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Spin-polarized quantum transport through a quantum dot in time-varying magnetic field
Physics Letters A 329 (2004) 55–59 www.elsevier.com/locate/pla
Spin-polarized quantum transport through a quantum dot in time-varying magnetic field Y.-P. Zhang ∗ , J.-Q. Liang Institute of Theoretical Physics and Department of Physics, Shanxi University, Taiyuan 030006, China Received 13 December 2003; accepted 24 June 2004 Available online 6 July 2004 Communicated by A.R. Bishop
Abstract Based on the many-body Schrödinger equation, we derive the spin-dependent rate equations for resonant transport through a quantum dot in time-varying magnetic field which leads to the spin flip. From the rate equations which resemble the optical Bloch equation we are able to identify the spin current viewed as the imbalance transport rates between spin up and down. It is shown that the stationary charge current I = I (t → ∞) does not depend on the magnetic field in agreement with the result in Phys. Rev. B 53 (1996) 15932, while the spin current oscillates with the rotating magnetic field. 2004 Elsevier B.V. All rights reserved. PACS: 72.25.-b; 73.21.La; 05.60.Gg; 68.65.Hb Keywords: Quantum dot; Spin-polarized currents
In recent years great attention has been paid to quantum transport through quantum dots with a number of discrete quantum levels due especially to recent advance in nanotechnology which allows the fabrication of very small dots weekly coupled to the macroscopic charge reservoirs [1]. The Coulomb blockade effects are focusing researches both theoretically and experimentally [2–8] with classical rate equations describing the transport through a single dot. The classical rate equations [9,10], however, cannot describe explicitly the quantum superposition of the states in adjacent wells and therefore ought to be modified. The Bloch-type rate equations [11–15] which manifest the quantum coherence in the off-diagonal elements of density-matrix coupled with the diagonal elements has been proposed for the quantum modification. The approach of deriving Bloch-type rate equations from the many-body Schrödinger equation can be applied only when resonant level E is within the bias, i.e., EFL E EFR , and the level width of the state in the quantum dots is much smaller than the bias such that ΓL(R) EFL − EFR , * Corresponding author.
E-mail address: [email protected] (Y.-P. Zhang). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.06.078
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Y.-P. Zhang, J.-Q. Liang / Physics Letters A 329 (2004) 55–59
Fig. 1. Spin-splitting of energy level in a single quantum dot due to magnetic field and resonant transport.
where EFL , EFR denote the Fermi energies of the left and right electron reservoirs, respectively, and E is the energy level in the dots. If the resonant level is near the edge of the band while with level width much smaller than the bandwidth, the method is valid only for the stationary case (t → ∞). Phenomena associated with spin-polarized currents have attracted considerable interest recently. It is certainly of importance to extend the charge transport through quantum dots to the spin-polarized transport. To this end we in this Letter following Gurvitz et al. [11,14,15] derive the spin-dependent Bloch-type rate equations from the manybody Schrödinger equation with Hubbard-type tunneling Hamiltonian, describing the entire system of reservoir and the quantum dots in the time-dependent magnetic fields which result in the spin flipping of electrons. Both the charge and spin-polarized transports are obtained simultaneously. The system what we considered includes a quantum dot coupled to two separate electron reservoirs (see Fig. 1). Longitudinal and rotating transversal magnetic fields B , B⊥ are applied in the dot, such that B = B ez , B⊥ = −B⊥ cos(ωt)ex + B⊥ sin(ωt)ey . The single level in the quantum dot splits to two levels E1 and E2 due to the Zeeman energy of spin in the static longitudinal field, the transversal field which is regarded as a perturbation results in the spin flipping. We furthermore assume that EFR E1,2 EFL , i.e., the energy levels are deeply inside the band. In the occupation number representation, the tunneling Hamiltonian of the entire system is seen to be + + H= El als als + Er ars ars + E1 a1+ a1 + E2 a2+ a2 + Ω0 a2+ a1 eiωt + a1+ a2 e−iωt l,s
+
r,s
+ Ω1l(r) a1+ al(r)↓ + al(r)↓ a1 +
l(r)
+ Ω2l(r) a2+ al(r)↑ + al(r)↑ a2 + U a1+ a1 a2+ a2 .
