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Shot noise in the hybrid systems with a quantum dot coupled to normal and superconducting leads
1 July 2002
Physics Letters A 299 (2002) 262–270 www.elsevier.com/locate/pla
Shot noise in the hybrid systems with a quantum dot coupled to normal and superconducting leads Hong-Kang Zhao a,b a Department of Physics, Beijing Institute of Technology, Beijing 100081, PR China 1 b School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, USA
Received 12 May 2002; received in revised form 12 May 2002; accepted 21 May 2002 Communicated by R.W. Wu
Abstract We have investigated the spectral density of shot noise in the tunneling systems composed of a quantum dot coupled with normal and superconducting leads. The spectral density of shot noise is obtained by introducing the Green’s functions of quantum dot. The zero-frequency shot noise in the same normal lead is given simply as the form of transmission coefficient associated with two leads. We have performed the numerical calculation on a two-terminal system at zero-temperature, and novel resonant structure due to Andreev reflection is observed. The sub-Poissonian shot noise is obtained by varying the parameters, which denotes that the shot noise can be suppressed by the external parameters. 2002 Published by Elsevier Science B.V.
The tunneling current through a conductor may fluctuate in time even if it is induced by a stationary voltage. The noise is usually characterized by its power spectrum at frequency ω, which is defined as the Fourier transform of the current correlation function [1]. The shot noise is a nonequilibrium fluctuation which is caused by the discreteness of the charge carriers [2]. When the size of an electronic system reaches the nanometer scale, noise becomes a very interesting problem. The investigation on the deviations from purely Poissonian shot noise in mesoscopic systems has been an increasingly interest subject. From the investigation of shot noise, we can learn additional information on electronic structure and transport properties, since it is directly related to the degree of randomness in carrier transfer. Due to the Poissonian distribution in a macroscopic system, the current shows the value of shot noise by the well-known Schottky formula [3] SP = 2eI . However, for a mesoscopic system, the electrons are correlated due to coherent transport, and they are governed by the Fermi distribution and Pauli principle. This quantum behaviour results in the deviation of shot noise to the Poissonian one. The suppression [4] and enhancement [5] of shot noise have been studied actively to classify what is the cause of the deviation of Poissonian form. Negative correlations between current pulses can lead to a complete suppression of the shot noise in quantum point contacts. The system of normal metal layer coupled with superconducting leads has been investigated extensively during the past decades [6]. The shot noise caused by Andreev reflection in such systems is certainly an interesting
E-mail address: [email protected] (H.-K. Zhao). 1 Permanent address.
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problem, for instance, the shot noise in a voltage biased superconducting quantum point contact was investigated in Ref. [7]. The shot noise in aluminum atomic point contacts containing a small number of conduction channels is measured, and the noise suppression is observed in the normal state [8]. Recently, it becomes of great interest to study the systems consisting of superconducting leads coupled with a quantum dot. Investigations involving such systems are superconductor-quantum-dot-superconductor (SDS), superconductor-quantum-dot-normal-metal lead (SDN), or superconducting quantum dot connected with normal leads [9–13]. In this Letter, we employ the nonequilibrium Green’s function (NGF) technique to study the shot noise spectra in the quantum dot coupled with normal metallic and superconducting leads. We study the situation that the quantum dot is subject to a magnetic field to induce Zeeman split. The Andreev reflection is included in the calculation, and the Andreev-reflected terms cause novel shot noise compared with the normal system. Sub-Poissonian shot noise is achieved to show novel structures, and Zeeman effect results in novel resonant shot noise structure due to the Zeeman splitting. Generally, the system is composed of N normal and superconducting leads coupled to a quantum dot. We consider the situation that the leads γ and β are biased by the dc voltage Vγ β which is the drop of chemical potential between the two leads µγ − µβ = eVγ β . There is a magnetic field B applying to the quantum dot through gate. The Hamiltonians of the superconducting leads are approximated by the mean-field BCS theory. The Hamiltonian of the system can be expressed as the sum of separate sub-Hamiltonians and the interaction term as H= ∆γ aγ† ,k↑aγ† ,−k↓ + ∆∗γ aγ ,−k↓aγ ,k↑ γ k aγ† ,kσ aγ ,kσ − γ kσ
+
σ
γk † εd,σ dσ dσ
+
T˜γ k (t)aγ† ,kσ dσ + h.c. .
(1)
γ kσ
† In the Hamiltonian (1), aγ† ,kσ (aγ ,kσ ), and dσ (dσ ) are the creation (annihilation) operators of electrons in the N leads and quantum dot. ∆γ is the energy gap of the superconducting lead. The energy gap is a complex quantity which is characterized by the phase φγ , ∆γ = |∆γ |eiφγ . We have neglected the Coulomb interaction in the Hamiltonian for simplicity. This approximation is valid in the case as the quantum dot is not extremely small. The interaction strengths is gauged by T˜γ ,k (t) = Tγ k exp(− hi¯ µγ 1 t), where µγ 1 = µγ − µ1 , and Tγ k are real constants satisfying the relation Tγ k = Tγ −k . For definiteness, we take the chemical potential µ1 of lead 1 as the energy reference point. In this way, physical quantities are not affected as the chemical potentials of the whole system are shifted the same value. µ = gµB /2 is the magnetic moment of electron, µB the Bohr magneton, γ k = ¯γ k − µ1 , εd,σ = E˜ d,σ − σ µB, E˜ d,σ = Ed, − eVg and Vg is the gate voltage which adjusts the energy of quantum dot. The magnetic field is squeezed in order not to affect the superconductors and normal leads. The tunneling current operator of the system can be derived by using the Heisenberg equation and continuity † equation, which gives Iˆγ (t) = − ie kσ [H, aγ ,kσ (t)aγ ,kσ (t)]. We make the Bogoliubov transformation to h¯ diagonalize the superconducting leads. The Hamiltonian of γ th superconducting lead is expressed as the diagonal form Hγ = kσ ξγ k αγ† ,kσ αγ ,kσ , where αγ† ,kσ and αγ ,kσ are the creation and annihilation operators of quasi-particle