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Probing the effective nuclear-spin magnetic field in a single quantum dot via full counting statistics
Annals of Physics 354 (2015) 375–384
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Probing the effective nuclear-spin magnetic field in a single quantum dot via full counting statistics Hai-Bin Xue a,∗ , Yi-Hang Nie b , Jingzhe Chen c , Wei Ren c,∗ a
College of Physics and Optoelectronics, Taiyuan University of Technology, Taiyuan 030024, China
b
Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China
c
International Centre for Quantum and Molecular Structures and Department of Physics, Shanghai University, 99 Shangda Road, Shanghai 200444, China
highlights • The effective nuclear-spin magnetic field gives rise to the off-diagonal elements of the reduced density matrix of single QD.
• The off-diagonal elements of reduced density matrix of the QD have a significant impact on the high-order current cumulants.
• The high-order current cumulants are sensitive to the orientation and magnitude of the effective nuclear-spin magnetic field.
• The FCS can be used to detect the orientation and magnitude of the effective nuclear-spin magnetic field in a single QD.
article
info
Article history: Received 14 November 2014 Accepted 5 January 2015 Available online 12 January 2015 Keywords: Single quantum dot Nuclear-spin magnetic field Full counting statistics
∗
abstract We study theoretically the full counting statistics of electron transport through a quantum dot weakly coupled to two ferromagnetic leads, in which an effective nuclear-spin magnetic field originating from the configuration of nuclear spins is considered. We demonstrate that the quantum coherence between the two singly-occupied eigenstates and the spin polarization of two ferromagnetic leads play an important role in the formation of superPoissonian noise. In particular, the orientation and magnitude of the effective field have a significant influence on the variations of the values of high-order cumulants, and the variations of the
Corresponding authors. Tel.: +86 351 6018030; fax: +86 351 6018030. E-mail addresses: [email protected] (H.-B. Xue), [email protected] (W. Ren).
http://dx.doi.org/10.1016/j.aop.2015.01.001 0003-4916/© 2015 Elsevier Inc. All rights reserved.
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skewness and kurtosis values are more sensitive to the orientation and magnitude of the effective field than the shot noise. Thus, the high-order cumulants of transport current can be used to qualitatively extract information on the orientation and magnitude of the effective nuclear-spin magnetic field in a single quantum dot. © 2015 Elsevier Inc. All rights reserved.
1. Introduction The controlled manipulation of the localized electron spins in semiconductor quantum dots (QDs) is critical for quantum information processing with confined electron spins [1–4]. Recently, the hyperfine interaction between localized electron and nuclear spins in the QDs has attracted much attention since it is the dominant mechanism of electron spin relaxation in QDs at low temperature [5–15], whereas the electron spin relaxation mechanism can be suppressed by an external magnetic field [16]. Due to the random configuration of nuclear spins, the quantum transport properties of electrons through the QD system can be strongly affected by the orientation and magnitude of the
− →
effective nuclear magnetic field B N . For example, the effective hyperfine field leads to the mixing between two-electron singlet and triplet states [17–21], while a small magnetic field suppresses such mixing [18]; and lifts the spin blockade in a double QDs. In fact, the quantum transport of electrons through the QD system is essentially a quantum stochastic process with the quantum fluctuations of transport current. It has been theoretically and experimentally demonstrated that the full counting statistics (FCS) [22,23] properties can allow one to identify the intrinsic properties of the QD system and access the information of electron correlation beyond what is obtainable from the differential conductance or the average current [24,25]. For instance, the high-order current cumulants [26] can be used to detect the quantum coherence in a side-coupled double QDs system [27] and the positions of the zeros of the generating function [28], reveal the intrinsic multistability [29], and extract the fractional charge of charge transfer through an impurity in a chiral Luttinger liquid [30]. Consequently, the FCS of electron transport through the QD system can not only provide a deeper insight into the nature of quantum transport mechanism, but also propose a new way to obtain the information on the orientation and magnitude of effective nuclear-spin magnetic field.
− →
The aim of this work is thus to study the influence of the effective nuclear-spin magnetic field B N on the first four cumulants of electron transport through a single QD system coupled weakly to two ferromagnetic leads, and then analyze the feasibility of extracting the information of the orientation
− →
and magnitude of the effective hyperfine field B N from the high-order cumulants of transport current. This paper is organized as follows. In Section 2, we introduce the considered QD system and outline the procedure to obtain the FCS formalism based on an effective particle-number-resolved quantum master equation [31–33] and the Rayleigh–Schrödinger perturbation theory developed in Refs. [34–36]. The numerical results are discussed in Section 3, where we discuss the effects of the orientation and magnitude of the effective nuclear-spin magnetic field on the first four current cumulants, and analyze the feasibility of extracting information about the effective nuclear-spin magnetic field from the high-order current cumulants. Finally, in Section 4 we summarize the work. 2. Model and formalism The system considered here consists of a single QD possessing an effective nuclear-spin magnetic field weakly coupled to two ferromagnetic leads, see Fig. 1. The system Hamiltonian is described by Htotal = Hdot + Hleads + HT , in which the QD Hamiltonian reads Hdot =
σ
− → → ε dĎσ dσ + UdĎ↑ d↑ dĎ↓ d↓ + − s · BN ,
(1)
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Fig. 1. (Color online) Schematic representation of the single QD system possessing an effective nuclear-spin magnetic field weakly coupled to two ferromagnetic leads.
