamin, New York, 1962). KC. Chou, Z.B. Su, B.L. Hao and L. Yu, Phys. Rep. 118 (1985) 1. A.R. Hibbs and RI? Feynman, Quantum mechanics and path integrals (McGraw-Hill, New York, 1965). RI? Feynman and FL. Vernon, Ann. Phys. (N.Y.) 24 (1963) 118. L.Y. Chen and C.S. Ting, Phys. Rev. B 43 (1991) 4534. AI? Jauho, N.S. Wingreen and Y. Meir, Phys. Rev. B 50 (1994) 5528. M.P. Anantram and S. Datta, Phys. Rev. B 51 (1995) 7632.
and Croucher Founda-