in the superconducting lead, and they satisfy the Fermi distribution. ξγ k = (γ2 k + |∆γ |2 )1/2 is the excitation energy of quasiparticle. For the γ th normal lead, we have ∆γ = 0, and ξγ k = γ k . The shot noise is caused by the nonequilibrium charged particle transport, and it is derived through the Fourier transformation of the current fluctuation correlation Πγ γ (t, t ) = δ Iˆγ (t)δ Iˆγ (t ) + δ Iˆγ (t )δ Iˆγ (t), where δ Iˆγ (t) = Iˆγ (t) − Iˆγ (t). The current correlation Iˆγ (t)Iˆγ (t ) contains products of four creation and annihilation operators of particle in quantum dot and leads. The expectation values of pair productor with one annihilation operator and one creation operator are associated with the normal Green’s functions. Employing the equations of motion (EOM) for the electron and quasiparticle operator, we can express the operators of γ th lead by the Green’s function of lead and the operator of electron in quantum dot. We consider the wide-band limit situation, and in this limit the linewidth is energy-independent, i.e., Γγ (E) = Γγ . The density of state (DOS) in a lead is given by Nγ () = ||/( 2 − |∆γ |2 )1/2 . We define the retarded Green’s
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function matrix of quantum dot as Grσ (t, t ) which satisfies the Dyson equation in the Nambu representation [12]. For the system of a quantum dot coupled with superconducting and normal leads, this equation provides the information originated from the scattering procedure. The retarded Green’s functions are obtained by solving the Fourier transformed versions of Dyson equation self-consistently. We present the results of the Green’s functions as follows for deriving the spectral density of the current noise of our system clearly. We define the (±)r (±) (±)r ¯r ()], quantity Q(±)r γ σ,αβ () to write Green’s functions conveniently as Qγ σ,αβ () = κσ ()Σγ σ,αβ ()/[1 − ζσ (±) (+)r (+) (−)r (−) r r −1 where κσ () = [ ± E˜ d, + σ µB] and ζσ () = κσ ()Σ¯ σ,22 (), ζσ () = κσ ()Σ¯ σ,11 (). In the definition above, Σ¯ r () is associated with the Fourier transformed retarded self-energy matrix given by γ σ,αβ
Wγ ( + µγ 1 )eiφγ (2) , Nγ ( − µγ 1 ) r () = Σ ¯r where Wγ () = |∆γ |/( 2 − |∆γ |2 )1/2 , and Σ¯ σ,jj γ γ σ,jj (). In the Andreev reflection regime where || < |∆γ | = 0, Nγ () and Wγ () are complex. This means that the tunneling current can penetrate through the superconducting barrier even if the source-drain bias is smaller than the energy gap. From the Fourier transformed version of Dyson equation, we find the normal retarded Green’s function of the quantum dot Grσ,11 () of our system self-consistently as ¯ rγ σ () = − i Γγ Σ 2
Nγ ( + µγ 1 ) Wγ ( − µγ 1 )e−iφγ
(−)
Grσ,11 () =
(−)
1 − κσ ()
κσ () () + Σ¯ r
¯r β∈{M} [Σβσ,11
(+)r βσ,12 ()Qβσ,21 (
+ 2µβ1 )]
.
(3)
In the above equation β sums over all superconductors and normal leads. This Green’s function is valid for the system composed of at least one normal metal lead. The energy of tunneling particle shifts a value as → ± 2µβ1 due to the Andreev reflection. This procedure affects the Green’s function of the quantum dot. The advanced Green’s functions are derived by the similar way. In fact, for the steady system, an advanced Green’s function is obtained by taking conjugation over corresponding retarded Green’s function. Substituting the corresponding operators into the Bogoliubov transformed version of current operator, we arrive at the current operator in the γ th normal metal lead i e γ † ()αβ σ , d d Aββ ,σ ()e− h¯ (− )t αβσ Iˆγ (t) = (4) h ββ σ
where
2 γ Aββ ,σ () = i Γ˜γ ()δβγ Grσ,11 () − i Γ˜β ()δγ β Gaσ,11 () − Γ˜γ ()Γ˜β () Grσ,11 () , (+)
(+)
ηβσ () =
[κσ ( + 2µβ1 )Γβ Wβ ( + µβ1 )]2 4|1 − ζσ(+) ( + 2µβ1 )|2
,
(+) ()]. Therefore, the tunneling current operator is determined by the and Γ˜β () = δ Γδ Re Nδ ( + µδ1 )[δβ,δ − ηβσ feature of equilibrium quasi-particles deep in the leads. It also contains resonant structure and Andreev reflection due to the connected quantum dot. The current operator formula appears the similar form as Büttiker formula given by scattering theory in Ref. [2] for the normal scattering contact sample system. The tunneling current is given by taking quantum average and ensemble expectation over the current operator Iγ = Iˆγ (t) by employing † the relation αβσ ()αβ σ ( ) = δββ δ( − )fβ (), where the Fermi–Dirac distribution function is defined by fβ () = 1/[exp(( − µβ1 )/kB T ) + 1]. Therefore, the tunneling current in the γ th lead is determined by the Landauer–Büttiker-like formula e d Tγ β,σ () fγ () − fβ () , Iγ = (5) h σ β=γ
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where Tαβ,σ () = Γ˜α ()Γ˜β ()|Grσ,11 ()|2 represents the transmission coefficient of electron tunneling from the αth lead to βth lead. For our system, there exists Tαβ,σ () = Tβα,σ (), which signifies that the probability of electron tunneling from the αth lead to βth lead is equal to the probability of electron tunneling from the βth lead to αth lead. The spectral density of shot noise Sγ γ (Ω) is determined by the quantum statistical expectation value of the 1 Fourier transformed current operator correlation through the relation [2] 2π Πγ γ (Ω, Ω ) = Sγ γ (Ω)δ(Ω + Ω ), where Πγ γ (Ω, Ω ) is the Fourier transformed version of Πγ γ (t, t ). Performing the Fourier transformation and employing the formula for the expectation value of four quasiparticles, finally we obtain the shot noise spectral density associated with the tunneling currents in the normal metal leads γ and γ e2 γ γ Sγ γ (Ω) = d Aβδ,σ ()Aδβ,σ ( − h¯ Ω)Fβδ (, − hΩ), ¯ h
(6)
βδσ
where Fβδ (, ) = fβ ()[1 − fδ ( )] + fδ ( )[1 − fβ ()]. This is a general formula for describing the frequencydependent current–current correlation fluctuation of multi-terminal-quantum-dot system, from which we can derive the spectral density of current noise Sγ γ (Ω). The terminal properties are determined by the Fourier transformed self-energy matrices of Eq. (2). Therefore, if we choose the energy gap ∆γ of the γ th lead to be zero, this lead is the normal metal. Otherwise, if ∆γ = 0, the lead is a superconductor. This denotes that the currents transporting into and out of quantum dot are correlated, and the spectral density of current noise S(Ω) is expressed by the Green’s function of quantum dot. Since the Zeeman field is introduced in the system, this causes the system to be nondegenerate. This means that the spin-up and spin-down states can cause different current correlations, and hence result in novel spectral density of shot noise. The current and shot noise formulas are the generalization of Landauer–Büttiker formulas, which can describe multiple electrons tunneling procedure in the large bias with the approximation that the electron reservoirs are large enough. The electrodes broaden immediately at the connections to the quantum dot, and the formulas are valid in the wide-band approximation. For the frequency-independent situation, the shot noise spectral density in the γ th lead is reduced from the general formula (6) by noticing Fγ δ (, ) = Fδγ (, ) that 2e2 Sγ γ (0) = h σ