where dĎσ (dσ ) creates (annihilates) an electron with spin σ and energy ε (which can be tuned by a gate voltage Vg ) in the QD. U is the intradot Coulomb interaction between two electrons in the QD system. The third term describes the conduction electron’s Zeeman splitting, where g µB has been
− →
− →
absorbed into B N . Here, the effective hyperfine field B N is regarded as stationary due to the nuclear fields change at time scale of nuclear spin relaxation that is much longer than any time scale associated with electron transport [5,16,18,19,37–39]. The relaxation in the electrodes is assumed to be sufficiently fast so that their electron distributions can be described by equilibrium Fermi functions. The electrodes are thus modeled as non-interacting Fermi gases and the corresponding Hamiltonians can be expressed as Hleads =
α ks
εαk aĎαks aαks ,
(2)
Ď
where aα ks (aα ks ) creates (annihilates) an electron with energy εα k , spin s and momentum k in α (α = L, R) electrode, and s = + (−) denotes the majority (minority) spin states with the density of states gα,s . The polarization vectors pL (left lead) and pR (right lead) are aligned parallel to each other, and the corresponding magnitude of the spin polarization is characterized by pα = |pα | = gα,+ − gα,− / gα,+ + gα,− . For the sake of simplicity, in the following discussion we set pL = pR = p. The tunneling between the QD and the two ferromagnetic electrodes are described by Ď
Ď
HT = tα k+ aα k+ d↑ + tα k− aα k− d↓ + H .c .,
(3)
where we define spin-up ↑ and spin-down ↓ to be the majority spin and minority spin of the ferromagnet, respectively. The QD-electrode coupling is assumed to be sufficiently weak, so that the sequential tunneling is dominant. The transitions are well described by quantum master equation of a reduced density matrix spanned by the eigenstates of the QD system. Under the second order Born approximation and Markov approximation, the particle-number-resolved quantum master equation for the reduced density matrix is given by [31–33] 1
ρ˙ (n) (t ) = −iLρ (n) (t ) − Rρ (n) (t ) , 2
(4)
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with
Rρ (n) (t ) =
ασ
(+) Ď Ď (−) (n) (n) Ď ρ (n) A(+) − A(−) dσ − dĎσ ρ (n) ALσ ασ dσ + dσ Aασ ρ Lσ ρ
Ď (n+1) (+) (n−1) Ď − A(−) ρ d − d ρ A + H .c ., σ σ Rσ Rσ
(5)
σ (+) + − where A(+) ασ = Γα nα (−L) dσ , nα = fα , nα = 1 − fα (fα is the Fermi function of the electrode α ), σ 2 Γα = 2π gα,s |tα | = (1 + spα )Γα /2 and Γα = Γα↑ + Γα↓ . Liouvillian superoperator L is defined as L (· · · ) = [Hdot , (· · · )]. ρ (n) (t ) is the reduced density matrix of the QD system conditioned by the number of electrons arriving at the right electrode up to time t. In order to calculate the inχ (n) first four cumulants, one can define S (χ , t ) = According to the definition of the (n)n ρ (intχ) e . −F (χ) inχ cumulant generating function e = Tr ρ t e = , we evidently have ( ) n n P (n, t ) e −F (χ) e = Tr [S (χ , t )], where the trace is over the eigenstates of the QD system. Since Eq. (4) has the following form
ρ˙ (n) = Aρ (n) + C ρ (n+1) + Dρ (n−1) ,
(6)
S (χ, t ) satisfies S˙ = AS + e−iχ CS + eiχ DS ≡ Lχ S .
(7)
In the low frequency limit, the counting time, namely, the time of measurement is much longer than the time of tunneling through the QD system. In this case, F (χ) can be written as [34–36,40–42] F (χ) = −λ1 (χ ) t ,
(8)
where λ1 (χ) is the eigenvalue of Lχ which goes to zero for χ → 0. According to the definition of the cumulants one can express λ1 (χ) as
λ1 (χ ) =
∞ Ck (iχ)k . t k! k=1
(9)
The low order cumulants can be calculated by the Rayleigh–Schrödinger perturbation theory in the counting parameter χ , which was developed in Refs. [34–36]. In order to calculate the first four current cumulants we expand Lχ to four order in χ Lχ = L0 + L1 χ +
1 2!
L2 χ 2 +
1 3!
L3 χ 3 +
1 4!
L4 χ 4 + · · · .
(10)
Along the lines of Refs. [34–36], we define the two projectors P = P 2 = |0⟩⟩⟨⟨0˜ | and Q = Q 2 = 1 − P, obeying the relations PL0 = L0 P = 0 and QL0 = L0 Q = L0 . Here, |0⟩⟩ being the steady state ρ stat is the right eigenvector of L0 , namely, L0 |0⟩⟩ = 0, and ⟨⟨0˜ | ≡ 1ˆ is the corresponding left eigenvector. 1 In view of L0 being singular, we also introduce the pseudoinverse according to R = QL− 0 Q , which is well-defined since the inversion being performed only in the subspace spanned by Q . After a careful calculation, λ1 (χ) reads
λ1 (χ) = ⟨⟨0˜ |L1 |0⟩⟩χ 1 + ⟨⟨0˜ |L2 |0⟩⟩ − 2⟨⟨0˜ |L1 RL1 |0⟩⟩ χ 2 2! 1 + ⟨⟨0˜ |L3 |0⟩⟩ − 3⟨⟨0˜ | (L2 RL1 + L1 RL2 ) |0⟩⟩ 3! − 6⟨⟨0˜ |L1 R (RL1 P − L1 R) L1 |0⟩⟩ χ 3 1 + ⟨⟨0˜ |L4 |0⟩⟩ − 6⟨⟨0˜ |L2 RL2 |0⟩⟩ 4! − 4⟨⟨0˜ | (L3 RL1 + L1 RL3 ) |0⟩⟩
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− 12⟨⟨0˜ |L2 R (RL1 P − L1 R) L1 |0⟩⟩ − 12⟨⟨0˜ |L1 R (RL2 P − L2 R) L1 |0⟩⟩ − 12⟨⟨0˜ |L1 R (RL1 P − L1 R) L2 |0⟩⟩ − 24⟨⟨0˜ |L1 R R2 L1 PL1 P − RL1 PL1 R − L1 R2 L1 P − RL1 RL1 P + L1 RL1 R L1 |0⟩⟩ χ 4 + · · · .