(1)
l(r)
Here, the subscripts l and r denote the levels in the left and right reservoirs s =↑, ↓, U is the Coulomb repulsive energy. Ω0 is the coupling constant of spin flipping due to the rotating field in the quantum dot and is proportional to the magnitude of transversal magnetic field B⊥ . Ωi,l(r) (i = 1, 2) denote the tunnel coupling at ith-level (i = 1, 2) between the quantum dot and left (right) reservoir. In the zero temperature, we assume that all the levels in the emitter and in the collector are initially filled up to the Fermi energies EFR , EFL , respectively. This is called the “vacuum” state |vac . The vacuum state is unstable and can decay to other states, which can be constructed form the many-body wave function describing the time evolution of this system as Ψ (t) = b0 (t) + bils (t)a + als + blsrs (t)a + als + b12lsl s (t)a + a + als al s + · · · |vac , (2) i
i,l,s
l,s,r,s
rs
l
1
2
where the coefficients b(t) are the time-dependent probability amplitudes of the corresponding states and can be determined from the Schrödinger equation d i Ψ (t) = H Ψ (t) (3) dt
Y.-P. Zhang, J.-Q. Liang / Physics Letters A 329 (2004) 55–59
57
along with the initial conditions such that b0 (0) = 1, while all the other coefficients in Eq. (2) vanish. This leads to a set of infinite coupled-linear-differential equations for the amplitudes b(t): i b˙ 1l↑(t) = (−El + E1 )b1l↑(t) + Ω0 b2l↑ (t)e−iωt + (4a) Ω1r bl↑r↓ (t) + Ω2l b12l↑l ↑ (t), l
r
i b˙ 2l↑(t) = (−El + E2 )b2l↓(t) + Ω2l b0 (t) + Ω0 b1l↓ (t)eiωt +
Ω2r bl↑r↑ (t) +
Ω2l b2r↓l↑l ↑ (t) +
l
(4b)
Ω1l b1r↓l↑l ↓ (t),
(4c)
l
i b˙12l↑l ↑ (t) = (−El − El + E1 + E2 + U )b12l↑l ↑ (t) + Ω2l b1l↑(t) + +
Ω2l b12l↓l ↑ (t),
l
r
i b˙ l↑r↓(t) = (−El + Er )bl↑r↓(t) + Ω1r b1l↑ (t) +
Ω2r br↑l↑l ↑ (t)
r
Ω1r b2r↓l↑l ↑ (t),
(4d)
r
.. . The Fock space of the quantum dot consists of four possible states, |0 —the levels E1,2 are empty, |1 —the level E1 is occupied, |2 —the level E2 is occupied, |3 —both levels E1,2 are occupied. It is easy to find the (k) density-matrix of the quantum dot σij (t) with the superscript index k denoting the number of electrons in the collector at time t, and the subscripts i, j = {0, 1, 2, 3} indicating the occupation states in the dot. The elements of density-matrix are seen to be 2 blsrs (t)2 + · · · = σ (0) + σ (1) + σ (2) + · · · , σ00 = b0 (t) + 00 00 00 l,r,s,s
b1ls (t)2 + σ11 = l,s
σ12 =
b1rsls l s
(t)2 + · · · = σ (0) + σ (1) + · · · , 11
l
∗ b1ls (t)b2ls (t) +
l
l,s
11
∗ b1rsls l s
(t)b2rsls
l s
(t) + · · · ,
(5)
.. . Using the same technique as in Refs. [11,14], we perform the Laplace transformation for b(t) in Eqs. (4) in order to convert the coupled-linear-differential equations into a set of infinite algebraic equations from which the Bloch-type rate equations for the elements of density-matrix are obtained as (k)
(k)
(k−1)
(k−1)
σ˙ 00 = −(Γ1L + Γ2L )σ00 + Γ1R σ11 + Γ2R σ22 , (k) (k) iωt (k) (k) (k−1) (k) −iωt
σ33 + iΩ0 σ12 e − σ21 e , σ˙ 11 = − Γ1R + Γ2L σ11 + Γ1L σ00 + Γ2R (k) (k) (k) (k−1) (k) (k)