d
Tγ δ,σ () 1 − Tβγ ,σ () Fγ δ (, )
δ=γ
β=γ
1 + Tγ β,σ ()Tγ δ,σ () Fγ γ (, ) + Fβδ (, ) . 2
(7)
δβ=γ
The frequency-independent shot noise formula (7) contains thermal noise which disappears as temperature approaches zero. This can be seen directly by noticing that the Fermi–Dirac function is a step function fγ () = 1 − θ ( − µγ 1 ) at zero temperature. It is expressed by the transmission coefficient Tγ β,σ () for concerning electrons tunneling from γ th lead to βth lead. For the multi-terminal system, the property of tunneling particles is affected by the differences of chemical potentials in whole terminals. This arises that the transmission coefficients are correlated to each other by the chemical potentials. This means that the tunneling electrons from the γ th lead to βth lead may feel the affections of potentials in the other terminals in addition to the leads γ and β. The current noise contains also the effect caused by Andreev reflection in the region |Vγ β | < |∆β |/e. Since the tunneling procedure can be controlled by the gate voltage and Zeeman field, the shot noise can also be controlled by these parameters. We perform the numerical calculation of shot noise S = S11 (0) of a two terminal hybrid system where the leads 1 and 2 are considered to be normal metal and superconductor, respectively. At zero-temperature, the shot noise of
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the normal lead is reduced from Eq. (7) to the following expression 2e2 S= d 1 − T12,σ () T12,σ (), h σ eV
(8)
0
in the region where the bias between the two leads µ2 − µ1 is positive, i.e., eV > 0. In the region eV < 0, the shot noise is given by taking integration from eV to zero in the above formula. |∆| = ∆ is chosen as the measurement scale for all energy quantities, such as linewidth Γγ and Zeeman energy µB. We consider the material of superconductor with small energy gap. The linewidth and Zeeman energy µB are comparable with the energy gap. The materials with such small energy gaps are Al and Cd. At zero temperature, their energy gaps are estimated as 1.7 × 10−1 and 7.5 × 10−2 meV, respectively. We deal with the symmetric coupling situation where the linewidth of the two leads are equal, i.e., Γ1 = Γ2 = 0.1∆. The single energy level and multi-level quantum-dot systems are investigated to display the physical properties. The single-level quantum dot system can explain the main behaviours and properties of the coherent tunneling approximately. We focus on the single-level quantum dot system first, and then present shot noise and Fano factor of the multi-level quantum-dot system as comparison. Fano factor F = SI (Ω)/SP is used to measure the suppression and enhancement of shot noise. As F < 1, the noise is sub-Poissonian, and as F > 1, it is super-Poissonian. Fig. 1 shows the shot noise of single-level system versus source-drain bias eV . The different curves depicted associated with different values of gate voltage and Zeeman field. We observe that the shot noise is sensitive to these parameters both in the Andreev reflection region 0 < eV < ∆, and in the normal tunneling region eV > ∆. In the Andreev reflection region, the shot noise varies rapidly to reach a considerable value. The solid curve represents the shot noise in the absence of Zeeman field as eVg = −0.5∆. It rises abruptly to its maximum value, and then declines to its saturate value. The dash-dotted curve shows the shot noise of both leads are normal metal (NDN) with the parameters taking as the solid curve. Obviously, the shot noise of the hybrid NDS system and the NDN system are quite different in the region 0 < eV < ∆, where the DOS of superconducting lead takes evident effect. They approach the same saturate value as eV ∆. For the NDN system, the shot noise increases monotonically in the region eV > 0. The shot noise of NDS system is smaller than that of the NDN system in the region 0 < eV < 0.5∆, and it increases to surpass the shot noise of NDN system as eV > 0.8∆. As the Zeeman field is applied to the quantum dot, the shot noise reduces its value, and some steps appear in the curve. This implies that the Zeeman field splits the energy level of the quantum dot to form multi-channel for electron to tunnel. The resonant tunneling procedure takes place in the channels. The shot noise also depends on the gate voltage, which can be seen by changing the gate voltage from eVg = −0.5∆ to 0.3∆ (dashed curve). We plot the Fano factor F = S(0)/2eI versus source-drain bias eV in Fig. 2 with the corresponding parameters as given in Fig. 1 for the single-level system. The dash-dotted curve is associated with the normal lead system (NDN). As eV < ∆, there is a minimum value F = 0.34 at the resonant point eV = 0.5∆, which is obtained in Ref. [14] by paying special attention to maintain gauge invariance at the nonlinear level. As eV ∆, the Fano factor approaches its saturate value F ≈ 0.45. The solid curve is related to the hybrid system (NDS) with gate voltage eVg = −0.5∆ in the absence of Zeeman field. One observes that as eV > ∆, the Fano factor has the same value as the NDN system. However, as 0 < eV < ∆, the Fano factor decreases abruptly to the minimum value F = 0.26 in the region 0.4∆ < eV < 0.5∆, which is quite different from the situation of NDN system. As the Zeeman field with µB = 0.8∆ is applied to the quantum dot, one observes that the Fano factor is strengthened in both of the regions 0 < eV < ∆, and eV > ∆ (dotted curve). This indicates that the Zeeman field may increase the Fano factor at special value of gate voltage. The saturate value reaches 0.5. The same behaviour has been observed experimentally away from the region of negative differential resistance by Iannaccone et al. in the normal resonant tunneling [5], which corresponds to the maximum theoretical suppression in Ref. [15]. As gate voltage is positive to be eVg = 0.3∆, and the Zeeman energy is µB = 0.5∆, one sees that the Fano factor varies drastically in the region 0 < eV < ∆ (dashed curve), and the minimum value is F = 0.26 at eV = 0.3∆. The saturate value F = 0.41 is smaller than the one in NDN system.
