Cooper pair splitting controlled by a temperature gradient

Let us consider the NSN structure depicted in Figure 2. Normal metallic leads are attached to a bulk superconductor with the aid of two junctions described by a set of conducting channel transmissions τ1,n and τ2,n with n being the integer number enumerating all conducting channels. The two junctions are located at a distance considerably shorter than the superconducting coherence length ξ. Normal electrodes are kept at different temperatures T1 and T2, thus creating a temperature gradient across our device. In addition, voltages V1 and V2 can be applied to two normal leads, as shown in Figure 2.

[2190-4286-14-7-2]

Figure 2: Schematics of the NSN structure under consideration. Normal electrodes are biased by external voltages V1 and V2 and are kept at different temperatures T1 and T2. The superconducting electrode is assumed to be thinner than the superconducting coherence length ξ.

The Hamiltonian of this structure can be chosen in the form

[2190-4286-14-7-i1](1)

where

[2190-4286-14-7-i2](2)

are the Hamiltonians of two normal leads, [Graphic 1] denote creation and annihilation operators for an electron with a spin projection α at a point x, m is the electron mass, and μ is the chemical potential,

[2190-4286-14-7-i3](3)

is the Hamiltonian of a superconducting electrode with the order parameter Δ and the terms

[2190-4286-14-7-i4](4)

account for electron transfer through the junctions between the superconductor and the normal leads. In Equation 4, the surface integrals are taken over the contact areas [Graphic 2], and tr(x) denote coordinate- and spin-independent tunneling amplitudes.

Let us denote the probability for N1 and N2 electrons to be transferred, respectively, through the junctions 1 and 2 during the observation time t as Pt(N1,N2). Introducing the so-called cumulant generating function [Graphic 3](χ1,χ2) by means of the formula

[2190-4286-14-7-i5](5)

with χ1,2 being the counting fields, one can express the average currents through the junctions [Graphic 4], and the current–current correlation functions

[2190-4286-14-7-i6](6)

in the following form

[2190-4286-14-7-i7](7)

The cumulant generating function [Graphic 5] in Equation 5 be evaluated in a general form with the aid of the path integral technique [22,25], which yields

[2190-4286-14-7-i8](8)

where [Graphic 6] is the Keldysh Green function of our system

[2190-4286-14-7-i9](9)

the 4 × 4 matrices [Graphic 7] represent the inverse Keldysh Green functions of isolated normal and superconducting leads and [Graphic 8] is the diagonal 4 × 4 matrix in the Nambu–Keldysh space describing tunneling between the leads,

[2190-4286-14-7-i10](10)

Further analysis of the general expression for the function [Graphic 9] (Equation 8) is essentially identical to that already carried out in [25]. Therefore, it is not necessary to go into details here. Employing Equation 7 and making use of the results [25], we recover general expressions for both the currents Ir across the junctions and the cross-correlated current noise S12 in the presence of a temperature gradient between two normal terminals.

In what follows we will be particularly interested in the limit of low voltages and temperatures eV1,2,T1,2 ≪ Δ. In this case, Ir (containing both local and non-local components) is practically insensitive to temperature and matches with the results [6,7,10] derived earlier in the corresponding limit.

For the non-local current noise in the same limit eV1,2,T1,2 ≪ Δ we obtain

[2190-4286-14-7-i11](11)

where we defined the Andreev reflection probabilities per conducting channel in both junctions

[2190-4286-14-7-i12](12)

introduced the function

[2190-4286-14-7-i13](13)

and employed Fermi distribution functions for electrons and holes in the normal leads

[2190-4286-14-7-i14](14)

Equation 11 defines the low-energy cross-correlated current noise in the presence of a temperature gradient and represents the main general result of the present work.

The expression (Equation 11) contains the integrals of the type [Graphic 10], which cannot be handled analytically except in some special limits, i.e.,

[2190-4286-14-7-i15](15)

for T1 = T2 = T and

[2190-4286-14-7-i16](16)

for T1 ≫ T2. In the opposite limit T2 ≫ T1 in Equation 16 one should simply interchange T1 ↔ T2.

In order to proceed, we note that there exists a very accurate interpolation formula between the above limits. It reads

[2190-4286-14-7-i17](17)

where we defined an effective temperature

[2190-4286-14-7-i18](18)

and introduced the notations T = (T1 + T2)/2 and δT = T1 − T2.

With the aid of this interpolation, the non-local noise (Equation 11) can be reduced to a much simpler form

[2190-4286-14-7-i19](19)

Here we have introduced the non-local conductance in the limit of zero temperature and zero bias voltage,

[2190-4286-14-7-i20](20)

[Graphic 11] being the normal state resistance of a superconducting island [25], as well as the parameters

[2190-4286-14-7-i21](21)

and the local Fano factors for two barriers in the Andreev reflection regime

[2190-4286-14-7-i22](22)

At zero bias voltages V1 = V2 = 0, we obtain the noise in the form

[2190-4286-14-7-i23](23)

Hence, for the excess non-local noise δS12 = S12(T,δT) − S12(T,0) induced by the temperature gradient we get

[2190-4286-14-7-i24](24)

This contribution is positive and reaches its maximum δS12 ≃ 0.44G12T at |δT| = T.

The effect of the temperature gradient on cross-correlated non-local noise remains appreciable also at non-zero bias voltages V1,2, in which case it essentially depends on transmission distributions in both junctions.

We start from the tunneling limit A1,n, A2,m ≪ 1, where one has

[2190-4286-14-7-i25](25)

Keeping only the terms ∝γ± in the expression (Equation 19), we obtain

[2190-4286-14-7-i26](26)

The first and the second terms on the right-hand side of this formula are attributed, respectively, to CAR and EC processes. We observe that, similarly to the limit δT = 0, the noise cross correlations remain positive, S12> 0, provided V1 and V2 have the same sign, and they turn negative, S12< 0, should V1 and V2 have different signs. The result is also illustrated in Figure 3.

[2190-4286-14-7-3]

Figure 3: Non-local noise S12 (Equation 26) in the tunneling limit (Equation 25). Left panel: T = 0; middle panel: T = 0.1Δ, δT = 0; right panel: |δT| = T = 0.1Δ.

In the opposite limit of perfectly conducting channels in both junctions with τ1,n = τ2,m = 1 one gets γ+ = 2, γ− = 0, β1 = β2 = 0. Hence, in this case, Equation 19 yields

[2190-4286-14-7-i27](27)

This result is always positive at non-zero bias and sufficiently low temperatures, indicating the importance of CAR processes in this limit, see also Figure 4.

[2190-4286-14-7-4]

Figure 4: Non-local noise S12 (Equation 27) in the case of fully transparent junctions. Left panel: T = 0; middle panel: T = 0.05Δ, δT = 0; right panel: |δT| = T = 0.05Δ.

Yet another important physical limit is realized provided the contact has the form of a short diffusive wire with the corresponding Thouless energy exceeding the superconducting gap Δ. In this diffusive limit the transmission probability distributions in both junctions are defined by the well-known formula

[2190-4286-14-7-i28](28)

with [Graphic 12] being the resistances of diffusive contacts in the normal state. Making use of this formula, one readily finds γ± = ±1 and β1 = β2 = 1/3. Then Equation 19 reduces to

[2190-4286-14-7-i29](29)

This result is also displayed in Figure 5.

[2190-4286-14-7-5]

Figure 5: Non-local noise S12 (Equation 29) in the case of diffusive barriers. Left panel: T = 0; middle panel: T = 0.03Δ, δT = 0; right panel: |δT| = T = 0.03Δ.

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