σ33 − iΩ0 σ12 eiωt − σ21 e−iωt , σ˙ 22 = − Γ2R + Γ1L σ22 + Γ2L σ00 + Γ1R (k) (k) (k) (k)
σ˙ 33 = − Γ1R σ33 + Γ1L + Γ2R σ22 + Γ2L σ11 , (k) (k) 1 (k) (k) (k) (k) −iωt
σ˙ 12 = −iεσ12 − Γ1R + Γ2R + Γ1L + Γ2L + iΩ0 σ11 − σ22 , σ12 + Γm σ00 e 2 (k) (k) 1 (k) (k) (k) (k)
σ21 + Γm σ00 − iΩ0 σ11 − σ22 eiωt . σ˙ 21 = iεσ21 − Γ1R + Γ2R + Γ1L + Γ2L 2
(6a) (6b) (6c) (6d) (6e) (6f)
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Y.-P. Zhang, J.-Q. Liang / Physics Letters A 329 (2004) 55–59
Where ε = E1 − E2 , which is level splitting induced by the longitudinal magnetic field, Γm = 2πρL (E1 ) × Ω1l (E1 )Ω2l (E1 ), and 2 2
ΓiL(R) = 2πρl(r)(Ei )Ωil(r) (Ei ) , (7) ΓiL(R) = 2πρl(r)(Ei + U )Ωil(r)(Ei + U ) (i = 1, 2) are the partial width of levels E1 and E2 resulted from the coupling to the emitter (collector). Where ρl(r) is the density of states in the left (right) reservoir [14]. We can obtain the following set of Bloch-type equations after summing over k in Eqs. (6): σ˙ 00 = −(Γ1L + Γ2L )σ00 + Γ1R σ11 + Γ2R σ22 , (8a)
iωt −iωt , σ˙ 11 = − Γ1R + Γ2L σ11 + Γ1L σ00 + Γ2R σ33 + iΩ0 σ12 e − σ21 e (8b)
iωt −iωt , σ˙ 22 = − Γ2R + Γ1L σ22 + Γ2L σ00 + Γ1R σ33 − iΩ0 σ12 e − σ21 e (8c)
σ˙ 33 = − Γ1R + Γ2R σ33 + Γ1L σ22 + Γ2Lσ11 , (8d) 1
σ12 + Γm σ00 + iΩ0 (σ11 − σ22 )e−iωt , + Γ2L σ˙ 12 = −iεσ12 − Γ1R + Γ2R + Γ1L (8e) 2 1
σ˙ 21 = iεσ21 − Γ1R + Γ2R + Γ1L (8f) σ21 + Γm σ00 − iΩ0(σ11 − σ22 )eiωt , + Γ2L 2 (k) where σij = σij (i, j = 0, 1, 2, 3). Eqs. (8) describe the time evolution of the density matrix for the single dot k
with two levels in the magnetic field. The current flowing through the system is determined by the arriving number rate of electrons in the collector, i.e., I = eN˙ R (t),
(9)
where NR is the number of electrons accumulated in the collector, the time-derivative of which is seen to be (k) (k) (k) (k)
. k σ˙ 00 + σ˙ 11 + σ˙ 22 + σ˙ 33 N˙ R (t) = (10) k
Using Eqs. (6) the total electron current is obtained as
I (t)
= Γ1R σ11 + Γ2R σ22 + Γ1R (11) σ33 . + Γ2R e The stationary (dc) current Idc = I (t → ∞), can be evaluated from Eqs. (8) by taking into account that σ˙ ij → 0 for t → ∞, thus Eqs. (8) reduce to a set of linear algebraic equations, supplemented by the probability conservation condition σ00 + σ11 + σ22 + σ33 = 1. For the sake of simplicity, we consider the case with the same partial widths
of the levels E1,2 , that means Γ1L(R) = Γ2L(R) = ΓL(R) , Γ1L(R) = Γ2L(R) = ΓL(R) , and then Γm = ΓL . Solving Eqs. (8) and (11), one finds the total dc current as 2ΓR ΓL (ΓL + ΓR ) I = . e ΓR ΓR + ΓL ΓL + 2ΓL ΓR
(12)
It is interesting to notice that the total current does not depend on the external magnetic field and is similar to the result in Ref. [11] while in our case there are two levels in the single dot. We can also evaluate the individual current for spin up and down, respectively, from the quantum dot to the right reservoir and the result is
ΓR ΓL (ΓL + ΓR ) I↑
= Γ2R σ22 + Γ2R σ33 = + A ΓL + ΓR sin ωt − ε cos ωt ,
e ΓR ΓR + ΓL ΓL + 2ΓL ΓR
(13a)
ΓR ΓL (ΓL + ΓR ) I↓
= Γ1R σ11 + Γ1R σ33 = − A ΓL + ΓR sin ωt − ε cos ωt ,