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Fig. 1. The shot noise of single-level system versus source-drain bias eV . The parameters are chosen as Ed = 0 and eVg = −0.5∆, µB = 0 for the solid curve; eVg = −0.5∆, µB = 0.8∆ for the dotted curve; eVg = −0.5∆, µB = 0 for the dash-dotted curve corresponding to NDN system; eVg = 0.3∆, µB = 0.5∆ for the dashed curve.
Fig. 2. The Fano factor of single-level system versus source-drain bias eV corresponding to Fig. 1. The parameters are chosen as Ed = 0 and eVg = −0.5∆, µB = 0 for the solid curve; eVg = −0.5∆, µB = 0.8∆ for the dotted curve; eVg = −0.5∆, µB = 0 for the dash-dotted curve associated with NDN system; eVg = 0.3∆, µB = 0.5∆ for the dashed curve.
Fig. 3 displays the resonant structure of shot noise versus Zeeman energy µB. The shot noise is symmetric about µB = 0 for the single-level system, and it is sensitive to the gate voltage and source-drain bias. In the Andreev reflection region as eV = 0.6∆ and eVg = 0.3∆, there are three peaks located on each main peak with the maximum magnitude S = 0.06(2e2/ h) (solid curve). The main peaks are separated further by increasing the gate voltage to eVg = 0.8∆. The two inner peaks are smeared to form two steps in the valley, and the magnitudes of the main peaks increase a little by the compensation of suppressing two peaks. As the source-drain bias is increased to the normal region eV = 1.2∆, subtle peaks located on each main peak are smeared, and the magnitudes of the resonant peaks increase to a large value S = 0.14(2e2/ h). This signifies that the Andreev reflection can cause novel shot noise structure due to changing the Zeeman field. We present the shot noise of NSN system versus Zeeman
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Fig. 3. The shot noise of single-level system versus Zeeman energy µB. The parameters are chosen as Ed = 0 and eV = 0.6∆, eVg = 0.3∆ for the solid curve; eV = 0.6∆, eVg = 0.8∆ for the dotted curve; eV = 1.2∆, eVg = 0.8∆ for the dashed curve; eV = 0.6∆, eVg = 0.3∆ for the dash-dotted curve of NDN system.
Fig. 4. The Fano factor of single-level system versus Zeeman energy µB. The parameters are chosen as Ed = 0 and eV = 0.6∆, eVg = 0.3∆ for the dotted curve; eV = 0.6∆, eVg = 0.8∆ for the dash-dotted curve; eV = 1.2∆, eVg = 0.8∆ for the solid curve.
energy by the dash-dotted curve with eV = 0.6∆, eVg = 0.3∆. Compared with the curves of NDS system in the Andreev reflection region, one observes that there is only one peak in each side of the curve, but the magnitude of each peak is larger than that of NDS system with the same parameters. We display the Fano factor of single-level system versus Zeeman energy in Fig. 4. The Fano factor is symmetric about µB = 0 and F < 1 in the whole region of µB. The dotted curve is plotted with the parameters eV = 0.6∆ and eVg = 0.3∆. There exist two valleys on each side with the minimum values F = 0.3 for the inner valley and F = 0.5 for the next one. As the gate voltage increases to 0.8∆, the two valleys are smeared to form one with the minimum value F = 0.28 (dash-dotted curve). As the source-drain bias increases to the normal region with eV = 1.2∆, the valleys become wider, and the minimum value becomes F = 0.365.
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Fig. 5. The shot noise of multi-level quantum-dot system versus source-drain bias eV . The parameters are chosen as Ed = 0, eVg = −0.8∆ and µB = 0 for the solid curve; eVg = 0, µB = 0.8∆ for the dotted curve; eVg = 0, µB = 0 for the dashed curve; eVg = 0, µB = 0 for the dash-dotted curve corresponding to NDN system.
Fig. 6. The Fano factor of multi-level quantum-dot system versus source-drain bias eV . The parameters are chosen as Ed = 0 and eVg = −0.8∆, µB = 0 for the solid curve; eVg = 0, µB = 0 for the dotted curve corresponding to NDN system; eVg = −0.2∆, µB = 0 for the dashed curve.
We show the shot noise and Fano factor of multi-level quantum-dot system in Figs. 5 and 6. To exhibit the main behaviours of these quantities, we choose the quantum dot containing five levels Ed = Ed + ∆E with equal level space ∆E = 0.5∆, and ( = 1, 2, . . . , 5). The steps in the shot noise are associated with the energy levels, and Zeeman effect can be observed. The Andreev reflection induced shot noise has the similar feature as that of singlelevel quantum dot system. However the Fano factor behaves quite differently for the single-level and multi-level quantum dot systems both in the suppression strengthes and shapes. The suppression of shot noise is much less in the multi-level system than the one in the single-level system. This result comes from the correlations of current in different channels. To summarize, the shot noise is sensitive to the external parameters eV , eVg and µB. The sub-Poissonian shot noise is obtained, which denotes that the shot noise can be suppressed to different extent by the external parameters
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in both the Andreev reflection region 0 < eV < ∆ and the normal region eV > ∆. The Zeeman field can split the resonant structure and to present steps in the shot noise. This signifies that due to Zeeman splitting, novel channels appear for electron to tunnel, and electrons can resonate in the channels. The shot noise is symmetric with respect to the Zeeman energy about µB = 0. Single-level and multi-level quantum dot systems exhibit different behaviours of shot noise and Fano factor.
Acknowledgements This work was supported by the National Natural Science Foundation of China under the Grant No. 19875004, and by the fellowship under the Distinguished Visiting Scholar Program of the Chinese Government.