(11)
From Eqs. (9) and (11) we can identify the first four current cumulants: C1 /t = ⟨⟨0˜ |L1 |0⟩⟩/i,
(12)
C2 /t = ⟨⟨0˜ |L2 |0⟩⟩ − 2⟨⟨0˜ |L1 RL1 |0⟩⟩ /i2 ,
(13)
C3 /t = ⟨⟨0˜ |L3 |0⟩⟩ − 3⟨⟨0˜ | (L2 RL1 + L1 RL2 ) |0⟩⟩ − 6⟨⟨0˜ |L1 R (RL1 P − L1 R) L1 |0⟩⟩ /i3 . C4 / t =
(14)
⟨⟨0˜ |L4 |0⟩⟩ − 6⟨⟨0˜ |L2 RL2 |0⟩⟩ − 4⟨⟨0˜ | (L3 RL1 + L1 RL3 ) |0⟩⟩ − 12⟨⟨0˜ |L2 R (RL1 P − L1 R) L1 |0⟩⟩ − 12⟨⟨0˜ |L1 R (RL2 P − L2 R) L1 |0⟩⟩ − 12⟨⟨0˜ |L1 R (RL1 P − L1 R) L2 |0⟩⟩ − 24⟨⟨0˜ |L1 R R2 L1 PL1 P − RL1 PL1 R − L1 R2 L1 P − RL1 RL1 P + L1 RL1 R L1 |0⟩⟩ /i4 .
(15)
The four equations above are the starting point of the calculation in the following. Here, the first four cumulants Ck are directly related to the transport characteristics. For example, the first-order cumulant (the peak position of the distribution of transferred-electron number) C1 = n¯ gives the average current ⟨I ⟩ = eC1 /t. The zero-frequency shot noise is relatedto the second-order cumulant (the peak-width of the distribution) S = 2e2 C2 /t = 2e2 n2 − n¯ 2 /t. The third-order cumulant 2
2 C3 = (n − n¯ )3 and four-order cumulant C4 = (n − n¯ )4 − 3(n − n¯ ) characterize, respectively, the skewness and kurtosis of the distribution. Here, (· · · ) = · · P (· ) (n, t ). In general, the shot noise, n skewness and kurtosis are represented by the Fano factor F2 = C2 /C1 , F3 = C3 /C1 and F4 = C4 /C1 , respectively.
3. Numerical results and discussion In the following numerical calculations, we assume the bias voltage (µL = −µR = Vb /2) is symmetrically and entirely dropped at the QD-electrode tunnel junctions, which implies that the levels of the QD are independent of the applied bias voltage, and choose meV as the unit of energy which corresponds to a typical experimental situation [43]. Here, the direction of the polarization vector pL is chosen to be the spin quantization axis (z axis) of the dot, and the corresponding angle of
− →
effective nuclear magnetic field B N is denoted by (θ , ϕ). In the basis {|0, 0⟩ , |↑, 0⟩ , |0, ↓⟩ , |↑, ↓⟩}, the QD Hamiltonian Hdot can be diagonalized, and the corresponding eigenstates and eigenvalues are given by
|Ψ0 ⟩ = |0, 0⟩ θ θ |Ψ1 ⟩+ = cos e−iϕ/2 |↑, 0⟩ + sin eiϕ/2 |0, ↓⟩ 2
2
θ θ |Ψ1 ⟩− = − sin e−iϕ/2 |↑, 0⟩ + cos eiϕ/2 |0, ↓⟩ 2 2 |Ψ2 ⟩ = |↑, ↓⟩
,
(16)
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Fig. 2. (Color online) The first four cumulants of zero-frequency current fluctuation vs the polar angle θ for different spin polarization p with a given effective nuclear-spin magnetic field BN = 0.0004. Here, the off-diagonal elements of the reduced density matrix are considered. The system parameters: ε = 1, U = 4, Vb = 6, kB T = 0.05, Γα↑ = Γα (1 + p) /2, Γα↓ = Γα (1 − p) /2, ΓL = ΓR = 0.004 and Γ = ΓL + ΓR .
ε0 = 0 1 ε1,+ = ε + BN 2
(17)
1 ε1,− = ε − BN 2 ε2 = (2ε + U ) .
Here, the spin-up and spin-down components of the two singly-occupied eigenstates are set to possess an anti-symmetric quantum phase, so that the influence of the off-diagonal elements of the reduced density matrix induced by the effective hyperfine field on the FCS of electron transport is independent of the azimuthal angle of the effective field ϕ . In addition, the magnitude of the effective hyperfine
− →
field B N is much smaller than the on-site energy ε [18,19,44], namely, 10−4 meV (about 5 mT) ≪1
− →
meV. As a result, the influence of the effective field B N on the FCS properties occurs mainly in the bias range where only the transition processes between the singly-occupied and empty eigenstates are involved in the quantum transport. The parameters of the QD are chosen as ε = 1, U = 4, Vb = 6, kB T = 0.05, Γα↑ = Γα (1 + p) /2, Γα↓ = Γα (1 − p) /2, ΓL = ΓR = 0.004 and Γ = ΓL + ΓR .
− →
In order to effectively extract the information of the effective field B N , we first study the effect of the spin polarization p on the FCS in the considered QD system with a given effective field BN = 0.0004, which corresponds to a typical magnitude of the effective field in single QD system [18,19,44]. Here, it is important to emphasize that the effective hyperfine field gives rise to the offdiagonal elements of the reduced density matrix. Thus, Figs. 2 and 3 show the first four current cumulants as a function of the polar angle θ for the cases of that the off-diagonal elements of the reduced density matrix at different values of spin polarization p are considered and not considered, respectively. It is demonstrated that the quantum coherence between the two singly-occupied eigenstates, namely, the off-diagonal elements of the reduced density matrix play an important
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381
Fig. 3. (Color online) The first four cumulants of zero-frequency current fluctuation vs the polar angle θ for different spin polarization p with a given effective nuclear-spin magnetic field BN = 0.0004. Here, the diagonal elements of the reduced density matrix are only considered. The other system parameters are the same as in Fig. 2.