e ΓR ΓR + ΓL ΓL + 2ΓL ΓR
(13b)
Y.-P. Zhang, J.-Q. Liang / Physics Letters A 329 (2004) 55–59
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where A=
2Ω0 ΓR2 ΓR ΓL
{(ΓL + ΓR )[(ΓL + ΓR )2 + ε2 ] + 4Ω02(ΓL + ΓR )}[ΓR ΓR + ΓL ΓL + 2ΓL ΓR ]
.
The spin current Is is defined as
IS = (I↑ − I↓ ) = 2A ΓL + ΓR sin ωt − ε cos ωt , e which oscillates with the transversal magnetic field. A steady spin current that
(14)
IS (15) = −2Aε e is approached when the frequency of transversal field vanishes, ω = 0. In order to have a better understanding of the spin current, we consider the simple case of the isolated quantum dot in the longitudinal and transversal magnetic fields. In this case we have
ΓL(R) = ΓL(R) = 0.
(16)
Solving Eqs. (8) and (15) with the initial condition σ11 (0) = 1, and σ22 (0) = 0, we obtain σ11 (t) =
Ω02 cos2 (ω1 t) + ε2 /4 Ω02 + ε2 /4
,
(17)
where ω1 = (1/2) 4Ω02 + ε2 . It is not surprising to see that the result Eq. (17) is similar to that in the Ref. [14] (see Eq. (5) there) where the “device” consists of two coupled dots with one level in each dot without taking into account of the spin degree of freedom while our single dot contains two coupled energy levels due to the spinmagnetic field interaction. The parameters Ω0 , ε depend on the transversal magnetic field B⊥ and longitudinal magnetic field B , respectively. Our observation shows that two coupled quantum dots with one level in each dot are equivalent to a single quantum dot with two coupled levels. In terms of the formalism in Refs. [11,14,15] we derive the spin-polarized rate equations for the quantum transport in a single quantum dot with the external magnetic fields which lead to the Zeeman split and spin flipping. The total electron current and the spin current are obtained for the stationary case. The total current agrees with the result of Gurvitz and Prager in Ref. [11], while the spin current oscillates with the rotating magnetic field and reduced to a steady current when the frequency of the rotating field vanishes.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
H. Park, et al., Nature 407 (2000) 57. C.W.J. Beenakker, Phys. Rev. B 44 (1991) 1646. N.S. Wingreen, K.W. Jacobsen, J.W. Wilkins, Phys. Rev. B 40 (1989) 11834. G. Klimeck, G. Chen, S. Datta, Phys. Rev. B 50 (1994) 2316. J.H.F. Scott-Thomas, S.B. Field, M.A. Kastner, H.I. Smith, D.A. Antoniadis, Phys. Rev. Lett. 62 (1989) 583. M. Field, C.G. Smith, M. Pepper, D.A. Ritchie, J.E.F. Frost, G.A.C. Jones, D.G. Hasko, Phys. Rev. Lett. 70 (1993) 1311. D. Boese, H. Schoeller, Europhys. Lett. 54 (2001) 668. A. Mitra, I. Aleiner, A.J. Mills, cond-mat/0302132. S.A. Gurvitz, H.J. Lipkin, Ya.S. Prager, Mod. Phys. Lett. B 8 (1994) 1377. J.H. Davies, S. Hershfield, P. Hyldgaard, J.W. Wilkins, Phys. Rev. B 47 (1993) 4603. S.A. Gurvitz, Ya.S. Prager, Phys. Rev. B 53 (1996) 15932. Yu.V. Nazarov, Physica B 189 (1993) 57. T.H. Stoof, Yu.V. Nazarov, Phys. Rev. B 53 (1996) 1050. S.A. Gurvitz, Phys. Rev. B 57 (1998) 6602. S.A. Gurvitz, Phys. Rev. B 56 (1997) 15215.