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Physics Letters A 299 (2002) 262–270 www.elsevier.com/locate/pla
Shot noise in the hybrid systems with a quantum dot coupled to normal and superconducting leads Hong-Kang Zhao a,b a Department of Physics, Beijing Institute of Technology, Beijing 100081, PR China 1 b School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, USA
Received 12 May 2002; received in revised form 12 May 2002; accepted 21 May 2002 Communicated by R.W. Wu
Abstract We have investigated the spectral density of shot noise in the tunneling systems composed of a quantum dot coupled with normal and superconducting leads. The spectral density of shot noise is obtained by introducing the Green’s functions of quantum dot. The zero-frequency shot noise in the same normal lead is given simply as the form of transmission coefficient associated with two leads. We have performed the numerical calculation on a two-terminal system at zero-temperature, and novel resonant structure due to Andreev reflection is observed. The sub-Poissonian shot noise is obtained by varying the parameters, which denotes that the shot noise can be suppressed by the external parameters. 2002 Published by Elsevier Science B.V.
The tunneling current through a conductor may fluctuate in time even if it is induced by a stationary voltage. The noise is usually characterized by its power spectrum at frequency ω, which is defined as the Fourier transform of the current correlation function [1]. The shot noise is a nonequilibrium fluctuation which is caused by the discreteness of the charge carriers [2]. When the size of an electronic system reaches the nanometer scale, noise becomes a very interesting problem. The investigation on the deviations from purely Poissonian shot noise in mesoscopic systems has been an increasingly interest subject. From the investigation of shot noise, we can learn additional information on electronic structure and transport properties, since it is directly related to the degree of randomness in carrier transfer. Due to the Poissonian distribution in a macroscopic system, the current shows the value of shot noise by the well-known Schottky formula [3] SP = 2eI . However, for a mesoscopic system, the electrons are correlated due to coherent transport, and they are governed by the Fermi distribution and Pauli principle. This quantum behaviour results in the deviation of shot noise to the Poissonian one. The suppression [4] and enhancement [5] of shot noise have been studied actively to classify what is the cause of the deviation of Poissonian form. Negative correlations between current pulses can lead to a complete suppression of the shot noise in quantum point contacts. The system of normal metal layer coupled with superconducting leads has been investigated extensively during the past decades [6]. The shot noise caused by Andreev reflection in such systems is certainly an interesting
E-mail address: [email protected] (H.-K. Zhao). 1 Permanent address.
0375-9601/02/$ – see front matter 2002 Published by Elsevier Science B.V. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 6 8 1 - 3
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problem, for instance, the shot noise in a voltage biased superconducting quantum point contact was investigated in Ref. [7]. The shot noise in aluminum atomic point contacts containing a small number of conduction channels is measured, and the noise suppression is observed in the normal state [8]. Recently, it becomes of great interest to study the systems consisting of superconducting leads coupled with a quantum dot. Investigations involving such systems are superconductor-quantum-dot-superconductor (SDS), superconductor-quantum-dot-normal-metal lead (SDN), or superconducting quantum dot connected with normal leads [9–13]. In this Letter, we employ the nonequilibrium Green’s function (NGF) technique to study the shot noise spectra in the quantum dot coupled with normal metallic and superconducting leads. We study the situation that the quantum dot is subject to a magnetic field to induce Zeeman split. The Andreev reflection is included in the calculation, and the Andreev-reflected terms cause novel shot noise compared with the normal system. Sub-Poissonian shot noise is achieved to show novel structures, and Zeeman effect results in novel resonant shot noise structure due to the Zeeman splitting. Generally, the system is composed of N normal and superconducting leads coupled to a quantum dot. We consider the situation that the leads γ and β are biased by the dc voltage Vγ β which is the drop of chemical potential between the two leads µγ − µβ = eVγ β . There is a magnetic field B applying to the quantum dot through gate. The Hamiltonians of the superconducting leads are approximated by the mean-field BCS theory. The Hamiltonian of the system can be expressed as the sum of separate sub-Hamiltonians and the interaction term as H= ∆γ aγ† ,k↑aγ† ,−k↓ + ∆∗γ aγ ,−k↓aγ ,k↑ γ k aγ† ,kσ aγ ,kσ − γ kσ
+
σ
γk † εd,σ dσ dσ
+
T˜γ k (t)aγ† ,kσ dσ + h.c. .
(1)
γ kσ
† In the Hamiltonian (1), aγ† ,kσ (aγ ,kσ ), and dσ (dσ ) are the creation (annihilation) operators of electrons in the N leads and quantum dot. ∆γ is the energy gap of the superconducting lead. The energy gap is a complex quantity which is characterized by the phase φγ , ∆γ = |∆γ |eiφγ . We have neglected the Coulomb interaction in the Hamiltonian for simplicity. This approximation is valid in the case as the quantum dot is not extremely small. The interaction strengths is gauged by T˜γ ,k (t) = Tγ k exp(− hi¯ µγ 1 t), where µγ 1 = µγ − µ1 , and Tγ k are real constants satisfying the relation Tγ k = Tγ −k . For definiteness, we take the chemical potential µ1 of lead 1 as the energy reference point. In this way, physical quantities are not affected as the chemical potentials of the whole system are shifted the same value. µ = gµB /2 is the magnetic moment of electron, µB the Bohr magneton, γ k = ¯γ k − µ1 , εd,σ = E˜ d,σ − σ µB, E˜ d,σ = Ed, − eVg and Vg is the gate voltage which adjusts the energy of quantum dot. The magnetic field is squeezed in order not to affect the superconductors and normal leads. The tunneling current operator of the system can be derived by using the Heisenberg equation and continuity † equation, which gives Iˆγ (t) = − ie kσ [H, aγ ,kσ (t)aγ ,kσ (t)]. We make the Bogoliubov transformation to h¯ diagonalize the superconducting leads. The Hamiltonian of γ th superconducting lead is expressed as the diagonal form Hγ = kσ ξγ k αγ† ,kσ αγ ,kσ , where αγ† ,kσ and αγ ,kσ are the creation and annihilation