role in determining not only the magnitude but also the occurrence of super-Poissonian noise, as shown in Figs. 2(b) and 3(b). The super-Poissonian noise characteristics can be understood in terms of the effective competition between fast and slow transport channels [27,33,45–49]. Furthermore, it is obvious that the first four current cumulants for a polar angle θ have the same bias-voltagedependence as that for π − θ , which originates from the symmetry of the QD Hamiltonian, so that we restrict our discussion to the case of θ ∈ [0, π /2]. We now analyze the influence of the orientation of the effective hyperfine field on the high-order current cumulants. For a non-zero polar angle θ , the effective nuclear-spin field can change the spin direction of conduction electron tunneling into the QD from the left lead, and the spin of conduction electron finally lies along the direction of the effective field. The relaxation time of the physical process depends on the magnitude of this effective field, and has a significant impact on the transport current noise. If a spin-up conduction electron tunnels into the QD, the eigenstate of the QD is characterized by the singly-occupied eigenstate |Ψ1 ⟩+ due to cos2 (θ /2) > sin2 (θ /2), here θ ∈ [0, π /2] (see Eq. (16)). In this situation, the θ -dependent probability of the singly-occupied eigenstate |Ψ1 ⟩+ collapsing to the electronic state |↑, 0⟩ is larger than that collapsing to the electronic state |0, ↓⟩. In par↑ ↓ ticular, the spin-up tunneling rate out of the QD is larger than the spin-down, namely, ΓR > ΓR , thus, + the transport channel of transferring the spin-up electrons, i.e., |Ψ1 ⟩ → |↑, 0⟩ → |Ψ0 ⟩, can form the fast transport channels. Whereas the spin-down transport channel, i.e., |Ψ1 ⟩+ → |0, ↓⟩ → |Ψ0 ⟩, forms the correlated slow transport channels. As for the spin-down case, the transport channels |Ψ1 ⟩− → |↑, 0⟩ → |Ψ0 ⟩ and |Ψ1 ⟩− → |0, ↓⟩ → |Ψ0 ⟩ form the corresponding fast and slow transport channels, respectively. Consequently, for a large enough spin polarization p, the fast transport channels |Ψ1 ⟩± → |↑, 0⟩ → |Ψ0 ⟩ can be effectively modulated by the corresponding correlated slow transport channels |Ψ1 ⟩± → |0, ↓⟩ → |Ψ0 ⟩, and leading to bunching of conduction electrons, which is responsible for the formation of super-Poissonian noise. Importantly, for a given effective field BN , the values of the high-order current cumulants show significant variations with increasing the polar angle θ (i.e., the orientation) of this effective hyperfine field for a large enough spin polarization p. In
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Fig. 4. (Color online) The first four cumulants of zero-frequency current fluctuation vs the polar angle θ for different effective nuclear-spin magnetic field BN with a given spin polarization p = 0.9. The other system parameters are the same as in Fig. 2.
particular, the corresponding values of the skewness and kurtosis are more sensitive to the variation of polar angle θ than the shot noise, see the short-dashed lines in Fig. 2(b)–(d). Thus, this remarkable
− →
property can be used to qualitatively characterize the orientation of the effective hyperfine field B N relative to the spin quantization z axis.
− →
On the other hand, the magnitude of this effective field B N determines the time for the spin of conduction electron being turned to the polar angle θ . If the magnitude of this effective field is much smaller than the QD-electrode coupling, e.g., BN ≪ ΓL (ΓL = ΓR ), the electron can tunnel out of the QD quickly before the conduction electron spin becomes lying along the direction of the effective field, so that the active competition between the fast and slow channel currents is slightly suppressed, resulting that the current noise’s maximal values have relatively small variation compared to a relatively large value BN , see Fig. 4. Whereas, for the BN ≫ ΓL case, the conduction electron spin can align up quickly before the electron tunneling out the QD. As a result, the active competition between the fast and slow channel currents is dramatically suppressed and even destroyed for the polar angle θ approaching π /2. Thus, the corresponding super-Poissonian values are significantly reduced even to the sub-Poissonian values, see the deep blue region in Fig. 5(a). Whereas for a given polar angle θ , which is not close to θ = 0 or θ = π , the values of the high-order current cumulants
− →
depend on the magnitude of this effective field B N , and the variations of the corresponding values of the skewness and kurtosis are more sensitive to the value BN than the shot noise, see Fig. 4(b)–(d). Consequently, this characteristic can be used to qualitatively determine the magnitude of this effective
− →
field B N . In addition, in the region of the polar angle θ approaching π /2, the values of the high-order current cumulants have a gentle variation with increasing the value BN , see the deep blue and deep red regions in Fig. 5. This feature is not conductive to extract the information of the magnitude of the effective field. In order to clearly extract the information on the magnitude of the effective field in this region, we should set the strength of coupling between the QD system and the electrodes several times larger than the magnitude of the effective field, see the dash-dotted and short-dashed lines in Fig. 4(b)–(d).
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Fig. 5. (Color online) (a) The shot noise, (b) the skewness and (c) the kurtosis as a function of the effective nuclear-spin magnetic field BN and the polar angle θ for a given spin polarization p = 0.9. The other system parameters are the same as in Fig. 2.
4. Conclusions We have numerically demonstrated that the orientation and magnitude of the effective nuclearspin magnetic field have a significant influence on the variations of the values of high-order cumulants, when only the transitions between the singly-occupied and empty eigenstates are involved in the electron transport. This effect can be qualitatively attributed to the effective competition between fast and slow transport channels. Of particular interest is that the variations of the skewness and kurtosis values are more sensitive to the orientation and magnitude of the effective field than the shot noise. Consequently, these remarkable properties provide a new way of extracting the orientation and magnitude information of the effective field via the current high-order cumulants. Acknowledgments This work was supported by the National Key Basic Research Program of China (Grant No. 2015CB921600), the National Natural Science Foundation of China (Grant Nos. 11204203, 11274208 and 11274222), QiMingXing Project (14QA1402000) from Shanghai Municipal Science and Technology Commission, Eastern Scholar Program and ShuGuang Program (No. 12SG34) from Shanghai Municipal Education Commission. References [1] D. Loss, D.P. DiVincenzo, Phys. Rev. A 57 (1998) 120. [2] G. Burkard, D. Loss, D.P. DiVincenzo, Phys. Rev. B 59 (1999) 2070.