Spin-polarized quantum transport through a quantum dot in time-varying magnetic field Y.-P. Zhang ∗ , J.-Q. Liang Institute of Theoretical Physics and Department of Physics, Shanxi University, Taiyuan 030006, China Received 13 December 2003; accepted 24 June 2004 Available online 6 July 2004 Communicated by A.R. Bishop
Abstract Based on the many-body Schrödinger equation, we derive the spin-dependent rate equations for resonant transport through a quantum dot in time-varying magnetic field which leads to the spin flip. From the rate equations which resemble the optical Bloch equation we are able to identify the spin current viewed as the imbalance transport rates between spin up and down. It is shown that the stationary charge current I = I (t → ∞) does not depend on the magnetic field in agreement with the result in Phys. Rev. B 53 (1996) 15932, while the spin current oscillates with the rotating magnetic field. 2004 Elsevier B.V. All rights reserved. PACS: 72.25.-b; 73.21.La; 05.60.Gg; 68.65.Hb Keywords: Quantum dot; Spin-polarized currents
In recent years great attention has been paid to quantum transport through quantum dots with a number of discrete quantum levels due especially to recent advance in nanotechnology which allows the fabrication of very small dots weekly coupled to the macroscopic charge reservoirs [1]. The Coulomb blockade effects are focusing researches both theoretically and experimentally [2–8] with classical rate equations describing the transport through a single dot. The classical rate equations [9,10], however, cannot describe explicitly the quantum superposition of the states in adjacent wells and therefore ought to be modified. The Bloch-type rate equations [11–15] which manifest the quantum coherence in the off-diagonal elements of density-matrix coupled with the diagonal elements has been proposed for the quantum modification. The approach of deriving Bloch-type rate equations from the many-body Schrödinger equation can be applied only when resonant level E is within the bias, i.e., EFL E EFR , and the level width of the state in the quantum dots is much smaller than the bias such that ΓL(R) EFL − EFR , * Corresponding author.
E-mail address: [email protected] (Y.-P. Zhang). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.06.078
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Y.-P. Zhang, J.-Q. Liang / Physics Letters A 329 (2004) 55–59
Fig. 1. Spin-splitting of energy level in a single quantum dot due to magnetic field and resonant transport.
