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Current noise in three-terminal hybrid quantum point contacts
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 025301 (http://iopscience.iop.org/0953-8984/26/2/025301) View the table of contents for this issue, or go to the journal homepage for more
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 025301 (8pp)
doi:10.1088/0953-8984/26/2/025301
Current noise in three-terminal hybrid quantum point contacts B H Wu1 , C R Wang1 , X S Chen1,2 and G J Xu3 1
Department of Applied Physics, Donghua University, 2999 North Renmin Road, Shanghai 201620, People’s Republic of China 2 National Laboratory for Infrared Physics, Shanghai Institute of Technical Physics, Chinese Academy of Science, Shanghai 200083, People’s Republic of China 3 Engineering Research Center of Optical Instrument and System, Ministry of Education, Shanghai Key Lab of Modern Optical System, University of Shanghai for Science and Technology, 516 Jungong Road, Shanghai 200093, People’s Republic of China E-mail: [email protected] Received 5 August 2013, in final form 17 October 2013 Published 5 December 2013 Abstract
We investigate the current noise of three-terminal hybrid structures at arbitrary bias voltages. Our results indicate that the noise can be a useful tool to extract dynamical information in multi-terminal hybrid structures. The zero-frequency noise is sensitive to the coupling with a normal lead. As a result, the characteristic multiple-step structure of the noise Fano factor due to multiple Andreev reflection will be suppressed as we increase this coupling. In addition, the internal dynamics due to processes of Andreev reflection and multiple Andreev reflection raises rich features in the noise spectrum corresponding to the energy differences of various dynamical processes. (Some figures may appear in colour only in the online journal)
1. Introduction
gap. In this multiple AR (MAR), the charge is transferred in amounts of ne, where n is an integer and determined by the threshold voltage Vn = 21/ne. While most previous studies on transport in hybrid structures have concerned two-terminal configurations, multiterminal geometries with several S or N leads are also under current interest for quantum information technology as they can provide additional degrees of freedom to externally modulate the transport properties. In multi-terminal structures, the interplay of AR and MAR involving several interfaces can give rise to rich subgap transport features [4, 5]. Among the various configurations, the simplest structure where both AR and MAR contribute to the transport properties is a three-terminal junction where two S leads and one N lead are connected by normal conducting channels. Jonckheere et al [6] have shown that the coupling to an additional N lead can generally be considered as a source of dephasing. By tuning the voltage at the N lead, the magnitude and sign of the Josephson current between two superconducting leads can be well controlled [7–11]. The N lead can also be a powerful probe to reveal the
Transport in hybrid structures with both normal (N) and superconductor (S) materials has attracted much recent interest due to their potential as the building blocks of solid state quantum information technology [1, 2]. When the energy of an injected charge carrier lies in the superconducting gap 1, a direct transfer of the charged particle is energetically prohibited. Nonequilibrium transport in the hybrid structures then evolves distinct mechanisms where charges are transferred in bunches in the low-bias V limit [3]. For a two-terminal structure formed by N and S terminals, the Andreev reflection (AR) dominates the charge transfer when V is larger than 1. In each AR process, an electron from N crossing the N–S interface is only permitted by leaving behind a hole in the N region and at the same time creating a Cooper pair in the S region. In that way, a total of 2e is transferred in one shot, where e is the electron charge. For devices where two S terminals are connected via a short normal conductor, charged particles can undergo several ARs at the two N–S interfaces to overcome the superconducting 0953-8984/14/025301+08$33.00
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c 2014 IOP Publishing Ltd Printed in the UK
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spectroscopic information of the central region [12]. A recent experiment [13] has demonstrated the first spectroscopy of the evolution of Andreev bound states in a carbon nanotube based three-terminal hybrid device. As explained above, AR and MAR play primary roles in the transport of hybrid structures and they have distinct effective charges during carrier transmission. The effective charge in AR is two, while it can, in principle, be any integer in MAR, depending on the bias voltage. A direct probe of the effective charge is beyond the usual current I measurement. In his seminal work, Schottky [14] has shown that the effective charge q of each transfer can be identified from the measurement of the zero-frequency power spectrum of the current fluctuation or shot noise S at the low transmission limit as q/e = S/2I. An enhanced shot noise due to the AR or MAR mechanisms has indeed been experimentally demonstrated in hybrid systems with various configurations [15–18]. For example, Jehl et al [17] have demonstrated doubled shot noise with respect to the normal case due to the AR at the N–S interface. A large effective charge due to MAR has also been observed by Cron et al [18] in structures of two superconductors connected by atomic point contact, where the effective charge displays staircase behavior at threshold voltages of the MAR processes. Good agreement between the experimental results and microscopic theory has also been achieved [19]. In a recent work, noise in N–S hybrid structures has been measured by the full counting statistics, where super-Poissonian noise is detected due to the interplay of single-particle tunneling and the AR [20]. By going beyond the zero-frequency limit, the current noise at finite frequency can provide more interesting information, such as the coherence and dynamics of nanostructures [21–25]. However, the noise spectrum in hybrid structures has raised little attention. In this direction, Cottet et al [26] have shown that the noise spectrum of a superconductor–ferromagnet quantum point contact is sensitive to the spin dependence of interfacial phase shift and can thus be used to characterize the scattering properties in this hybrid structure. More recently, Badiane et al [27] found that the noise spectrum can directly give evidence of the 4π periodic in topological Josephson junctions while signatures of such a fractional Josephson effect are absent from the measurement of the averaged current or the zero-frequency noise. Regarding the noise in multi-terminal hybrid structures, it has been studied both theoretically and experimentally for the purpose of creating nonlocal entangled electron pairs in a configuration with two S leads and one N lead [28–30]. There, the main concern is the zero-frequency noise and the transport is stationary, with the AR process dominating the low-bias regime. However, the noise and its spectral information in multi-terminal structures with more than one S lead has not been considered so far. The analysis is complicated, since at arbitrary bias in these structures, the MAR processes contribute to the current and the transport is no longer stationary due to the ac Josephson effect. It is the purpose of the present paper to investigate the noise properties in multi-terminal hybrid structures. We will
Figure 1. Schematic plot of the multi-terminal hybrid structure with
superconducting left (L) and right (R) leads and a normal (N) lead. The three terminals are connected via conducting channels of 0 quantum point contact with coupling parameters tαβ , where α, β = L, R or N.
consider a minimal model with three terminals connected via point contact, as shown in figure 1. All the leads are connected via coherent point contact and at arbitrary bias voltages. As a result, the transport will be time periodic, with frequencies determined by the Josephson relation, in contrast to the stationary transport in previous studies. A similar structure where a quantum dot is coupled to a pair of SC leads and a normal lead has been recently thoroughly investigated by Jonckheere et al [6]. There, the transport properties are restricted to the dc and ac current profile. It is shown that the normal lead can act as a source of dephasing. With increasing dot coupling to the N lead, the transport displays crossover from zero dephasing to the incoherent case. As a result, the harmonics of the ac current will be reduced. However, much less attention has been paid to studying the noise of the Josephson current. By using a microscopic Hamiltonian approach developed by Cuevas et al [31], we present results for the noise properties of the hybrid junction. Our results show that the interplay of AR and MAR can induce rich features in both the zero-frequency noise power and the noise spectrum. It is found that the noise signatures of MAR, in particularly the higher order processes, are sensitive to the coupling to the N lead. In addition, the noise spectrum in the hybrid structures clearly exhibits energy differences associated with the AR and MAR processes. Therefore, the noise properties of hybrid structures can be a useful tool to extract the dynamical information in multi-terminal hybrid structures. The paper is organized as follows. In section 2, we explain our model structure and the theoretical formalism for the noise properties of hybrid structures. Section 3 presents and discusses our numerical results of the zero-frequency noise and noise spectrum for the hybrid structures. Finally, section 4 briefly summarizes our results. 2. Model and formalism
The device under consideration is a three-terminal hybrid structure. A schematic representation of the model is depicted in figure 1. The left (L) and right (R) terminals are S and the 2
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order. Within our model, the current from the α lead, for example, is given by the time evolution of its total electron number. By evaluating the equation of motion of the total electron number in the α lead, we arrive at the operator for the current from the α lead as ie X X Iα (τ ) = −eN˙ α (τ ) = [tαβ c†ασ (τ )cβσ (τ ) h¯ σ β6=α
third one is N. All the terminals are connected via quantum transport channels. Experimentally, these quantum transport channels can, for example, be either atomic size contact realized in break junctions or quantum point contact involving constricted two-dimensional electron gases in semiconductor heterostructures. For the sake of simplicity, we represent the quantum transport channels between two terminals with a single conduction channel. The Hamiltonian for the system can be written as [31] H = HN + HL + HR + HT ,
− tβα c†βσ (τ )cασ (τ )].