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Probing the effective nuclear-spin magnetic field in a single quantum dot via full counting statistics Hai-Bin Xue a,∗ , Yi-Hang Nie b , Jingzhe Chen c , Wei Ren c,∗ a
College of Physics and Optoelectronics, Taiyuan University of Technology, Taiyuan 030024, China
b
Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China
c
International Centre for Quantum and Molecular Structures and Department of Physics, Shanghai University, 99 Shangda Road, Shanghai 200444, China
highlights • The effective nuclear-spin magnetic field gives rise to the off-diagonal elements of the reduced density matrix of single QD.
• The off-diagonal elements of reduced density matrix of the QD have a significant impact on the high-order current cumulants.
• The high-order current cumulants are sensitive to the orientation and magnitude of the effective nuclear-spin magnetic field.
• The FCS can be used to detect the orientation and magnitude of the effective nuclear-spin magnetic field in a single QD.
article
info
Article history: Received 14 November 2014 Accepted 5 January 2015 Available online 12 January 2015 Keywords: Single quantum dot Nuclear-spin magnetic field Full counting statistics
∗
abstract We study theoretically the full counting statistics of electron transport through a quantum dot weakly coupled to two ferromagnetic leads, in which an effective nuclear-spin magnetic field originating from the configuration of nuclear spins is considered. We demonstrate that the quantum coherence between the two singly-occupied eigenstates and the spin polarization of two ferromagnetic leads play an important role in the formation of superPoissonian noise. In particular, the orientation and magnitude of the effective field have a significant influence on the variations of the values of high-order cumulants, and the variations of the
Corresponding authors. Tel.: +86 351 6018030; fax: +86 351 6018030. E-mail addresses: [email protected] (H.-B. Xue), [email protected] (W. Ren).
http://dx.doi.org/10.1016/j.aop.2015.01.001 0003-4916/© 2015 Elsevier Inc. All rights reserved.
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skewness and kurtosis values are more sensitive to the orientation and magnitude of the effective field than the shot noise. Thus, the high-order cumulants of transport current can be used to qualitatively extract information on the orientation and magnitude of the effective nuclear-spin magnetic field in a single quantum dot. © 2015 Elsevier Inc. All rights reserved.
1. Introduction The controlled manipulation of the localized electron spins in semiconductor quantum dots (QDs) is critical for quantum information processing with confined electron spins [1–4]. Recently, the hyperfine interaction between localized electron and nuclear spins in the QDs has attracted much attention since it is the dominant mechanism of electron spin relaxation in QDs at low temperature [5–15], whereas the electron spin relaxation mechanism can be suppressed by an external magnetic field [16]. Due to the random configuration of nuclear spins, the quantum transport properties of electrons through the QD system can be strongly affected by the orientation and magnitude of the
− →
effective nuclear magnetic field B N . For example, the effective hyperfine field leads to the mixing between two-electron singlet and triplet states [17–21], while a small magnetic field suppresses such mixing [18]; and lifts the spin blockade in a double QDs. In fact, the quantum transport of electrons through the QD system is essentially a quantum stochastic process with the quantum fluctuations of transport current. It has been theoretically and experimentally demonstrated that the full counting statistics (FCS) [22,23] properties can allow one to identify the intrinsic properties of the QD system and access the information of electron correlation beyond what is obtainable from the differential conductance or the average current [24,25]. For instance, the high-order current cumulants [26] can be used to detect the quantum coherence in a side-coupled double QDs system [27] and the positions of the zeros of the generating function [28], reveal the intrinsic multistability [29], and extract the fractional charge of charge transfer through an impurity in a chiral Luttinger liquid [30]. Consequently, the FCS of electron transport through the QD system can not only provide a deeper insight into the nature of quantum transport mechanism, but also propose a new way to obtain the information on the orientation and magnitude of effective nuclear-spin magnetic field.
− →
The aim of this work is thus to study the influence of the effective nuclear-spin magnetic field B N on the first four cumulants of electron transport through a single QD system coupled weakly to two ferromagnetic leads, and then analyze the feasibility of extracting the information of the orientation
− →
and magnitude of the effective hyperfine field B N from the high-order cumulants of transport current. This paper is organized as follows. In Section 2, we introduce the considered QD system and outline the procedure to obtain the FCS formalism based on an effective particle-number-resolved quantum master equation [31–33] and the Rayleigh–Schrödinger perturbation theory developed in Refs. [34–36]. The numerical results are discussed in Section 3, where we discuss the effects of the orientation and magnitude of the effective nuclear-spin magnetic field on the first four current cumulants, and analyze the feasibility of extracting information about the effective nuclear-spin magnetic field from the high-order current cumulants. Finally, in Section 4 we summarize the work. 2. Model and formalism The system considered here consists of a single QD possessing an effective nuclear-spin magnetic field weakly coupled to two ferromagnetic leads, see Fig. 1. The system Hamiltonian is described by Htotal = Hdot + Hleads + HT , in which the QD Hamiltonian reads Hdot =
σ
− → → ε dĎσ dσ + UdĎ↑ d↑ dĎ↓ d↓ + − s · BN ,
(1)
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377
Fig. 1. (Color online) Schematic representation of the single QD system possessing an effective nuclear-spin magnetic field weakly coupled to two ferromagnetic leads.