operators of quasi-particle in the superconducting lead, and they satisfy the Fermi distribution. ξγ k = (γ2 k + |∆γ |2 )1/2 is the excitation energy of quasiparticle. For the γ th normal lead, we have ∆γ = 0, and ξγ k = γ k . The shot noise is caused by the nonequilibrium charged particle transport, and it is derived through the Fourier transformation of the current fluctuation correlation Πγ γ (t, t ) = δ Iˆγ (t)δ Iˆγ (t ) + δ Iˆγ (t )δ Iˆγ (t), where δ Iˆγ (t) = Iˆγ (t) − Iˆγ (t). The current correlation Iˆγ (t)Iˆγ (t ) contains products of four creation and annihilation operators of particle in quantum dot and leads. The expectation values of pair productor with one annihilation operator and one creation operator are associated with the normal Green’s functions. Employing the equations of motion (EOM) for the electron and quasiparticle operator, we can express the operators of γ th lead by the Green’s function of lead and the operator of electron in quantum dot. We consider the wide-band limit situation, and in this limit the linewidth is energy-independent, i.e., Γγ (E) = Γγ . The density of state (DOS) in a lead is given by Nγ () = ||/( 2 − |∆γ |2 )1/2 . We define the retarded Green’s
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function matrix of quantum dot as Grσ (t, t ) which satisfies the Dyson equation in the Nambu representation [12]. For the system of a quantum dot coupled with superconducting and normal leads, this equation provides the information originated from the scattering procedure. The retarded Green’s functions are obtained by solving the Fourier transformed versions of Dyson equation self-consistently. We present the results of the Green’s functions as follows for deriving the spectral density of the current noise of our system clearly. We define the (±)r (±) (±)r ¯r ()], quantity Q(±)r γ σ,αβ () to write Green’s functions conveniently as Qγ σ,αβ () = κσ ()Σγ σ,αβ ()/[1 − ζσ (±) (+)r (+) (−)r (−) r r −1 where κσ () = [ ± E˜ d, + σ µB] and ζσ () = κσ ()Σ¯ σ,22 (), ζσ () = κσ ()Σ¯ σ,11 (). In the definition above, Σ¯ r () is associated with the Fourier transformed retarded self-energy matrix given by γ σ,αβ
Wγ ( + µγ 1 )eiφγ (2) , Nγ ( − µγ 1 ) r () = Σ ¯r where Wγ () = |∆γ |/( 2 − |∆γ |2 )1/2 , and Σ¯ σ,jj γ γ σ,jj (). In the Andreev reflection regime where || < |∆γ | = 0, Nγ () and Wγ () are complex. This means that the tunneling current can penetrate through the superconducting barrier even if the source-drain bias is smaller than the energy gap. From the Fourier transformed version of Dyson equation, we find the normal retarded Green’s function of the quantum dot Grσ,11 () of our system self-consistently as ¯ rγ σ () = − i Γγ Σ 2
Nγ ( + µγ 1 ) Wγ ( − µγ 1 )e−iφγ
(−)
Grσ,11 () =
(−)
1 − κσ ()
κσ () () + Σ¯ r
¯r β∈{M} [Σβσ,11
(+)r βσ,12 ()Qβσ,21 (
+ 2µβ1 )]
.
(3)
In the above equation β sums over all superconductors and normal leads. This Green’s function is valid for the system composed of at least one normal metal lead. The energy of tunneling particle shifts a value as → ± 2µβ1 due to the Andreev reflection. This procedure affects the Green’s function of the quantum dot. The advanced Green’s functions are derived by the similar way. In fact, for the steady system, an advanced Green’s function is obtained by taking conjugation over corresponding retarded Green’s function. Substituting the corresponding operators into the Bogoliubov transformed version of current operator, we arrive at the current operator in the γ th normal metal lead i e γ † ()αβ σ , d d Aββ ,σ ()e− h¯ (− )t αβσ Iˆγ (t) = (4) h ββ σ
where
2 γ Aββ ,σ () = i Γ˜γ ()δβγ Grσ,11 () − i Γ˜β ()δγ β Gaσ,11 () − Γ˜γ ()Γ˜β () Grσ,11 () , (+)
(+)
ηβσ () =
[κσ ( + 2µβ1 )Γβ Wβ ( + µβ1 )]2 4|1 − ζσ(+) ( + 2µβ1 )|2
,
(+) ()]. Therefore, the tunneling current operator is determined by the and Γ˜β () = δ Γδ Re Nδ ( + µδ1 )[δβ,δ − ηβσ feature of equilibrium quasi-particles deep in the leads. It also contains resonant structure and Andreev reflection due to the connected quantum dot. The current operator formula appears the similar form as Büttiker formula given by scattering theory in Ref. [2] for the normal scattering contact sample system. The tunneling current is given by taking quantum average and ensemble expectation over the current operator Iγ = Iˆγ (t) by employing † the relation αβσ ()αβ σ ( ) = δββ δ( − )fβ (), where the Fermi–Dirac distribution function is defined by fβ () = 1/[exp(( − µβ1 )/kB T ) + 1]. Therefore, the tunneling current in the γ th lead is determined by the Landauer–Büttiker-like formula e d Tγ β,σ () fγ () − fβ () , Iγ = (5) h σ β=γ
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where Tαβ,σ () = Γ˜α ()Γ˜β ()|Grσ,11 ()|2 represents the transmission coefficient of electron tunneling from the αth lead to βth lead. For our system, there exists Tαβ,σ () = Tβα,σ (), which signifies that the probability of electron tunneling from the αth lead to βth lead is equal to the probability of electron tunneling from the βth lead to αth lead. The spectral density of shot noise Sγ γ (Ω) is determined by the quantum statistical expectation value of the 1 Fourier transformed current operator correlation through the relation [2] 2π Πγ γ (Ω, Ω ) = Sγ γ (Ω)δ(Ω + Ω ), where Πγ γ (Ω, Ω ) is the Fourier transformed version of Πγ γ (t, t ). Performing the Fourier transformation and employing the formula for the expectation value of four quasiparticles, finally we obtain the shot noise spectral density associated with the tunneling currents in the normal metal leads γ and γ e2 γ γ Sγ γ (Ω) = d Aβδ,σ ()Aδβ,σ ( − h¯ Ω)Fβδ (, − hΩ), ¯ h
(6)
βδσ
where Fβδ (, ) = fβ ()[1 − fδ ( )] + fδ ( )[1 − fβ ()]. This is a general formula for describing the frequencydependent current–current correlation fluctuation of multi-terminal-quantum-dot system, from which we can derive the spectral density of current noise Sγ γ (Ω). The terminal properties are determined by the Fourier transformed self-energy matrices of Eq. (2). Therefore, if we choose the energy gap ∆γ of the γ th lead to be zero, this lead is the normal metal. Otherwise, if ∆γ = 0, the lead is a superconductor. This denotes that the currents transporting into and out of quantum dot are correlated, and the spectral density of current noise S(Ω) is expressed by the Green’s function of quantum dot. Since the Zeeman field is introduced in the system, this causes the system to be nondegenerate. This means that the spin-up and spin-down states can cause different current correlations, and hence result in novel spectral density of shot noise. The current and shot noise formulas are the generalization of Landauer–Büttiker formulas, which can describe multiple electrons tunneling procedure in the large bias with the approximation that the electron reservoirs are large enough. The electrodes broaden immediately at the connections to the quantum dot, and the formulas are valid in the wide-band approximation. For the frequency-independent situation, the shot noise spectral density in the γ th lead is reduced from the general formula (6) by noticing Fγ δ (, ) = Fδγ (, ) that 2e2 Sγ γ (0) = h σ