(1)
To proceed further, it is convenient to make use of the operator in Nambu space by 9α = (c†α↑ , cα↓ )† . The corresponding Green’s function in the Keldysh contour is then ˆ αβ (t1 , t2 ) = ihTC (9α (t1 ) ⊗ defined as a 2 × 2 matrix as G † 9β (t2 ))i, where TC is the contour-order operator. This contour-ordered Green’s function is related to four real-time nonequilibrium Green’s functions. More specifically, the retarded (r), advanced (a), lesser () ˆ r/a (τ, τ 0 ) = Green’s functions in Nambu space are given as G αβ ˆ < (τ, τ 0 ) = ih9 † (τ 0 ) ⊗ ∓iθ (±τ ∓ τ 0 )h{9α (τ ), 9 † (τ 0 )}i, G
where HN is the Hamiltonian for an ideal N lead without coupling to the other terminals. HL and HR are the BCS Hamiltonians for the uncoupled superconducting leads, with characteristic gap parameters 1L and 1R , respectively. In the following discussions, we assume the L and R leads are identical and 1L = 1R = 1. In the presence of an applied bias, without losing generality, we set the chemical potential of the R lead to be zero as an energy reference. The chemical potential of the L or N terminals can be tuned by bias voltages as µα = eVα with α = L, N, where e (e < 0) is the charge of the electron. The bias across the two superconductors reads VS = VL − VR . Due to the ac Josephson effect, the bias difference between the two superconductors will induce a time-dependent phase factor in the gap parameter. For the sake of simplicity, we have assumed the constant phase difference between the two leads is zero. It is then convenient to perform a gauge transformation and make the extra phase factor appear in the coupling element between different leads [8, 31]. The gap parameters then become real. The Hamiltonian HT for the tunnel coupling between the leads adopts the following time-dependent form: X X HT = [tαβ (τ )c†ασ cβσ + tβα (τ )c†βσ cασ ], (2)
β
αβ
β
ˆ > (τ, τ 0 ) = −ih9α (τ )⊗9 † (τ 0 )i. Here, the curly 9α (τ )i and G αβ β brackets denote the anticommutator and θ is the Heaviside function. One can easily verify that the greater Green’s ˆ > (τ, τ 0 ) = −σˆ x [G ˆ < (τ 0 , τ )]T , function holds the relation G αβ βα where T stands for transpose and σˆ x is the x component of the Pauli matrix. In terms of the nonequilibrium Green’s function in the Nambu representation, the expectation value of the current operator can be given as ( X e I¯α (τ ) = Tr [ˆvαβ (τ )G< βα (τ, τ ) h¯ β6=α ) − vˆ βα (τ )G< αβ (τ, τ )] ,
σ α,β=L,R,N
∗ (τ ) contains where σ =↑, ↓ is the spin index and tαβ (τ ) = tβα the time-dependent phase factor between the α and β lead. Since the chemical potential of L is shifted by the external bias VS , the coupling element between L and the other leads takes the time-dependent form as 0 iφ(τ )/2 tLα (τ ) = tLα e ,
(4)
(5)
where vˆ αβ (τ ) = σˆ z ˆtαβ (τ ) with σˆ z the z component Pauli matrix and ! tαβ (τ ) 0 ˆtαβ (τ ) = . 0 −tβα (τ )
(3)
The current usually fluctuates around this expectation value. The reasons for this fluctuation include the thermal effect as well as the particle nature of charge transport. As discussed above, the current fluctuation is a versatile tool to reveal desirable information such as the effective charge or the carrier dynamics which are not contained in the conventional current measurement. The magnitude of current fluctuation or current noise can be characterized by the correlation function [14]
with the time-dependent phase factor φ(τ ) = 2eVS τ/h¯ appearing in the coupling elements. As the chemical potential of R is zero, the coupling parameters tNR and tRN are independent of time, i.e., ∗ = t0 , where t0 tNR = tNR αβ is the coupling parameter NR characterizing the normal transmission coefficient of the conducting channel. As discussed by Cuevas et al [31], this coupling parameter characterizes the transparency of the channel for charge transmission. By tuning the parameter, the channel transmission can vary from the tunneling limit (low transparency) to the ballistic limit. To investigate the transport properties of the hybrid junction, we apply the perturbative Green function approach, following Cuevas et al [31]. This approach has the ability to take account of both AR and MAR processes of arbitrary
Sαβ (τ, τ 0 ) = {hδ Iˆα (τ ), δ Iˆβ (τ 0 )i},
(6)
where the δ Iˆα (τ ) = Iˆα (τ ) − I¯α (τ ) depicts the deviation from its expectation value. This correlation function can be either the auto-correlation function for α = β or the cross-correlation function for α 6= β. It is known that, in the presence of correlation effects, the auto-correlation can be 3
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B H Wu et al
super-Poissonian and the cross-correlation can be tuned from negative to positive values. In this work, we focus on the auto-correlation function, where the noise Fano factor can be super-Poissonian since electrons can be transferred in bunches due to the AR or MAR in hybrid structures. By inserting the expression of the current operator, the noise spectral density of the α lead reads Z 0 Sαα (ω, τ ) = h¯ dτ 0 eiω(τ −τ ) hδ Iˆα (τ )δ Iˆα (τ 0 )
over the {0, 0} or dc component of the Floquet matrix. The above equation is the central result of the present study. One can verify that, by setting one coupling parameter to zero, the above noise formula for three-terminal hybrid structures can be reduced to the existing results for S–N or S–S two-terminal hybrid structures. Now the task remaining is to obtain the various nonequilibrium Green’s functions. The lesser or greater Green’s function in Floquet space appearing in evaluating the noise spectrum is obtained by the Keldysh equation in Nambu space as
+ δ Iˆα (τ 0 )δ Iˆα (τ )i Z 0 = h¯ dτ 0 eiω(τ −τ ) Kαα (τ, τ 0 ) + Kαα (τ 0 , τ ) ,
ˆ (ε) = [1 + G ˆ r (ε)6 ˆ a (ε)], (11) ˆ r ]ˆg (ε)[1 + 6 ˆ aG G where the retarded and advanced self-energies are due to the coupling between different leads. In the Nambu space, the self-energies are simply given by
(7)
where Kαα (τ, τ 0 ) =
XXXX β β 0 6=β γ γ 0 6=γ
Tr{ˆvββ 0 (τ )
ˆ r,a = vˆ αβ . 6 αβ
ˆ 0 (τ 0 , τ ) ×G βγ γ β × δβα + δβ 0 α · δγ α + δγ 0 α · ββ 0 γ γ 0 }, (8)
According to the Dyson equation, the retarded and advanced Green’s functions satisfy ˆ r,a (ε) = [(ˆgr,a (ε))−1 − 6 ˆ r,a ]−1 . G
with
ββ 0 γ γ 0
( 1 = −1
In obtaining the noise spectral density, we have to perform the Fourier transform of a double-time function. Due to the ac Josephson effect, the double-time function is not a mere function of its time difference. By observing the fact that the time dependence in the Hamiltonian is a phase factor with a frequency of ωS = eVS /h¯ , the physical quantities will vary in time with the same frequency. According to the Floquet theorem [32], it is convenient to expand the time-dependent quantities in the Floquet basis: |ni = einωS t with n integers. In the same spirit, the Fourier components fn of a double-time function f (τ, τ 0 ) can be given by Z Z 1 T 0 ∞ 0 0 dτ dτ eiω(τ −τ ) e−inωS τ f (τ, τ 0 ), (9) fn (ω) = T 0 −∞
Here, W is an energy scale related to the normal density of states at the Fermi level of the α lead as ρ(F ) ∼ 1/π W and η is a positive small energy relaxation rate that takes into account the damping of the quasiparticle states due to inelastic processes inside the leads [31]. Here, we have assumed W and η are the same for all the three leads. For the normal leads, one can simply set the gap parameter 1α to zero in the above expressions. The unperturbed lesser Green’s functions for the α superconducting lead are given by gˆ < gaα (ε) − gˆ rα (ε)]fα (ε), α (ε) = [ˆ
(15)
where f (ε) is the Fermi function for the α lead after unitary transformation. For the normal lead, the unperturbed lesser Green’s function in Nambu space is given by ! f (ε − µN ) 0 < . (16) gˆ N (ε) = 2π iρN (ε) 0 f (ε + µN )
where T = 2π/ωS is the period determined by the bias voltage across the superconducting leads. By making use of the relation fm,n (ω) = fm−n (ω + mωS ), the Fourier components can be written in a concise matrix form in the Floquet basis. Then, the time-averaged noise spectral power for the periodic system [33] can be written as Z 1 T S¯ αα (ω) = Sαα (ω, τ )dτ T 0 Z X X 2e2 = Tr{ˆvββ 0 h ββ 0 γγ0
While the unperturbed greater Green’s function for the leads can be easily found from the general relation for Green’s functions gˆ r − gˆ a = gˆ < − gˆ > . One can observe that by writing the Green’s functions in Floquet space, the numerical evaluation of the current and noise in hybrid structures requires matrix inversion and matrix multiplication. In principle, the matrix size which is related to the Floquet index should be infinity. In practical calculations, we usually truncate the matrix size to a finite number and make the numerical simulation feasible. However, this truncation relies on the ratio of 1 and VS , which
β6=β 0 γ 6=γ 0
×
(13)
Here, the unperturbed retarded and advanced Green’s functions for the α lead are given in the Nambu space by ! ε ± iη 1α W r,a −1 . (14) [ˆgα (ε)] = p 1α ε ± iη 12α − (ε ± iη)2
for β = γ 0 = α or β 0 = γ = α for β = γ = α or β 0 = γ 0 = α.