where dĎσ (dσ ) creates (annihilates) an electron with spin σ and energy ε (which can be tuned by a gate voltage Vg ) in the QD. U is the intradot Coulomb interaction between two electrons in the QD system. The third term describes the conduction electron’s Zeeman splitting, where g µB has been
− →
− →
absorbed into B N . Here, the effective hyperfine field B N is regarded as stationary due to the nuclear fields change at time scale of nuclear spin relaxation that is much longer than any time scale associated with electron transport [5,16,18,19,37–39]. The relaxation in the electrodes is assumed to be sufficiently fast so that their electron distributions can be described by equilibrium Fermi functions. The electrodes are thus modeled as non-interacting Fermi gases and the corresponding Hamiltonians can be expressed as Hleads =
α ks
εαk aĎαks aαks ,
(2)
Ď
where aα ks (aα ks ) creates (annihilates) an electron with energy εα k , spin s and momentum k in α (α = L, R) electrode, and s = + (−) denotes the majority (minority) spin states with the density of states gα,s . The polarization vectors pL (left lead) and pR (right lead) are aligned parallel to each other, and the corresponding magnitude of the spin polarization is characterized by pα = |pα | = gα,+ − gα,− / gα,+ + gα,− . For the sake of simplicity, in the following discussion we set pL = pR = p. The tunneling between the QD and the two ferromagnetic electrodes are described by Ď
Ď
HT = tα k+ aα k+ d↑ + tα k− aα k− d↓ + H .c .,
(3)
where we define spin-up ↑ and spin-down ↓ to be the majority spin and minority spin of the ferromagnet, respectively. The QD-electrode coupling is assumed to be sufficiently weak, so that the sequential tunneling is dominant. The transitions are well described by quantum master equation of a reduced density matrix spanned by the eigenstates of the QD system. Under the second order Born approximation and Markov approximation, the particle-number-resolved quantum master equation for the reduced density matrix is given by [31–33] 1
ρ˙ (n) (t ) = −iLρ (n) (t ) − Rρ (n) (t ) , 2
(4)
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with
Rρ (n) (t ) =
ασ
(+) Ď Ď (−) (n) (n) Ď ρ (n) A(+) − A(−) dσ − dĎσ ρ (n) ALσ ασ dσ + dσ Aασ ρ Lσ ρ
Ď (n+1) (+) (n−1) Ď − A(−) ρ d − d ρ A + H .c ., σ σ Rσ Rσ
(5)
σ (+) + − where A(+) ασ = Γα nα (−L) dσ , nα = fα , nα = 1 − fα (fα is the Fermi function of the electrode α ), σ 2 Γα = 2π gα,s |tα | = (1 + spα )Γα /2 and Γα = Γα↑ + Γα↓ . Liouvillian superoperator L is defined as L (· · · ) = [Hdot , (· · · )]. ρ (n) (t ) is the reduced density matrix of the QD system conditioned by the number of electrons arriving at the right electrode up to time t. In order to calculate the inχ (n) first four cumulants, one can define S (χ , t ) = According to the definition of the (n)n ρ (intχ) e . −F (χ) inχ cumulant generating function e = Tr ρ t e = , we evidently have ( ) n n P (n, t ) e −F (χ) e = Tr [S (χ , t )], where the trace is over the eigenstates of the QD system. Since Eq. (4) has the following form
ρ˙ (n) = Aρ (n) + C ρ (n+1) + Dρ (n−1) ,
(6)
S (χ, t ) satisfies S˙ = AS + e−iχ CS + eiχ DS ≡ Lχ S .
(7)
In the low frequency limit, the counting time, namely, the time of measurement is much longer than the time of tunneling through the QD system. In this case, F (χ) can be written as [34–36,40–42] F (χ) = −λ1 (χ ) t ,
(8)
where λ1 (χ) is the eigenvalue of Lχ which goes to zero for χ → 0. According to the definition of the cumulants one can express λ1 (χ) as
λ1 (χ ) =
∞ Ck (iχ)k . t k! k=1
(9)
The low order cumulants can be calculated by the Rayleigh–Schrödinger perturbation theory in the counting parameter χ , which was developed in Refs. [34–36]. In order to calculate the first four current cumulants we expand Lχ to four order in χ Lχ = L0 + L1 χ +
1 2!
L2 χ 2 +
1 3!
L3 χ 3 +
1 4!
L4 χ 4 + · · · .
(10)
Along the lines of Refs. [34–36], we define the two projectors P = P 2 = |0⟩⟩⟨⟨0˜ | and Q = Q 2 = 1 − P, obeying the relations PL0 = L0 P = 0 and QL0 = L0 Q = L0 . Here, |0⟩⟩ being the steady state ρ stat is the right eigenvector of L0 , namely, L0 |0⟩⟩ = 0, and ⟨⟨0˜ | ≡ 1ˆ is the corresponding left eigenvector. 1 In view of L0 being singular, we also introduce the pseudoinverse according to R = QL− 0 Q , which is well-defined since the inversion being performed only in the subspace spanned by Q . After a careful calculation, λ1 (χ) reads
λ1 (χ) = ⟨⟨0˜ |L1 |0⟩⟩χ 1 + ⟨⟨0˜ |L2 |0⟩⟩ − 2⟨⟨0˜ |L1 RL1 |0⟩⟩ χ 2 2! 1 + ⟨⟨0˜ |L3 |0⟩⟩ − 3⟨⟨0˜ | (L2 RL1 + L1 RL2 ) |0⟩⟩ 3! − 6⟨⟨0˜ |L1 R (RL1 P − L1 R) L1 |0⟩⟩ χ 3 1 + ⟨⟨0˜ |L4 |0⟩⟩ − 6⟨⟨0˜ |L2 RL2 |0⟩⟩ 4! − 4⟨⟨0˜ | (L3 RL1 + L1 RL3 ) |0⟩⟩
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379
− 12⟨⟨0˜ |L2 R (RL1 P − L1 R) L1 |0⟩⟩ − 12⟨⟨0˜ |L1 R (RL2 P − L2 R) L1 |0⟩⟩ − 12⟨⟨0˜ |L1 R (RL1 P − L1 R) L2 |0⟩⟩ − 24⟨⟨0˜ |L1 R R2 L1 PL1 P − RL1 PL1 R − L1 R2 L1 P − RL1 RL1 P + L1 RL1 R L1 |0⟩⟩ χ 4 + · · · .