d
Tγ δ,σ () 1 − Tβγ ,σ () Fγ δ (, )
δ=γ
β=γ
1 + Tγ β,σ ()Tγ δ,σ () Fγ γ (, ) + Fβδ (, ) . 2
(7)
δβ=γ
The frequency-independent shot noise formula (7) contains thermal noise which disappears as temperature approaches zero. This can be seen directly by noticing that the Fermi–Dirac function is a step function fγ () = 1 − θ ( − µγ 1 ) at zero temperature. It is expressed by the transmission coefficient Tγ β,σ () for concerning electrons tunneling from γ th lead to βth lead. For the multi-terminal system, the property of tunneling particles is affected by the differences of chemical potentials in whole terminals. This arises that the transmission coefficients are correlated to each other by the chemical potentials. This means that the tunneling electrons from the γ th lead to βth lead may feel the affections of potentials in the other terminals in addition to the leads γ and β. The current noise contains also the effect caused by Andreev reflection in the region |Vγ β | < |∆β |/e. Since the tunneling procedure can be controlled by the gate voltage and Zeeman field, the shot noise can also be controlled by these parameters. We perform the numerical calculation of shot noise S = S11 (0) of a two terminal hybrid system where the leads 1 and 2 are considered to be normal metal and superconductor, respectively. At zero-temperature, the shot noise of
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the normal lead is reduced from Eq. (7) to the following expression 2e2 S= d 1 − T12,σ () T12,σ (), h σ eV
(8)
0
in the region where the bias between the two leads µ2 − µ1 is positive, i.e., eV > 0. In the region eV < 0, the shot noise is given by taking integration from eV to zero in the above formula. |∆| = ∆ is chosen as the measurement scale for all energy quantities, such as linewidth Γγ and Zeeman energy µB. We consider the material of superconductor with small energy gap. The linewidth and Zeeman energy µB are comparable with the energy gap. The materials with such small energy gaps are Al and Cd. At zero temperature, their energy gaps are estimated as 1.7 × 10−1 and 7.5 × 10−2 meV, respectively. We deal with the symmetric coupling situation where the linewidth of the two leads are equal, i.e., Γ1 = Γ2 = 0.1∆. The single energy level and multi-level quantum-dot systems are investigated to display the physical properties. The single-level quantum dot system can explain the main behaviours and properties of the coherent tunneling approximately. We focus on the single-level quantum dot system first, and then present shot noise and Fano factor of the multi-level quantum-dot system as comparison. Fano factor F = SI (Ω)/SP is used to measure the suppression and enhancement of shot noise. As F < 1, the noise is sub-Poissonian, and as F > 1, it is super-Poissonian. Fig. 1 shows the shot noise of single-level system versus source-drain bias eV . The different curves depicted associated with different values of gate voltage and Zeeman field. We observe that the shot noise is sensitive to these parameters both in the Andreev reflection region 0 < eV < ∆, and in the normal tunneling region eV > ∆. In the Andreev reflection region, the shot noise varies rapidly to reach a considerable value. The solid curve represents the shot noise in the absence of Zeeman field as eVg = −0.5∆. It rises abruptly to its maximum value, and then declines to its saturate value. The dash-dotted curve shows the shot noise of both leads are normal metal (NDN) with the parameters taking as the solid curve. Obviously, the shot noise of the hybrid NDS system and the NDN system are quite different in the region 0 < eV < ∆, where the DOS of superconducting lead takes evident effect. They approach the same saturate value as eV ∆. For the NDN system, the shot noise increases monotonically in the region eV > 0. The shot noise of NDS system is smaller than that of the NDN system in the region 0 < eV < 0.5∆, and it increases to surpass the shot noise of NDN system as eV > 0.8∆. As the Zeeman field is applied to the quantum dot, the shot noise reduces its value, and some steps appear in the curve. This implies that the Zeeman field splits the energy level of the quantum dot to form multi-channel for electron to tunnel. The resonant tunneling procedure takes place in the channels. The shot noise also depends on the gate voltage, which can be seen by changing the gate voltage from eVg = −0.5∆ to 0.3∆ (dashed curve). We plot the Fano factor F = S(0)/2eI versus source-drain bias eV in Fig. 2 with the corresponding parameters as given in Fig. 1 for the single-level system. The dash-dotted curve is associated with the normal lead system (NDN). As eV < ∆, there is a minimum value F = 0.34 at the resonant point eV = 0.5∆, which is obtained in Ref. [14] by paying special attention to maintain gauge invariance at the nonlinear level. As eV ∆, the Fano factor approaches its saturate value F ≈ 0.45. The solid curve is related to the hybrid system (NDS) with gate voltage eVg = −0.5∆ in the absence of Zeeman field. One observes that as eV > ∆, the Fano factor has the same value as the NDN system. However, as 0 < eV < ∆, the Fano factor decreases abruptly to the minimum value F = 0.26 in the region 0.4∆ < eV < 0.5∆, which is quite different from the situation of NDN system. As the Zeeman field with µB = 0.8∆ is applied to the quantum dot, one observes that the Fano factor is strengthened in both of the regions 0 < eV < ∆, and eV > ∆ (dotted curve). This indicates that the Zeeman field may increase the Fano factor at special value of gate voltage. The saturate value reaches 0.5. The same behaviour has been observed experimentally away from the region of negative differential resistance by Iannaccone et al. in the normal resonant tunneling [5], which corresponds to the maximum theoretical suppression in Ref. [15]. As gate voltage is positive to be eVg = 0.3∆, and the Zeeman energy is µB = 0.5∆, one sees that the Fano factor varies drastically in the region 0 < eV < ∆ (dashed curve), and the minimum value is F = 0.26 at eV = 0.3∆. The saturate value F = 0.41 is smaller than the one in NDN system.