ˆ 0 (ε)}00 + ε) · vˆ γ γ 0 · G γ β
· ββ 0 γ γ 0 · (δβα + δβ 0 α )(δγ α + δγ 0 α )dε, (10) ˆ and the hopping element in where the Green’s functions G Nambu space vˆ are given in the Floquet basis. The trace is 4
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B H Wu et al
approximately characterizes the effective MAR order. For very small VS , the matrix size is too large and beyond our numerical ability. Therefore, our interest will be restricted to the range of VS > 0.21 and the truncation of Floquet states is carefully checked to ensure the convergence of the numerical results. 3. Numerical results and discussion
Our main concern here is the current noise, which can provide information of internal dynamics beyond the current measurement. The current noise generally contains contributions from both the thermal fluctuation at nonzero temperature and the shot noise which arises due to the discreteness of charge carriers. In the low-transmission regime, and neglecting the thermal noise, the zero-frequency noise Fano factor F = S/2eI can be associated with the effective charge at each transfer. For instance, for an S–N–S junction and a subgap voltage V, the main contribution to the current is the MAR process of the order of n ∼ 21/eV, where an effective charge of ne is transferred in each process. This multiple effective charge can be clearly visualized in the enhancement of the noise Fano factor [18]. In the following, we will investigate the nonequilibrium noise properties of the multi-terminal hybrid point contact. Both the zero-frequency noise power and the noise spectra will be analyzed. In these numerical studies, we assume the two SC leads and their coupling to the N lead are symmetric, i.e., 1L = 1R = 1 and 0 = t0 = t . We take a small temperature k T = 10−3 1 tLN N B RN to reduce the dephasing effect due to thermal fluctuations. The phenomenological parameter which accounts for the damping effect due to inelastic processes in the leads is η = 10−4 1. The energy scale W = 101 is used throughout the calculation. Without losing generality, we fix the chemical potential of the R lead at zero and the voltages across the L–R and N–R interfaces are denoted as VS and VN , respectively. For the parameters used here, the structures work in the low-transmission regime [31]. We present results of the current noise from the N lead and L lead, respectively. In figure 2, we present numerical results of the Fano factor for the L lead: FS = 2eSLI¯ , where SL and I¯L are the L current noise and the time-averaged current from the L lead. We have set VN = 0 and tS = 31. Without coupling to an additional N lead (tN = 0), our results show multi-step structures in the Fano factor curve, in agreement with the existing results [19]. The steps are direct consequences of MAR and the step edge indicates the onset of the MAR process with different transmission effective charge. However, the coherent MAR, in particular the higher order MAR, becomes very sensitive to the existence of the N lead. By increasing tN , the step structure at lower VS is suppressed while the features at the onset voltage of MAR are still visible. One can notice that a moderate amount (tN = 0.21) of coupling is sufficient to provoke an important deviation with respect to the staircase structure of the Fano factor without coupling. This deviation is most significant at low VS , where the higher order MAR relies on phase coherence in the process. As a result, they become more vulnerable to the
Figure 2. The zero-frequency noise Fano factor FS measured at the
L lead as a function of eVS , where VS is the voltage across the L and R superconducting leads for different coupling parameters tN to the N lead. See text for parameters.
presence of AR at the N–S interface. With a further increase tN , the features at the onset of MAR are smeared out. In our results, the features of FS in the lower VS regime become indistinguishable for tN = tS = 31. Only step structures with reduced heights at VS = 21 and 1 remain visible, since these steps are associated with MAR with fewer reflections at the interface and therefore more robust against the N–S coupling than the higher order processes. These findings are in agreement with previous findings [6] for the current in three-terminal hybrid structures, where the normal lead behaves as a dephasing probe [34–36] to suppress the current due to MAR. Figure 2 shows the coupling to the normal lead will generally destroy the step structures in the Fano factor of the S lead. It would be interesting to investigate how the presence of the pair of S leads affects the noise in the normal lead. For a two-terminal S–N structure, a doubling of the noise Fano factor has long been predicted and observed due to the AR at the N–S interface where twice the electron charge is transmitted. However, in the presence of a pair of S leads at arbitrary bias, the noise behavior is more intriguing since the MAR takes place in addition to the AR. Our results for the noise Fano factor at the N lead FN = 2eSNI¯ as a function of VN N are shown in figure 3. Here, SN and I¯N are respectively the zero frequency noise power and the average current in the N lead. We first take a look at situations where the pair of SC leads are unbiased. The structure is equivalent to an N–S junction. In the calculations, we work in the low-transmission regime and have set tN = 1 and tS = 1.51. In this situation, MAR processes have no contribution to the noise in the N lead. As expected, the numerical results shown in figure 3 are identical to those of an N–S junction. One can see that a step appears at VN = ±1, where FN is doubled, indicating the AR takes place. The Fano factor FN is symmetric with respect to VN due to the electron–hole symmetry. Moreover, a singular value of FN at VN = 0 is observed. This is due to the finite thermal noise at VN = 0, while the current I¯N is 5
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Figure 4. The noise spectra SN (ω) measured at the N lead for
Figure 3. The zero-frequency noise Fano factor FN measured at the
N lead as a function of eVN , where VN is the voltage applied to the N lead for different eVS . See text for the other parameters.
different eVS . The voltage applied to the N lead is fixed at eVN = 0.31. See text for the other parameters.