(11)
From Eqs. (9) and (11) we can identify the first four current cumulants: C1 /t = ⟨⟨0˜ |L1 |0⟩⟩/i,
(12)
C2 /t = ⟨⟨0˜ |L2 |0⟩⟩ − 2⟨⟨0˜ |L1 RL1 |0⟩⟩ /i2 ,
(13)
C3 /t = ⟨⟨0˜ |L3 |0⟩⟩ − 3⟨⟨0˜ | (L2 RL1 + L1 RL2 ) |0⟩⟩ − 6⟨⟨0˜ |L1 R (RL1 P − L1 R) L1 |0⟩⟩ /i3 . C4 / t =
(14)
⟨⟨0˜ |L4 |0⟩⟩ − 6⟨⟨0˜ |L2 RL2 |0⟩⟩ − 4⟨⟨0˜ | (L3 RL1 + L1 RL3 ) |0⟩⟩ − 12⟨⟨0˜ |L2 R (RL1 P − L1 R) L1 |0⟩⟩ − 12⟨⟨0˜ |L1 R (RL2 P − L2 R) L1 |0⟩⟩ − 12⟨⟨0˜ |L1 R (RL1 P − L1 R) L2 |0⟩⟩ − 24⟨⟨0˜ |L1 R R2 L1 PL1 P − RL1 PL1 R − L1 R2 L1 P − RL1 RL1 P + L1 RL1 R L1 |0⟩⟩ /i4 .
(15)
The four equations above are the starting point of the calculation in the following. Here, the first four cumulants Ck are directly related to the transport characteristics. For example, the first-order cumulant (the peak position of the distribution of transferred-electron number) C1 = n¯ gives the average current ⟨I ⟩ = eC1 /t. The zero-frequency shot noise is relatedto the second-order cumulant (the peak-width of the distribution) S = 2e2 C2 /t = 2e2 n2 − n¯ 2 /t. The third-order cumulant 2
2 C3 = (n − n¯ )3 and four-order cumulant C4 = (n − n¯ )4 − 3(n − n¯ ) characterize, respectively, the skewness and kurtosis of the distribution. Here, (· · · ) = · · P (· ) (n, t ). In general, the shot noise, n skewness and kurtosis are represented by the Fano factor F2 = C2 /C1 , F3 = C3 /C1 and F4 = C4 /C1 , respectively.
3. Numerical results and discussion In the following numerical calculations, we assume the bias voltage (µL = −µR = Vb /2) is symmetrically and entirely dropped at the QD-electrode tunnel junctions, which implies that the levels of the QD are independent of the applied bias voltage, and choose meV as the unit of energy which corresponds to a typical experimental situation [43]. Here, the direction of the polarization vector pL is chosen to be the spin quantization axis (z axis) of the dot, and the corresponding angle of
− →
effective nuclear magnetic field B N is denoted by (θ , ϕ). In the basis {|0, 0⟩ , |↑, 0⟩ , |0, ↓⟩ , |↑, ↓⟩}, the QD Hamiltonian Hdot can be diagonalized, and the corresponding eigenstates and eigenvalues are given by
|Ψ0 ⟩ = |0, 0⟩ θ θ |Ψ1 ⟩+ = cos e−iϕ/2 |↑, 0⟩ + sin eiϕ/2 |0, ↓⟩ 2
2
θ θ |Ψ1 ⟩− = − sin e−iϕ/2 |↑, 0⟩ + cos eiϕ/2 |0, ↓⟩ 2 2 |Ψ2 ⟩ = |↑, ↓⟩
,
(16)
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Fig. 2. (Color online) The first four cumulants of zero-frequency current fluctuation vs the polar angle θ for different spin polarization p with a given effective nuclear-spin magnetic field BN = 0.0004. Here, the off-diagonal elements of the reduced density matrix are considered. The system parameters: ε = 1, U = 4, Vb = 6, kB T = 0.05, Γα↑ = Γα (1 + p) /2, Γα↓ = Γα (1 − p) /2, ΓL = ΓR = 0.004 and Γ = ΓL + ΓR .
ε0 = 0 1 ε1,+ = ε + BN 2
(17)
1 ε1,− = ε − BN 2 ε2 = (2ε + U ) .
Here, the spin-up and spin-down components of the two singly-occupied eigenstates are set to possess an anti-symmetric quantum phase, so that the influence of the off-diagonal elements of the reduced density matrix induced by the effective hyperfine field on the FCS of electron transport is independent of the azimuthal angle of the effective field ϕ . In addition, the magnitude of the effective hyperfine
− →
field B N is much smaller than the on-site energy ε [18,19,44], namely, 10−4 meV (about 5 mT) ≪1
− →
meV. As a result, the influence of the effective field B N on the FCS properties occurs mainly in the bias range where only the transition processes between the singly-occupied and empty eigenstates are involved in the quantum transport. The parameters of the QD are chosen as ε = 1, U = 4, Vb = 6, kB T = 0.05, Γα↑ = Γα (1 + p) /2, Γα↓ = Γα (1 − p) /2, ΓL = ΓR = 0.004 and Γ = ΓL + ΓR .
− →
In order to effectively extract the information of the effective field B N , we first study the effect of the spin polarization p on the FCS in the considered QD system with a given effective field BN = 0.0004, which corresponds to a typical magnitude of the effective field in single QD system [18,19,44]. Here, it is important to emphasize that the effective hyperfine field gives rise to the offdiagonal elements of the reduced density matrix. Thus, Figs. 2 and 3 show the first four current cumulants as a function of the polar angle θ for the cases of that the off-diagonal elements of the reduced density matrix at different values of spin polarization p are considered and not considered, respectively. It is demonstrated that the quantum coherence between the two singly-occupied eigenstates, namely, the off-diagonal elements of the reduced density matrix play an important
H.-B. Xue et al. / Annals of Physics 354 (2015) 375–384
381
Fig. 3. (Color online) The first four cumulants of zero-frequency current fluctuation vs the polar angle θ for different spin polarization p with a given effective nuclear-spin magnetic field BN = 0.0004. Here, the diagonal elements of the reduced density matrix are only considered. The other system parameters are the same as in Fig. 2.