where EFL , EFR denote the Fermi energies of the left and right electron reservoirs, respectively, and E is the energy level in the dots. If the resonant level is near the edge of the band while with level width much smaller than the bandwidth, the method is valid only for the stationary case (t → ∞). Phenomena associated with spin-polarized currents have attracted considerable interest recently. It is certainly of importance to extend the charge transport through quantum dots to the spin-polarized transport. To this end we in this Letter following Gurvitz et al. [11,14,15] derive the spin-dependent Bloch-type rate equations from the manybody Schrödinger equation with Hubbard-type tunneling Hamiltonian, describing the entire system of reservoir and the quantum dots in the time-dependent magnetic fields which result in the spin flipping of electrons. Both the charge and spin-polarized transports are obtained simultaneously. The system what we considered includes a quantum dot coupled to two separate electron reservoirs (see Fig. 1). Longitudinal and rotating transversal magnetic fields B , B⊥ are applied in the dot, such that B = B ez , B⊥ = −B⊥ cos(ωt)ex + B⊥ sin(ωt)ey . The single level in the quantum dot splits to two levels E1 and E2 due to the Zeeman energy of spin in the static longitudinal field, the transversal field which is regarded as a perturbation results in the spin flipping. We furthermore assume that EFR E1,2 EFL , i.e., the energy levels are deeply inside the band. In the occupation number representation, the tunneling Hamiltonian of the entire system is seen to be + + H= El als als + Er ars ars + E1 a1+ a1 + E2 a2+ a2 + Ω0 a2+ a1 eiωt + a1+ a2 e−iωt l,s
+
r,s
+ Ω1l(r) a1+ al(r)↓ + al(r)↓ a1 +
l(r)
+ Ω2l(r) a2+ al(r)↑ + al(r)↑ a2 + U a1+ a1 a2+ a2 .
(1)
l(r)
Here, the subscripts l and r denote the levels in the left and right reservoirs s =↑, ↓, U is the Coulomb repulsive energy. Ω0 is the coupling constant of spin flipping due to the rotating field in the quantum dot and is proportional to the magnitude of transversal magnetic field B⊥ . Ωi,l(r) (i = 1, 2) denote the tunnel coupling at ith-level (i = 1, 2) between the quantum dot and left (right) reservoir. In the zero temperature, we assume that all the levels in the emitter and in the collector are initially filled up to the Fermi energies EFR , EFL , respectively. This is called the “vacuum” state |vac . The vacuum state is unstable and can decay to other states, which can be constructed form the many-body wave function describing the time evolution of this system as Ψ (t) = b0 (t) + bils (t)a + als + blsrs (t)a + als + b12lsl s (t)a + a + als al s + · · · |vac , (2) i
i,l,s
l,s,r,s
rs
l
1
2
where the coefficients b(t) are the time-dependent probability amplitudes of the corresponding states and can be determined from the Schrödinger equation d i Ψ (t) = H Ψ (t) (3) dt
Y.-P. Zhang, J.-Q. Liang / Physics Letters A 329 (2004) 55–59
57
along with the initial conditions such that b0 (0) = 1, while all the other coefficients in Eq. (2) vanish. This leads to a set of infinite coupled-linear-differential equations for the amplitudes b(t): i b˙ 1l↑(t) = (−El + E1 )b1l↑(t) + Ω0 b2l↑ (t)e−iωt + (4a) Ω1r bl↑r↓ (t) + Ω2l b12l↑l ↑ (t), l
r
i b˙ 2l↑(t) = (−El + E2 )b2l↓(t) + Ω2l b0 (t) + Ω0 b1l↓ (t)eiωt +