suppressed. When the pair of S leads are biased at an arbitrary voltage, charge carriers transmitted to or from the N lead can experience several AR with different characteristic energies. As a result, the Fano factor FN displays rich structures as a function of VN . As can be seen from figure 3, FN becomes asymmetric with respect to VN for arbitrary VS . This asymmetry originates from the difference of dominant transport processes. For eVN ∼ 1, the main transport process is the AR between the N–R interface where the effective charge is doubled. For eVN ∼ −1, the main contribution to the current IN is the direct tunneling of electrons from the L lead to the N lead, where the effective charge remains unchanged. As a result, the Fano factor at eVN = −1 becomes smaller than that at eVN = 1. A similar analysis shows that the other features in the FN curve as a function of VN can be ascribed to particular processes at the N–S interfaces. For example, the moving peak for eVN > 1 in figure 3 is related to the onset of a process where an electron from the normal lead can directly tunnel to the upper edge of the energy gap at the left lead. The corresponding onset bias voltage eVN is 1 + eVS , which shifts with increasing VS as shown in the figure. From figure 3, one can also identify pronounced features appearing at eVN = eVS − 1. This feature arises due to the fact that direct tunneling of electrons from the L lead to the N lead is energetically allowed at this bias voltage. Due to the three-terminal configuration, the electrons injected from the normal lead can undergo several AR at the interfaces of the terminals. For instance, an electron from the normal lead can take two AR at the N–R and L–R interface before it arrives at the upper gap edge of the R lead. During this process, the electron first obtains an energy of −eVN and then 2eVS to overcome the SC gap 1. Signatures of these processes can clearly be identified in figure 3 at eVN = 2eVS − 1. Another interesting observation is the large FN for 0 < eVN < eVS . This large Fano factor is not mainly due to higher order MAR as in figure 2. For this bias scheme, the chemical potential of the normal lead is between those of the L and R leads. The current from the N lead to the L lead and the
R lead can partly cancel each other. However, the current fluctuation in the N lead remains pronounced. In this sense, the pair of S leads act as a nonequilibrium thermal reservoir. As a result, the Fano factor in the N lead is greatly enhanced. It should be noted that similar behaviors, such as the shift of the divergence of the noise Fano factor as well as its enhancement in three-terminal devices, have also been found in a quantum dot coupled to two electronic leads and a thermal lead [37]. In the above, we have shown that the Fano factor of zero-frequency noise can clearly characterize the change of effective charge at the onset voltages of either AR or MAR processes. Now we present results of the noise spectrum of multi-terminal hybrid structures. As most previous studies in hybrid structures focus on the current and zero-frequency noise properties, the noise spectrum S(ω) receives little attention. In the following, we show the noise spectrum can give direct information on the carrier dynamics in the nanostructures. We take the following parameters as tN = 0.51 and tS = 1.51 in numerical simulations. The voltage bias at the normal lead is VN = 0.31/|e|. By varying the voltage across the SC leads VS , we present the noise spectrum for auto-correlations of the current at the N and L lead, respectively, in figures 4 and 5. In figure 4, we show the noise spectra SN (ω) for different eVS at fixed eVN . One can see that in the large-frequency limit, the noise spectra are almost independent of the variation of the voltage applied at the S leads. However, some distinct features arise in the low-frequency regime. Dotted lines are present to indicate the evolution of the positions of both fixed and moving features. Analysis shows that these features are related to the resonant frequencies in the nanostructure. As indicated in figure 4, one can show that the features at 1 ± eVN are related to the resonances of the Fermi level of the N lead and the lower and upper gap edges of the R lead, respectively. Similarly, the features at 1 ± e(VN − eVS ) are due to the resonances of the normal lead Fermi level with the gap edges of the L lead. These features move with changing 6
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rich features corresponding to the energy difference of the dynamical processes. The interplay of the AR and MAR in the multi-terminal hybrid devices can be clearly identified in its noise spectrum properties. Acknowledgments
This work was supported by the Natural Science Foundation of China (Grant Nos 11074266, 11174049, 61290301) and the Fundamental Research Funds for the Central Universities. References [1] Lambert C J and Raimondi R 1998 J. Phys.: Condens. Matter 10 901 [2] Martin-Rodero A and Yeyati A L 2011 Adv. Phys. 60 899 [3] Tinkham M 1996 Introduction to Superconductivity (Singapore: McGraw-Hill) [4] Houzet M and Samuelsson P 2010 Phys. Rev. B 82 060517 [5] Jonckheere T, Rech J, Martin T, Doucot B, Feinberg D and Melin R 2013 Phys. Rev. B 87 214501 [6] Jonckheere T, Zazunov A, Bayandin K V, Shumeiko V and Martin T 2009 Phys. Rev. B 80 184510 [7] van Wees B J, Lenssen K-M H and Harmans C J P M 1991 Phys. Rev. B 44 470 [8] Sun Q-F, Wang J and Lin T-H 2000 Phys. Rev. B 62 648 [9] Morpurgo A F, Klapwijk T M and van Wees B J 1998 Appl. Phys. Lett. 72 966 [10] Baselmans J J A, Morpurgo A F, van Wees B J and Klapwijk T M 1999 Nature 397 43 [11] Huang J, Pierre F, Heikkil¨a T T, Wilhelm F K and Birge N O 2002 Phys. Rev. B 66 020507 [12] Ilhan H T, Demir H V and Bagwell P F 1998 Phys. Rev. B 58 15120 [13] Pillet J-D, Quay C, Morfin P, Bena C, Yeyati A L and Joyez P 2010 Nature Phys. 6 965 [14] Blanter Ya M and B¨uttiker M 2000 Phys. Rep. 336 1 [15] Reulet B, Prober D E and Belzig W 2003 Quantum Noise in Mesoscopic Physics ed Y V Nazarov (Dordrecht: Kluwer) [16] Dieleman P, Bukkems H G, Klapwijk T M, Schicke M and Gundlach K H 1997 Phys. Rev. Lett. 79 3486 [17] Jehl X, Sanquer M, Calemczuk R and Mailly D 2000 Nature 405 50 [18] Cron R, Goffman M F, Esteve D and Urbina C 2001 Phys. Rev. Lett. 86 4104 [19] Cuevas J, Martin-Rodero A and Yeyati A L 1999 Phys. Rev. Lett. 82 4086 [20] Maisi V F, Kambly D, Flindt C and Pekola J P 2013 in preparation (arXiv:1307.4176) [21] Aguado R and Kouwenhoven L P 2000 Phys. Rev. Lett. 84 1986 [22] Luo J Y, Li X Q and Yan Y J 2007 Phys. Rev. B 76 085325 [23] Dong B, Lei X L and Horing N J M 2008 J. Appl. Phys. 104 033532 [24] Entin-Wohlman O, Imry Y, Gurvitz S A and Aharony A 2007 Phys. Rev. B 75 193308 [25] Wu B H and Cao J C 2010 Phys. Rev. B 81 125326 [26] Cottet A, Doucot B and Belzig W 2008 Phys. Rev. Lett. 101 257001 [27] Badiane D M, Houzet M and Meyer J S 2011 Phys. Rev. Lett. 107 177002 [28] Recher P, Sukhorukov E V and Loss D 2001 Phys. Rev. B 63 165314
Figure 5. The noise spectra SS (ω) measured at the L lead for
different eVS . The curves are vertically shifted for the sake of clarity. The parameters are the same as in figure 4.