role in determining not only the magnitude but also the occurrence of super-Poissonian noise, as shown in Figs. 2(b) and 3(b). The super-Poissonian noise characteristics can be understood in terms of the effective competition between fast and slow transport channels [27,33,45–49]. Furthermore, it is obvious that the first four current cumulants for a polar angle θ have the same bias-voltagedependence as that for π − θ , which originates from the symmetry of the QD Hamiltonian, so that we restrict our discussion to the case of θ ∈ [0, π /2]. We now analyze the influence of the orientation of the effective hyperfine field on the high-order current cumulants. For a non-zero polar angle θ , the effective nuclear-spin field can change the spin direction of conduction electron tunneling into the QD from the left lead, and the spin of conduction electron finally lies along the direction of the effective field. The relaxation time of the physical process depends on the magnitude of this effective field, and has a significant impact on the transport current noise. If a spin-up conduction electron tunnels into the QD, the eigenstate of the QD is characterized by the singly-occupied eigenstate |Ψ1 ⟩+ due to cos2 (θ /2) > sin2 (θ /2), here θ ∈ [0, π /2] (see Eq. (16)). In this situation, the θ -dependent probability of the singly-occupied eigenstate |Ψ1 ⟩+ collapsing to the electronic state |↑, 0⟩ is larger than that collapsing to the electronic state |0, ↓⟩. In par↑ ↓ ticular, the spin-up tunneling rate out of the QD is larger than the spin-down, namely, ΓR > ΓR , thus, + the transport channel of transferring the spin-up electrons, i.e., |Ψ1 ⟩ → |↑, 0⟩ → |Ψ0 ⟩, can form the fast transport channels. Whereas the spin-down transport channel, i.e., |Ψ1 ⟩+ → |0, ↓⟩ → |Ψ0 ⟩, forms the correlated slow transport channels. As for the spin-down case, the transport channels |Ψ1 ⟩− → |↑, 0⟩ → |Ψ0 ⟩ and |Ψ1 ⟩− → |0, ↓⟩ → |Ψ0 ⟩ form the corresponding fast and slow transport channels, respectively. Consequently, for a large enough spin polarization p, the fast transport channels |Ψ1 ⟩± → |↑, 0⟩ → |Ψ0 ⟩ can be effectively modulated by the corresponding correlated slow transport channels |Ψ1 ⟩± → |0, ↓⟩ → |Ψ0 ⟩, and leading to bunching of conduction electrons, which is responsible for the formation of super-Poissonian noise. Importantly, for a given effective field BN , the values of the high-order current cumulants show significant variations with increasing the polar angle θ (i.e., the orientation) of this effective hyperfine field for a large enough spin polarization p. In
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Fig. 4. (Color online) The first four cumulants of zero-frequency current fluctuation vs the polar angle θ for different effective nuclear-spin magnetic field BN with a given spin polarization p = 0.9. The other system parameters are the same as in Fig. 2.
particular, the corresponding values of the skewness and kurtosis are more sensitive to the variation of polar angle θ than the shot noise, see the short-dashed lines in Fig. 2(b)–(d). Thus, this remarkable
− →
property can be used to qualitatively characterize the orientation of the effective hyperfine field B N relative to the spin quantization z axis.
− →
On the other hand, the magnitude of this effective field B N determines the time for the spin of conduction electron being turned to the polar angle θ . If the magnitude of this effective field is much smaller than the QD-electrode coupling, e.g., BN ≪ ΓL (ΓL = ΓR ), the electron can tunnel out of the QD quickly before the conduction electron spin becomes lying along the direction of the effective field, so that the active competition between the fast and slow channel currents is slightly suppressed, resulting that the current noise’s maximal values have relatively small variation compared to a relatively large value BN , see Fig. 4. Whereas, for the BN ≫ ΓL case, the conduction electron spin can align up quickly before the electron tunneling out the QD. As a result, the active competition between the fast and slow channel currents is dramatically suppressed and even destroyed for the polar angle θ approaching π /2. Thus, the corresponding super-Poissonian values are significantly reduced even to the sub-Poissonian values, see the deep blue region in Fig. 5(a). Whereas for a given polar angle θ , which is not close to θ = 0 or θ = π , the values of the high-order current cumulants
− →
depend on the magnitude of this effective field B N , and the variations of the corresponding values of the skewness and kurtosis are more sensitive to the value BN than the shot noise, see Fig. 4(b)–(d). Consequently, this characteristic can be used to qualitatively determine the magnitude of this effective
− →
field B N . In addition, in the region of the polar angle θ approaching π /2, the values of the high-order current cumulants have a gentle variation with increasing the value BN , see the deep blue and deep red regions in Fig. 5. This feature is not conductive to extract the information of the magnitude of the effective field. In order to clearly extract the information on the magnitude of the effective field in this region, we should set the strength of coupling between the QD system and the electrodes several times larger than the magnitude of the effective field, see the dash-dotted and short-dashed lines in Fig. 4(b)–(d).
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Fig. 5. (Color online) (a) The shot noise, (b) the skewness and (c) the kurtosis as a function of the effective nuclear-spin magnetic field BN and the polar angle θ for a given spin polarization p = 0.9. The other system parameters are the same as in Fig. 2.
4. Conclusions We have numerically demonstrated that the orientation and magnitude of the effective nuclearspin magnetic field have a significant influence on the variations of the values of high-order cumulants, when only the transitions between the singly-occupied and empty eigenstates are involved in the electron transport. This effect can be qualitatively attributed to the effective competition between fast and slow transport channels. Of particular interest is that the variations of the skewness and kurtosis values are more sensitive to the orientation and magnitude of the effective field than the shot noise. Consequently, these remarkable properties provide a new way of extracting the orientation and magnitude information of the effective field via the current high-order cumulants. Acknowledgments This work was supported by the National Key Basic Research Program of China (Grant No. 2015CB921600), the National Natural Science Foundation of China (Grant Nos. 11204203, 11274208 and 11274222), QiMingXing Project (14QA1402000) from Shanghai Municipal Science and Technology Commission, Eastern Scholar Program and ShuGuang Program (No. 12SG34) from Shanghai Municipal Education Commission. References [1] D. Loss, D.P. DiVincenzo, Phys. Rev. A 57 (1998) 120. [2] G. Burkard, D. Loss, D.P. DiVincenzo, Phys. Rev. B 59 (1999) 2070.
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