Ω2r bl↑r↑ (t) +
Ω2l b2r↓l↑l ↑ (t) +
l
(4b)
Ω1l b1r↓l↑l ↓ (t),
(4c)
l
i b˙12l↑l ↑ (t) = (−El − El + E1 + E2 + U )b12l↑l ↑ (t) + Ω2l b1l↑(t) + +
Ω2l b12l↓l ↑ (t),
l
r
i b˙ l↑r↓(t) = (−El + Er )bl↑r↓(t) + Ω1r b1l↑ (t) +
Ω2r br↑l↑l ↑ (t)
r
Ω1r b2r↓l↑l ↑ (t),
(4d)
r
.. . The Fock space of the quantum dot consists of four possible states, |0 —the levels E1,2 are empty, |1 —the level E1 is occupied, |2 —the level E2 is occupied, |3 —both levels E1,2 are occupied. It is easy to find the (k) density-matrix of the quantum dot σij (t) with the superscript index k denoting the number of electrons in the collector at time t, and the subscripts i, j = {0, 1, 2, 3} indicating the occupation states in the dot. The elements of density-matrix are seen to be 2 blsrs (t)2 + · · · = σ (0) + σ (1) + σ (2) + · · · , σ00 = b0 (t) + 00 00 00 l,r,s,s
b1ls (t)2 + σ11 = l,s
σ12 =
b1rsls l s
(t)2 + · · · = σ (0) + σ (1) + · · · , 11
l
∗ b1ls (t)b2ls (t) +
l
l,s
11
∗ b1rsls l s
(t)b2rsls
l s
(t) + · · · ,
(5)
.. . Using the same technique as in Refs. [11,14], we perform the Laplace transformation for b(t) in Eqs. (4) in order to convert the coupled-linear-differential equations into a set of infinite algebraic equations from which the Bloch-type rate equations for the elements of density-matrix are obtained as (k)
(k)
(k−1)
(k−1)
σ˙ 00 = −(Γ1L + Γ2L )σ00 + Γ1R σ11 + Γ2R σ22 , (k) (k) iωt (k) (k) (k−1) (k) −iωt
σ33 + iΩ0 σ12 e − σ21 e , σ˙ 11 = − Γ1R + Γ2L σ11 + Γ1L σ00 + Γ2R (k) (k) (k) (k−1) (k) (k)
σ33 − iΩ0 σ12 eiωt − σ21 e−iωt , σ˙ 22 = − Γ2R + Γ1L σ22 + Γ2L σ00 + Γ1R (k) (k) (k) (k)
σ˙ 33 = − Γ1R σ33 + Γ1L + Γ2R σ22 + Γ2L σ11 , (k) (k) 1 (k) (k) (k) (k) −iωt
σ˙ 12 = −iεσ12 − Γ1R + Γ2R + Γ1L + Γ2L + iΩ0 σ11 − σ22 , σ12 + Γm σ00 e 2 (k) (k) 1 (k) (k) (k) (k)
σ21 + Γm σ00 − iΩ0 σ11 − σ22 eiωt . σ˙ 21 = iεσ21 − Γ1R + Γ2R + Γ1L + Γ2L 2
(6a) (6b) (6c) (6d) (6e) (6f)
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Y.-P. Zhang, J.-Q. Liang / Physics Letters A 329 (2004) 55–59
Where ε = E1 − E2 , which is level splitting induced by the longitudinal magnetic field, Γm = 2πρL (E1 ) × Ω1l (E1 )Ω2l (E1 ), and 2 2
ΓiL(R) = 2πρl(r)(Ei )Ωil(r) (Ei ) , (7) ΓiL(R) = 2πρl(r)(Ei + U )Ωil(r)(Ei + U ) (i = 1, 2) are the partial width of levels E1 and E2 resulted from the coupling to the emitter (collector). Where ρl(r) is the density of states in the left (right) reservoir [14]. We can obtain the following set of Bloch-type equations after summing over k in Eqs. (6): σ˙ 00 = −(Γ1L + Γ2L )σ00 + Γ1R σ11 + Γ2R σ22 , (8a)
iωt −iωt , σ˙ 11 = − Γ1R + Γ2L σ11 + Γ1L σ00 + Γ2R σ33 + iΩ0 σ12 e − σ21 e (8b)
iωt −iωt , σ˙ 22 = − Γ2R + Γ1L σ22 + Γ2L σ00 + Γ1R σ33 − iΩ0 σ12 e − σ21 e (8c)
σ˙ 33 = − Γ1R + Γ2R σ33 + Γ1L σ22 + Γ2Lσ11 , (8d) 1
σ12 + Γm σ00 + iΩ0 (σ11 − σ22 )e−iωt , + Γ2L σ˙ 12 = −iεσ12 − Γ1R + Γ2R + Γ1L (8e) 2 1
σ˙ 21 = iεσ21 − Γ1R + Γ2R + Γ1L (8f) σ21 + Γm σ00 − iΩ0(σ11 − σ22 )eiωt , + Γ2L 2 (k) where σij = σij (i, j = 0, 1, 2, 3). Eqs. (8) describe the time evolution of the density matrix for the single dot k
with two levels in the magnetic field. The current flowing through the system is determined by the arriving number rate of electrons in the collector, i.e., I = eN˙ R (t),
(9)
where NR is the number of electrons accumulated in the collector, the time-derivative of which is seen to be (k) (k) (k) (k)