bias voltage VS in our numerical results. A smaller feature at 21 is also identified in the results. This feature is insensitive to the voltage change and can be ascribed to the resonance of the two gap edges of the SC leads. Now, we investigate the noise spectra at the L lead. The numerical results of the noise spectra for varying VS are presented in figure 5. For the sake of clarity, the curves are vertically shifted and the evolutions of the fixed or moving feature positions are indicated by dotted lines. One can see that more pronounced peak features arise in the noise spectra. These pronounced peaks can be ascribed to the contribution from MAR processes. The corresponding frequencies are the differences of energies for MAR and the superconducting gap as h¯ ω = 21 − neVS , where n = ±1 and 2 are shown in the figure. The fixed feature at h¯ ω = 21 is significant here, in contrast to the results for the N lead shown in figure 4. There are also features due to the AR in the three leads which involve the energy scale of the N lead. However, these features are weak for the present parameters. One can find features at a frequency hω ¯ = 1−e(VN +VS ), as indicated in the figure after careful investigation. The features discussed above have also been checked by varying eVN in our simulation. Therefore, the noise spectra give clear information on the energy scales of internal dynamics and can serve as a novel probe for this information. 4. Conclusion
In conclusion, we investigated the noise properties of three-terminal hybrid structures at arbitrary bias voltages. The zero-frequency noise shows that the higher order MAR processes are sensitive to the presence of the AR at the N–S interface. As a result, the multiple-step structures in the noise Fano factor will be suppressed. The AR at the interfaces of the N and S leads gives rise to new features in the noise Fano factor measured at the N lead. We also investigated the noise spectrum at finite frequency. Our results show that the internal dynamics of the multi-terminal hybrid structure raises 7
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[34] B¨uttiker M 1988 IBM J. Res. Dev. 32 63 [35] de Jong M J M and Beenakker C W J 1996 Physica A 230 219 [36] F¨orster H, Samuelsson P, Pilgram S and B¨uttiker M 2007 Phys. Rev. B 75 035340 [37] S´anchez R, Sothmann B, Jordan A N and B¨uttiker M 2013 in preparation (arXiv:1307.0598)
[29] B¨orlin J, Belzig W and Bruder C 2002 Phys. Rev. Lett. 88 197001 [30] Wei J and Chandrasekhar V 2010 Nature Phys. 6 494 [31] Cuevas J, Martin-Rodero A and Yeyati A L 1996 Phys. Rev. B 54 7366 [32] Shirly J H 1965 Phys. Rev. 138 B979 [33] Kohler S, Lehnman J and H¨anggi P 2005 Phys. Rep. 406 379
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Current noise in three-terminal hybrid quantum point contacts
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 025301 (8pp)
doi:10.1088/0953-8984/26/2/025301
Current noise in three-terminal hybrid quantum point contacts B H Wu1 , C R Wang1 , X S Chen1,2 and G J Xu3 1
Department of Applied Physics, Donghua University, 2999 North Renmin Road, Shanghai 201620, People’s Republic of China 2 National Laboratory for Infrared Physics, Shanghai Institute of Technical Physics, Chinese Academy of Science, Shanghai 200083, People’s Republic of China 3 Engineering Research Center of Optical Instrument and System, Ministry of Education, Shanghai Key Lab of Modern Optical System, University of Shanghai for Science and Technology, 516 Jungong Road, Shanghai 200093, People’s Republic of China E-mail: [email protected] Received 5 August 2013, in final form 17 October 2013 Published 5 December 2013 Abstract
We investigate the current noise of three-terminal hybrid structures at arbitrary bias voltages. Our results indicate that the noise can be a useful tool to extract dynamical information in multi-terminal hybrid structures. The zero-frequency noise is sensitive to the coupling with a normal lead. As a result, the characteristic multiple-step structure of the noise Fano factor due to multiple Andreev reflection will be suppressed as we increase this coupling. In addition, the internal dynamics due to processes of Andreev reflection and multiple Andreev reflection raises rich features in the noise spectrum corresponding to the energy differences of various dynamical processes. (Some figures may appear in colour only in the online journal)
1. Introduction
gap. In this multiple AR (MAR), the charge is transferred in amounts of ne, where n is an integer and determined by the threshold voltage Vn = 21/ne. While most previous studies on transport in hybrid structures have concerned two-terminal configurations, multiterminal geometries with several S or N leads are also under current interest for quantum information technology as they can provide additional degrees of freedom to externally modulate the transport properties. In multi-terminal structures, the interplay of AR and MAR involving several interfaces can give rise to rich subgap transport features [4, 5]. Among the various configurations, the simplest structure where both AR and MAR contribute to the transport properties is a three-terminal junction where two S leads and one N lead are connected by normal conducting channels. Jonckheere et al [6] have shown that the coupling to an additional N lead can generally be considered as a source of dephasing. By tuning the voltage at the N lead, the magnitude and sign of the Josephson current between two superconducting leads can be well controlled [7–11]. The N lead can also be a powerful probe to reveal the
Transport in hybrid structures with both normal (N) and superconductor (S) materials has attracted much recent interest due to their potential as the building blocks of solid state quantum information technology [1, 2]. When the energy of an injected charge carrier lies in the superconducting gap 1, a direct transfer of the charged particle is energetically prohibited. Nonequilibrium transport in the hybrid structures then evolves distinct mechanisms where charges are transferred in bunches in the low-bias V limit [3]. For a two-terminal structure formed by N and S terminals, the Andreev reflection (AR) dominates the charge transfer when V is larger than 1. In each AR process, an electron from N crossing the N–S interface is only permitted by leaving behind a hole in the N region and at the same time creating a Cooper pair in the S region. In that way, a total of 2e is transferred in one shot, where e is the electron charge. For devices where two S terminals are connected via a short normal conductor, charged particles can undergo several ARs at the two N–S interfaces to overcome the superconducting 0953-8984/14/025301+08$33.00
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spectroscopic information of the central region [12]. A recent experiment [13] has demonstrated the first spectroscopy of the evolution of Andreev bound states in a carbon nanotube based three-terminal hybrid device. As explained above, AR and MAR play primary roles in the transport of hybrid structures and they have distinct effective charges during carrier transmission. The effective charge in AR is two, while it can, in principle, be any integer in MAR, depending on the bias voltage. A direct probe of the effective charge is beyond the usual current I measurement. In his seminal work, Schottky [14] has shown that the effective charge q of each transfer can be identified from the measurement of the zero-frequency power spectrum of the current fluctuation or shot noise S at the low transmission limit as q/e = S/2I. An enhanced shot noise due to the AR or MAR mechanisms has indeed been experimentally demonstrated in hybrid systems with various configurations [15–18]. For example, Jehl et al [17] have demonstrated doubled shot noise with respect to the normal case due to the AR at the N–S interface. A large effective charge due to MAR has also been observed by Cron et al [18] in structures of two superconductors connected by atomic point contact, where the effective charge displays staircase behavior at threshold voltages of the MAR processes. Good agreement between the experimental results and microscopic theory has also been achieved [19]. In a recent work, noise in N–S hybrid structures has been measured by the full counting statistics, where super-Poissonian noise is detected due to the interplay of single-particle tunneling and the AR [20]. By going beyond the zero-frequency limit, the current noise at finite frequency can provide more interesting information, such as the coherence and dynamics of nanostructures [21–25]. However, the noise spectrum in hybrid structures has raised little attention. In this direction, Cottet et al [26] have shown that the noise spectrum of a superconductor–ferromagnet quantum point contact is sensitive to the spin dependence of interfacial phase shift and can thus be used to characterize the scattering properties in this hybrid structure. More recently, Badiane et al [27] found that the noise spectrum can directly give evidence of the 4π periodic in topological Josephson junctions while signatures of such a fractional Josephson effect are absent from the measurement of the averaged current or the zero-frequency noise. Regarding the noise in multi-terminal hybrid structures, it has been studied both theoretically and experimentally for the purpose of creating nonlocal entangled electron pairs in a configuration with two S leads and one N lead [28–30]. There, the main concern is the zero-frequency noise and the transport is stationary, with the AR process dominating the low-bias regime. However, the noise and its spectral information in multi-terminal structures with more than one S lead has not been considered so far. The analysis is complicated, since at arbitrary bias in these structures, the MAR processes contribute to the current and the transport is no longer stationary due to the ac Josephson effect. It is the purpose of the present paper to investigate the noise properties in multi-terminal hybrid structures. We will
Figure 1. Schematic plot of the multi-terminal hybrid structure with
superconducting left (L) and right (R) leads and a normal (N) lead. The three terminals are connected via conducting channels of 0 quantum point contact with coupling parameters tαβ , where α, β = L, R or N.
consider a minimal model with three terminals connected via point contact, as shown in figure 1. All the leads are connected via coherent point contact and at arbitrary bias voltages. As a result, the transport will be time periodic, with frequencies determined by the Josephson relation, in contrast to the stationary transport in previous studies. A similar structure where a quantum dot is coupled to a pair of SC leads and a normal lead has been recently thoroughly investigated by Jonckheere et al [6]. There, the transport properties are restricted to the dc and ac current profile. It is shown that the normal lead can act as a source of dephasing. With increasing dot coupling to the N lead, the transport displays crossover from zero dephasing to the incoherent case. As a result, the harmonics of the ac current will be reduced. However, much less attention has been paid to studying the noise of the Josephson current. By using a microscopic Hamiltonian approach developed by Cuevas et al [31], we present results for the noise properties of the hybrid junction. Our results show that the interplay of AR and MAR can induce rich features in both the zero-frequency noise power and the noise spectrum. It is found that the noise signatures of MAR, in particularly the higher order processes, are sensitive to the coupling to the N lead. In addition, the noise spectrum in the hybrid structures clearly exhibits energy differences associated with the AR and MAR processes. Therefore, the noise properties of hybrid structures can be a useful tool to extract the dynamical information in multi-terminal hybrid structures. The paper is organized as follows. In section 2, we explain our model structure and the theoretical formalism for the noise properties of hybrid structures. Section 3 presents and discusses our numerical results of the zero-frequency noise and noise spectrum for the hybrid structures. Finally, section 4 briefly summarizes our results. 2. Model and formalism
The device under consideration is a three-terminal hybrid structure. A schematic representation of the model is depicted in figure 1. The left (L) and right (R) terminals are S and the 2
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order. Within our model, the current from the α lead, for example, is given by the time evolution of its total electron number. By evaluating the equation of motion of the total electron number in the α lead, we arrive at the operator for the current from the α lead as ie X X Iα (τ ) = −eN˙ α (τ ) = [tαβ c†ασ (τ )cβσ (τ ) h¯ σ β6=α
third one is N. All the terminals are connected via quantum transport channels. Experimentally, these quantum transport channels can, for example, be either atomic size contact realized in break junctions or quantum point contact involving constricted two-dimensional electron gases in semiconductor heterostructures. For the sake of simplicity, we represent the quantum transport channels between two terminals with a single conduction channel. The Hamiltonian for the system can be written as [31] H = HN + HL + HR + HT ,
− tβα c†βσ (τ )cασ (τ )].