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Fig. 1. The shot noise of single-level system versus source-drain bias eV . The parameters are chosen as Ed = 0 and eVg = −0.5∆, µB = 0 for the solid curve; eVg = −0.5∆, µB = 0.8∆ for the dotted curve; eVg = −0.5∆, µB = 0 for the dash-dotted curve corresponding to NDN system; eVg = 0.3∆, µB = 0.5∆ for the dashed curve.
Fig. 2. The Fano factor of single-level system versus source-drain bias eV corresponding to Fig. 1. The parameters are chosen as Ed = 0 and eVg = −0.5∆, µB = 0 for the solid curve; eVg = −0.5∆, µB = 0.8∆ for the dotted curve; eVg = −0.5∆, µB = 0 for the dash-dotted curve associated with NDN system; eVg = 0.3∆, µB = 0.5∆ for the dashed curve.
Fig. 3 displays the resonant structure of shot noise versus Zeeman energy µB. The shot noise is symmetric about µB = 0 for the single-level system, and it is sensitive to the gate voltage and source-drain bias. In the Andreev reflection region as eV = 0.6∆ and eVg = 0.3∆, there are three peaks located on each main peak with the maximum magnitude S = 0.06(2e2/ h) (solid curve). The main peaks are separated further by increasing the gate voltage to eVg = 0.8∆. The two inner peaks are smeared to form two steps in the valley, and the magnitudes of the main peaks increase a little by the compensation of suppressing two peaks. As the source-drain bias is increased to the normal region eV = 1.2∆, subtle peaks located on each main peak are smeared, and the magnitudes of the resonant peaks increase to a large value S = 0.14(2e2/ h). This signifies that the Andreev reflection can cause novel shot noise structure due to changing the Zeeman field. We present the shot noise of NSN system versus Zeeman
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Fig. 3. The shot noise of single-level system versus Zeeman energy µB. The parameters are chosen as Ed = 0 and eV = 0.6∆, eVg = 0.3∆ for the solid curve; eV = 0.6∆, eVg = 0.8∆ for the dotted curve; eV = 1.2∆, eVg = 0.8∆ for the dashed curve; eV = 0.6∆, eVg = 0.3∆ for the dash-dotted curve of NDN system.
Fig. 4. The Fano factor of single-level system versus Zeeman energy µB. The parameters are chosen as Ed = 0 and eV = 0.6∆, eVg = 0.3∆ for the dotted curve; eV = 0.6∆, eVg = 0.8∆ for the dash-dotted curve; eV = 1.2∆, eVg = 0.8∆ for the solid curve.
energy by the dash-dotted curve with eV = 0.6∆, eVg = 0.3∆. Compared with the curves of NDS system in the Andreev reflection region, one observes that there is only one peak in each side of the curve, but the magnitude of each peak is larger than that of NDS system with the same parameters. We display the Fano factor of single-level system versus Zeeman energy in Fig. 4. The Fano factor is symmetric about µB = 0 and F < 1 in the whole region of µB. The dotted curve is plotted with the parameters eV = 0.6∆ and eVg = 0.3∆. There exist two valleys on each side with the minimum values F = 0.3 for the inner valley and F = 0.5 for the next one. As the gate voltage increases to 0.8∆, the two valleys are smeared to form one with the minimum value F = 0.28 (dash-dotted curve). As the source-drain bias increases to the normal region with eV = 1.2∆, the valleys become wider, and the minimum value becomes F = 0.365.
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Fig. 5. The shot noise of multi-level quantum-dot system versus source-drain bias eV . The parameters are chosen as Ed = 0, eVg = −0.8∆ and µB = 0 for the solid curve; eVg = 0, µB = 0.8∆ for the dotted curve; eVg = 0, µB = 0 for the dashed curve; eVg = 0, µB = 0 for the dash-dotted curve corresponding to NDN system.
Fig. 6. The Fano factor of multi-level quantum-dot system versus source-drain bias eV . The parameters are chosen as Ed = 0 and eVg = −0.8∆, µB = 0 for the solid curve; eVg = 0, µB = 0 for the dotted curve corresponding to NDN system; eVg = −0.2∆, µB = 0 for the dashed curve.
We show the shot noise and Fano factor of multi-level quantum-dot system in Figs. 5 and 6. To exhibit the main behaviours of these quantities, we choose the quantum dot containing five levels Ed = Ed + ∆E with equal level space ∆E = 0.5∆, and ( = 1, 2, . . . , 5). The steps in the shot noise are associated with the energy levels, and Zeeman effect can be observed. The Andreev reflection induced shot noise has the similar feature as that of singlelevel quantum dot system. However the Fano factor behaves quite differently for the single-level and multi-level quantum dot systems both in the suppression strengthes and shapes. The suppression of shot noise is much less in the multi-level system than the one in the single-level system. This result comes from the correlations of current in different channels. To summarize, the shot noise is sensitive to the external parameters eV , eVg and µB. The sub-Poissonian shot noise is obtained, which denotes that the shot noise can be suppressed to different extent by the external parameters
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in both the Andreev reflection region 0 < eV < ∆ and the normal region eV > ∆. The Zeeman field can split the resonant structure and to present steps in the shot noise. This signifies that due to Zeeman splitting, novel channels appear for electron to tunnel, and electrons can resonate in the channels. The shot noise is symmetric with respect to the Zeeman energy about µB = 0. Single-level and multi-level quantum dot systems exhibit different behaviours of shot noise and Fano factor.
Acknowledgements This work was supported by the National Natural Science Foundation of China under the Grant No. 19875004, and by the fellowship under the Distinguished Visiting Scholar Program of the Chinese Government.
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