. k σ˙ 00 + σ˙ 11 + σ˙ 22 + σ˙ 33 N˙ R (t) = (10) k
Using Eqs. (6) the total electron current is obtained as
I (t)
= Γ1R σ11 + Γ2R σ22 + Γ1R (11) σ33 . + Γ2R e The stationary (dc) current Idc = I (t → ∞), can be evaluated from Eqs. (8) by taking into account that σ˙ ij → 0 for t → ∞, thus Eqs. (8) reduce to a set of linear algebraic equations, supplemented by the probability conservation condition σ00 + σ11 + σ22 + σ33 = 1. For the sake of simplicity, we consider the case with the same partial widths
of the levels E1,2 , that means Γ1L(R) = Γ2L(R) = ΓL(R) , Γ1L(R) = Γ2L(R) = ΓL(R) , and then Γm = ΓL . Solving Eqs. (8) and (11), one finds the total dc current as 2ΓR ΓL (ΓL + ΓR ) I = . e ΓR ΓR + ΓL ΓL + 2ΓL ΓR
(12)
It is interesting to notice that the total current does not depend on the external magnetic field and is similar to the result in Ref. [11] while in our case there are two levels in the single dot. We can also evaluate the individual current for spin up and down, respectively, from the quantum dot to the right reservoir and the result is
ΓR ΓL (ΓL + ΓR ) I↑
= Γ2R σ22 + Γ2R σ33 = + A ΓL + ΓR sin ωt − ε cos ωt ,
e ΓR ΓR + ΓL ΓL + 2ΓL ΓR
(13a)
ΓR ΓL (ΓL + ΓR ) I↓
= Γ1R σ11 + Γ1R σ33 = − A ΓL + ΓR sin ωt − ε cos ωt ,
e ΓR ΓR + ΓL ΓL + 2ΓL ΓR
(13b)
Y.-P. Zhang, J.-Q. Liang / Physics Letters A 329 (2004) 55–59
59
where A=
2Ω0 ΓR2 ΓR ΓL
{(ΓL + ΓR )[(ΓL + ΓR )2 + ε2 ] + 4Ω02(ΓL + ΓR )}[ΓR ΓR + ΓL ΓL + 2ΓL ΓR ]
.
The spin current Is is defined as
IS = (I↑ − I↓ ) = 2A ΓL + ΓR sin ωt − ε cos ωt , e which oscillates with the transversal magnetic field. A steady spin current that
(14)
IS (15) = −2Aε e is approached when the frequency of transversal field vanishes, ω = 0. In order to have a better understanding of the spin current, we consider the simple case of the isolated quantum dot in the longitudinal and transversal magnetic fields. In this case we have
ΓL(R) = ΓL(R) = 0.
(16)
Solving Eqs. (8) and (15) with the initial condition σ11 (0) = 1, and σ22 (0) = 0, we obtain σ11 (t) =
Ω02 cos2 (ω1 t) + ε2 /4 Ω02 + ε2 /4
,
(17)
where ω1 = (1/2) 4Ω02 + ε2 . It is not surprising to see that the result Eq. (17) is similar to that in the Ref. [14] (see Eq. (5) there) where the “device” consists of two coupled dots with one level in each dot without taking into account of the spin degree of freedom while our single dot contains two coupled energy levels due to the spinmagnetic field interaction. The parameters Ω0 , ε depend on the transversal magnetic field B⊥ and longitudinal magnetic field B , respectively. Our observation shows that two coupled quantum dots with one level in each dot are equivalent to a single quantum dot with two coupled levels. In terms of the formalism in Refs. [11,14,15] we derive the spin-polarized rate equations for the quantum transport in a single quantum dot with the external magnetic fields which lead to the Zeeman split and spin flipping. The total electron current and the spin current are obtained for the stationary case. The total current agrees with the result of Gurvitz and Prager in Ref. [11], while the spin current oscillates with the rotating magnetic field and reduced to a steady current when the frequency of the rotating field vanishes.
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