(1)
To proceed further, it is convenient to make use of the operator in Nambu space by 9α = (c†α↑ , cα↓ )† . The corresponding Green’s function in the Keldysh contour is then ˆ αβ (t1 , t2 ) = ihTC (9α (t1 ) ⊗ defined as a 2 × 2 matrix as G † 9β (t2 ))i, where TC is the contour-order operator. This contour-ordered Green’s function is related to four real-time nonequilibrium Green’s functions. More specifically, the retarded (r), advanced (a), lesser () ˆ r/a (τ, τ 0 ) = Green’s functions in Nambu space are given as G αβ ˆ < (τ, τ 0 ) = ih9 † (τ 0 ) ⊗ ∓iθ (±τ ∓ τ 0 )h{9α (τ ), 9 † (τ 0 )}i, G
where HN is the Hamiltonian for an ideal N lead without coupling to the other terminals. HL and HR are the BCS Hamiltonians for the uncoupled superconducting leads, with characteristic gap parameters 1L and 1R , respectively. In the following discussions, we assume the L and R leads are identical and 1L = 1R = 1. In the presence of an applied bias, without losing generality, we set the chemical potential of the R lead to be zero as an energy reference. The chemical potential of the L or N terminals can be tuned by bias voltages as µα = eVα with α = L, N, where e (e < 0) is the charge of the electron. The bias across the two superconductors reads VS = VL − VR . Due to the ac Josephson effect, the bias difference between the two superconductors will induce a time-dependent phase factor in the gap parameter. For the sake of simplicity, we have assumed the constant phase difference between the two leads is zero. It is then convenient to perform a gauge transformation and make the extra phase factor appear in the coupling element between different leads [8, 31]. The gap parameters then become real. The Hamiltonian HT for the tunnel coupling between the leads adopts the following time-dependent form: X X HT = [tαβ (τ )c†ασ cβσ + tβα (τ )c†βσ cασ ], (2)
β
αβ
β
ˆ > (τ, τ 0 ) = −ih9α (τ )⊗9 † (τ 0 )i. Here, the curly 9α (τ )i and G αβ β brackets denote the anticommutator and θ is the Heaviside function. One can easily verify that the greater Green’s ˆ > (τ, τ 0 ) = −σˆ x [G ˆ < (τ 0 , τ )]T , function holds the relation G αβ βα where T stands for transpose and σˆ x is the x component of the Pauli matrix. In terms of the nonequilibrium Green’s function in the Nambu representation, the expectation value of the current operator can be given as ( X e I¯α (τ ) = Tr [ˆvαβ (τ )G< βα (τ, τ ) h¯ β6=α ) − vˆ βα (τ )G< αβ (τ, τ )] ,
σ α,β=L,R,N
∗ (τ ) contains where σ =↑, ↓ is the spin index and tαβ (τ ) = tβα the time-dependent phase factor between the α and β lead. Since the chemical potential of L is shifted by the external bias VS , the coupling element between L and the other leads takes the time-dependent form as 0 iφ(τ )/2 tLα (τ ) = tLα e ,
(4)
(5)
where vˆ αβ (τ ) = σˆ z ˆtαβ (τ ) with σˆ z the z component Pauli matrix and ! tαβ (τ ) 0 ˆtαβ (τ ) = . 0 −tβα (τ )
(3)
The current usually fluctuates around this expectation value. The reasons for this fluctuation include the thermal effect as well as the particle nature of charge transport. As discussed above, the current fluctuation is a versatile tool to reveal desirable information such as the effective charge or the carrier dynamics which are not contained in the conventional current measurement. The magnitude of current fluctuation or current noise can be characterized by the correlation function [14]
with the time-dependent phase factor φ(τ ) = 2eVS τ/h¯ appearing in the coupling elements. As the chemical potential of R is zero, the coupling parameters tNR and tRN are independent of time, i.e., ∗ = t0 , where t0 tNR = tNR αβ is the coupling parameter NR characterizing the normal transmission coefficient of the conducting channel. As discussed by Cuevas et al [31], this coupling parameter characterizes the transparency of the channel for charge transmission. By tuning the parameter, the channel transmission can vary from the tunneling limit (low transparency) to the ballistic limit. To investigate the transport properties of the hybrid junction, we apply the perturbative Green function approach, following Cuevas et al [31]. This approach has the ability to take account of both AR and MAR processes of arbitrary
Sαβ (τ, τ 0 ) = {hδ Iˆα (τ ), δ Iˆβ (τ 0 )i},
(6)
where the δ Iˆα (τ ) = Iˆα (τ ) − I¯α (τ ) depicts the deviation from its expectation value. This correlation function can be either the auto-correlation function for α = β or the cross-correlation function for α 6= β. It is known that, in the presence of correlation effects, the auto-correlation can be 3
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B H Wu et al
super-Poissonian and the cross-correlation can be tuned from negative to positive values. In this work, we focus on the auto-correlation function, where the noise Fano factor can be super-Poissonian since electrons can be transferred in bunches due to the AR or MAR in hybrid structures. By inserting the expression of the current operator, the noise spectral density of the α lead reads Z 0 Sαα (ω, τ ) = h¯ dτ 0 eiω(τ −τ ) hδ Iˆα (τ )δ Iˆα (τ 0 )
over the {0, 0} or dc component of the Floquet matrix. The above equation is the central result of the present study. One can verify that, by setting one coupling parameter to zero, the above noise formula for three-terminal hybrid structures can be reduced to the existing results for S–N or S–S two-terminal hybrid structures. Now the task remaining is to obtain the various nonequilibrium Green’s functions. The lesser or greater Green’s function in Floquet space appearing in evaluating the noise spectrum is obtained by the Keldysh equation in Nambu space as
+ δ Iˆα (τ 0 )δ Iˆα (τ )i Z 0 = h¯ dτ 0 eiω(τ −τ ) Kαα (τ, τ 0 ) + Kαα (τ 0 , τ ) ,
ˆ (ε) = [1 + G ˆ r (ε)6 ˆ a (ε)], (11) ˆ r ]ˆg (ε)[1 + 6 ˆ aG G where the retarded and advanced self-energies are due to the coupling between different leads. In the Nambu space, the self-energies are simply given by
(7)
where Kαα (τ, τ 0 ) =
XXXX β β 0 6=β γ γ 0 6=γ
Tr{ˆvββ 0 (τ )
ˆ r,a = vˆ αβ . 6 αβ
ˆ 0 (τ 0 , τ ) ×G βγ γ β × δβα + δβ 0 α · δγ α + δγ 0 α · ββ 0 γ γ 0 }, (8)
According to the Dyson equation, the retarded and advanced Green’s functions satisfy ˆ r,a (ε) = [(ˆgr,a (ε))−1 − 6 ˆ r,a ]−1 . G
with
ββ 0 γ γ 0
( 1 = −1
In obtaining the noise spectral density, we have to perform the Fourier transform of a double-time function. Due to the ac Josephson effect, the double-time function is not a mere function of its time difference. By observing the fact that the time dependence in the Hamiltonian is a phase factor with a frequency of ωS = eVS /h¯ , the physical quantities will vary in time with the same frequency. According to the Floquet theorem [32], it is convenient to expand the time-dependent quantities in the Floquet basis: |ni = einωS t with n integers. In the same spirit, the Fourier components fn of a double-time function f (τ, τ 0 ) can be given by Z Z 1 T 0 ∞ 0 0 dτ dτ eiω(τ −τ ) e−inωS τ f (τ, τ 0 ), (9) fn (ω) = T 0 −∞
Here, W is an energy scale related to the normal density of states at the Fermi level of the α lead as ρ(F ) ∼ 1/π W and η is a positive small energy relaxation rate that takes into account the damping of the quasiparticle states due to inelastic processes inside the leads [31]. Here, we have assumed W and η are the same for all the three leads. For the normal leads, one can simply set the gap parameter 1α to zero in the above expressions. The unperturbed lesser Green’s functions for the α superconducting lead are given by gˆ < gaα (ε) − gˆ rα (ε)]fα (ε), α (ε) = [ˆ
(15)
where f (ε) is the Fermi function for the α lead after unitary transformation. For the normal lead, the unperturbed lesser Green’s function in Nambu space is given by ! f (ε − µN ) 0 < . (16) gˆ N (ε) = 2π iρN (ε) 0 f (ε + µN )
where T = 2π/ωS is the period determined by the bias voltage across the superconducting leads. By making use of the relation fm,n (ω) = fm−n (ω + mωS ), the Fourier components can be written in a concise matrix form in the Floquet basis. Then, the time-averaged noise spectral power for the periodic system [33] can be written as Z 1 T S¯ αα (ω) = Sαα (ω, τ )dτ T 0 Z X X 2e2 = Tr{ˆvββ 0 h ββ 0 γγ0
While the unperturbed greater Green’s function for the leads can be easily found from the general relation for Green’s functions gˆ r − gˆ a = gˆ < − gˆ > . One can observe that by writing the Green’s functions in Floquet space, the numerical evaluation of the current and noise in hybrid structures requires matrix inversion and matrix multiplication. In principle, the matrix size which is related to the Floquet index should be infinity. In practical calculations, we usually truncate the matrix size to a finite number and make the numerical simulation feasible. However, this truncation relies on the ratio of 1 and VS , which
β6=β 0 γ 6=γ 0
×
(13)
Here, the unperturbed retarded and advanced Green’s functions for the α lead are given in the Nambu space by ! ε ± iη 1α W r,a −1 . (14) [ˆgα (ε)] = p 1α ε ± iη 12α − (ε ± iη)2
for β = γ 0 = α or β 0 = γ = α for β = γ = α or β 0 = γ 0 = α.
ˆ 0 (ε)}00 + ε) · vˆ γ γ 0 · G γ β
· ββ 0 γ γ 0 · (δβα + δβ 0 α )(δγ α + δγ 0 α )dε, (10) ˆ and the hopping element in where the Green’s functions G Nambu space vˆ are given in the Floquet basis. The trace is 4
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B H Wu et al
approximately characterizes the effective MAR order. For very small VS , the matrix size is too large and beyond our numerical ability. Therefore, our interest will be restricted to the range of VS > 0.21 and the truncation of Floquet states is carefully checked to ensure the convergence of the numerical results. 3. Numerical results and discussion
Our main concern here is the current noise, which can provide information of internal dynamics beyond the current measurement. The current noise generally contains contributions from both the thermal fluctuation at nonzero temperature and the shot noise which arises due to the discreteness of charge carriers. In the low-transmission regime, and neglecting the thermal noise, the zero-frequency noise Fano factor F = S/2eI can be associated with the effective charge at each transfer. For instance, for an S–N–S junction and a subgap voltage V, the main contribution to the current is the MAR process of the order of n ∼ 21/eV, where an effective charge of ne is transferred in each process. This multiple effective charge can be clearly visualized in the enhancement of the noise Fano factor [18]. In the following, we will investigate the nonequilibrium noise properties of the multi-terminal hybrid point contact. Both the zero-frequency noise power and the noise spectra will be analyzed. In these numerical studies, we assume the two SC leads and their coupling to the N lead are symmetric, i.e., 1L = 1R = 1 and 0 = t0 = t . We take a small temperature k T = 10−3 1 tLN N B RN to reduce the dephasing effect due to thermal fluctuations. The phenomenological parameter which accounts for the damping effect due to inelastic processes in the leads is η = 10−4 1. The energy scale W = 101 is used throughout the calculation. Without losing generality, we fix the chemical potential of the R lead at zero and the voltages across the L–R and N–R interfaces are denoted as VS and VN , respectively. For the parameters used here, the structures work in the low-transmission regime [31]. We present results of the current noise from the N lead and L lead, respectively. In figure 2, we present numerical results of the Fano factor for the L lead: FS = 2eSLI¯ , where SL and I¯L are the L current noise and the time-averaged current from the L lead. We have set VN = 0 and tS = 31. Without coupling to an additional N lead (tN = 0), our results show multi-step structures in the Fano factor curve, in agreement with the existing results [19]. The steps are direct consequences of MAR and the step edge indicates the onset of the MAR process with different transmission effective charge. However, the coherent MAR, in particular the higher order MAR, becomes very sensitive to the existence of the N lead. By increasing tN , the step structure at lower VS is suppressed while the features at the onset voltage of MAR are still visible. One can notice that a moderate amount (tN = 0.21) of coupling is sufficient to provoke an important deviation with respect to the staircase structure of the Fano factor without coupling. This deviation is most significant at low VS , where the higher order MAR relies on phase coherence in the process. As a result, they become more vulnerable to the
Figure 2. The zero-frequency noise Fano factor FS measured at the
L lead as a function of eVS , where VS is the voltage across the L and R superconducting leads for different coupling parameters tN to the N lead. See text for parameters.
presence of AR at the N–S interface. With a further increase tN , the features at the onset of MAR are smeared out. In our results, the features of FS in the lower VS regime become indistinguishable for tN = tS = 31. Only step structures with reduced heights at VS = 21 and 1 remain visible, since these steps are associated with MAR with fewer reflections at the interface and therefore more robust against the N–S coupling than the higher order processes. These findings are in agreement with previous findings [6] for the current in three-terminal hybrid structures, where the normal lead behaves as a dephasing probe [34–36] to suppress the current due to MAR. Figure 2 shows the coupling to the normal lead will generally destroy the step structures in the Fano factor of the S lead. It would be interesting to investigate how the presence of the pair of S leads affects the noise in the normal lead. For a two-terminal S–N structure, a doubling of the noise Fano factor has long been predicted and observed due to the AR at the N–S interface where twice the electron charge is transmitted. However, in the presence of a pair of S leads at arbitrary bias, the noise behavior is more intriguing since the MAR takes place in addition to the AR. Our results for the noise Fano factor at the N lead FN = 2eSNI¯ as a function of VN N are shown in figure 3. Here, SN and I¯N are respectively the zero frequency noise power and the average current in the N lead. We first take a look at situations where the pair of SC leads are unbiased. The structure is equivalent to an N–S junction. In the calculations, we work in the low-transmission regime and have set tN = 1 and tS = 1.51. In this situation, MAR processes have no contribution to the noise in the N lead. As expected, the numerical results shown in figure 3 are identical to those of an N–S junction. One can see that a step appears at VN = ±1, where FN is doubled, indicating the AR takes place. The Fano factor FN is symmetric with respect to VN due to the electron–hole symmetry. Moreover, a singular value of FN at VN = 0 is observed. This is due to the finite thermal noise at VN = 0, while the current I¯N is 5
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B H Wu et al
Figure 4. The noise spectra SN (ω) measured at the N lead for
Figure 3. The zero-frequency noise Fano factor FN measured at the
N lead as a function of eVN , where VN is the voltage applied to the N lead for different eVS . See text for the other parameters.
different eVS . The voltage applied to the N lead is fixed at eVN = 0.31. See text for the other parameters.
suppressed. When the pair of S leads are biased at an arbitrary voltage, charge carriers transmitted to or from the N lead can experience several AR with different characteristic energies. As a result, the Fano factor FN displays rich structures as a function of VN . As can be seen from figure 3, FN becomes asymmetric with respect to VN for arbitrary VS . This asymmetry originates from the difference of dominant transport processes. For eVN ∼ 1, the main transport process is the AR between the N–R interface where the effective charge is doubled. For eVN ∼ −1, the main contribution to the current IN is the direct tunneling of electrons from the L lead to the N lead, where the effective charge remains unchanged. As a result, the Fano factor at eVN = −1 becomes smaller than that at eVN = 1. A similar analysis shows that the other features in the FN curve as a function of VN can be ascribed to particular processes at the N–S interfaces. For example, the moving peak for eVN > 1 in figure 3 is related to the onset of a process where an electron from the normal lead can directly tunnel to the upper edge of the energy gap at the left lead. The corresponding onset bias voltage eVN is 1 + eVS , which shifts with increasing VS as shown in the figure. From figure 3, one can also identify pronounced features appearing at eVN = eVS − 1. This feature arises due to the fact that direct tunneling of electrons from the L lead to the N lead is energetically allowed at this bias voltage. Due to the three-terminal configuration, the electrons injected from the normal lead can undergo several AR at the interfaces of the terminals. For instance, an electron from the normal lead can take two AR at the N–R and L–R interface before it arrives at the upper gap edge of the R lead. During this process, the electron first obtains an energy of −eVN and then 2eVS to overcome the SC gap 1. Signatures of these processes can clearly be identified in figure 3 at eVN = 2eVS − 1. Another interesting observation is the large FN for 0 < eVN < eVS . This large Fano factor is not mainly due to higher order MAR as in figure 2. For this bias scheme, the chemical potential of the normal lead is between those of the L and R leads. The current from the N lead to the L lead and the
R lead can partly cancel each other. However, the current fluctuation in the N lead remains pronounced. In this sense, the pair of S leads act as a nonequilibrium thermal reservoir. As a result, the Fano factor in the N lead is greatly enhanced. It should be noted that similar behaviors, such as the shift of the divergence of the noise Fano factor as well as its enhancement in three-terminal devices, have also been found in a quantum dot coupled to two electronic leads and a thermal lead [37]. In the above, we have shown that the Fano factor of zero-frequency noise can clearly characterize the change of effective charge at the onset voltages of either AR or MAR processes. Now we present results of the noise spectrum of multi-terminal hybrid structures. As most previous studies in hybrid structures focus on the current and zero-frequency noise properties, the noise spectrum S(ω) receives little attention. In the following, we show the noise spectrum can give direct information on the carrier dynamics in the nanostructures. We take the following parameters as tN = 0.51 and tS = 1.51 in numerical simulations. The voltage bias at the normal lead is VN = 0.31/|e|. By varying the voltage across the SC leads VS , we present the noise spectrum for auto-correlations of the current at the N and L lead, respectively, in figures 4 and 5. In figure 4, we show the noise spectra SN (ω) for different eVS at fixed eVN . One can see that in the large-frequency limit, the noise spectra are almost independent of the variation of the voltage applied at the S leads. However, some distinct features arise in the low-frequency regime. Dotted lines are present to indicate the evolution of the positions of both fixed and moving features. Analysis shows that these features are related to the resonant frequencies in the nanostructure. As indicated in figure 4, one can show that the features at 1 ± eVN are related to the resonances of the Fermi level of the N lead and the lower and upper gap edges of the R lead, respectively. Similarly, the features at 1 ± e(VN − eVS ) are due to the resonances of the normal lead Fermi level with the gap edges of the L lead. These features move with changing 6
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rich features corresponding to the energy difference of the dynamical processes. The interplay of the AR and MAR in the multi-terminal hybrid devices can be clearly identified in its noise spectrum properties. Acknowledgments
This work was supported by the Natural Science Foundation of China (Grant Nos 11074266, 11174049, 61290301) and the Fundamental Research Funds for the Central Universities. References [1] Lambert C J and Raimondi R 1998 J. Phys.: Condens. Matter 10 901 [2] Martin-Rodero A and Yeyati A L 2011 Adv. Phys. 60 899 [3] Tinkham M 1996 Introduction to Superconductivity (Singapore: McGraw-Hill) [4] Houzet M and Samuelsson P 2010 Phys. Rev. B 82 060517 [5] Jonckheere T, Rech J, Martin T, Doucot B, Feinberg D and Melin R 2013 Phys. Rev. B 87 214501 [6] Jonckheere T, Zazunov A, Bayandin K V, Shumeiko V and Martin T 2009 Phys. Rev. B 80 184510 [7] van Wees B J, Lenssen K-M H and Harmans C J P M 1991 Phys. Rev. B 44 470 [8] Sun Q-F, Wang J and Lin T-H 2000 Phys. Rev. B 62 648 [9] Morpurgo A F, Klapwijk T M and van Wees B J 1998 Appl. Phys. Lett. 72 966 [10] Baselmans J J A, Morpurgo A F, van Wees B J and Klapwijk T M 1999 Nature 397 43 [11] Huang J, Pierre F, Heikkil¨a T T, Wilhelm F K and Birge N O 2002 Phys. Rev. B 66 020507 [12] Ilhan H T, Demir H V and Bagwell P F 1998 Phys. Rev. B 58 15120 [13] Pillet J-D, Quay C, Morfin P, Bena C, Yeyati A L and Joyez P 2010 Nature Phys. 6 965 [14] Blanter Ya M and B¨uttiker M 2000 Phys. Rep. 336 1 [15] Reulet B, Prober D E and Belzig W 2003 Quantum Noise in Mesoscopic Physics ed Y V Nazarov (Dordrecht: Kluwer) [16] Dieleman P, Bukkems H G, Klapwijk T M, Schicke M and Gundlach K H 1997 Phys. Rev. Lett. 79 3486 [17] Jehl X, Sanquer M, Calemczuk R and Mailly D 2000 Nature 405 50 [18] Cron R, Goffman M F, Esteve D and Urbina C 2001 Phys. Rev. Lett. 86 4104 [19] Cuevas J, Martin-Rodero A and Yeyati A L 1999 Phys. Rev. Lett. 82 4086 [20] Maisi V F, Kambly D, Flindt C and Pekola J P 2013 in preparation (arXiv:1307.4176) [21] Aguado R and Kouwenhoven L P 2000 Phys. Rev. Lett. 84 1986 [22] Luo J Y, Li X Q and Yan Y J 2007 Phys. Rev. B 76 085325 [23] Dong B, Lei X L and Horing N J M 2008 J. Appl. Phys. 104 033532 [24] Entin-Wohlman O, Imry Y, Gurvitz S A and Aharony A 2007 Phys. Rev. B 75 193308 [25] Wu B H and Cao J C 2010 Phys. Rev. B 81 125326 [26] Cottet A, Doucot B and Belzig W 2008 Phys. Rev. Lett. 101 257001 [27] Badiane D M, Houzet M and Meyer J S 2011 Phys. Rev. Lett. 107 177002 [28] Recher P, Sukhorukov E V and Loss D 2001 Phys. Rev. B 63 165314
Figure 5. The noise spectra SS (ω) measured at the L lead for
different eVS . The curves are vertically shifted for the sake of clarity. The parameters are the same as in figure 4.
bias voltage VS in our numerical results. A smaller feature at 21 is also identified in the results. This feature is insensitive to the voltage change and can be ascribed to the resonance of the two gap edges of the SC leads. Now, we investigate the noise spectra at the L lead. The numerical results of the noise spectra for varying VS are presented in figure 5. For the sake of clarity, the curves are vertically shifted and the evolutions of the fixed or moving feature positions are indicated by dotted lines. One can see that more pronounced peak features arise in the noise spectra. These pronounced peaks can be ascribed to the contribution from MAR processes. The corresponding frequencies are the differences of energies for MAR and the superconducting gap as h¯ ω = 21 − neVS , where n = ±1 and 2 are shown in the figure. The fixed feature at h¯ ω = 21 is significant here, in contrast to the results for the N lead shown in figure 4. There are also features due to the AR in the three leads which involve the energy scale of the N lead. However, these features are weak for the present parameters. One can find features at a frequency hω ¯ = 1−e(VN +VS ), as indicated in the figure after careful investigation. The features discussed above have also been checked by varying eVN in our simulation. Therefore, the noise spectra give clear information on the energy scales of internal dynamics and can serve as a novel probe for this information. 4. Conclusion
In conclusion, we investigated the noise properties of three-terminal hybrid structures at arbitrary bias voltages. The zero-frequency noise shows that the higher order MAR processes are sensitive to the presence of the AR at the N–S interface. As a result, the multiple-step structures in the noise Fano factor will be suppressed. The AR at the interfaces of the N and S leads gives rise to new features in the noise Fano factor measured at the N lead. We also investigated the noise spectrum at finite frequency. Our results show that the internal dynamics of the multi-terminal hybrid structure raises 7
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