Axel Freyn

Martina Flöser

Régis Mélin
email: regis.melin@grenoble.cnrs.fr

Positive current cross-correlations in a highly transparent normal-superconducting beam splitter due to synchronized Andreev and inverse Andreev reflections

Predictions are established for linear differential current-current cross-correlations dS a,b /dV in a symmetrically biased three-terminal normal metal-superconductor-normal metal (NSN) device. Highly transparent contacts turn out to be especially interesting because they feature positive dS a,b /dV . At high transparency, processes based on Crossed Andreev Reflection (CAR) contribute only negligibly to the current and to dS a,b /dV . Under these circumstances, current-current crosscorrelations can be plausibly interpreted as a coherent coupling between the two NS interfaces in the form of synchronized Andreev and inverse Andreev reflections (AR-AR), corresponding to the process where a pair of electron-like quasi-particles and a pair of hole-like quasi-particles arrive from the normal electrodes and annihilate in the superconductor. Hence, positive dS a,b /dV does not automatically imply CAR. For tunnel contacts, dS a,b /dV is positive because of CAR. In between these two extremities, at intermediate transparencies, dS a,b /dV is negative because both processes which cause positive correlations, occur only with small amplitude. We use scattering theory to obtain analytic expressions for current and noise, and microscopic calculation using a tight binding model in order to obtain a clear interpretation of the physical processes.

I. INTRODUCTION

Beautiful experiments on transport and noise in normal metal-superconducting (NS) hybrids allow probing the microscopic physics associated to the superconducting condensate and quasi-particles. For instance, the superconducting gap ∆ was revealed by tunnel spectroscopy on a normal metal-insulator-superconductor tunnel junction. In NS structures with highly transparent contacts, Andreev reflection 1 is the phenomenon by which pairs of electron-like quasi-particles from the normal electrode N can enter the superconductor S and join the condensate. Additional processes appear in N a SN b structures with two normal electrodes N a and N b : an electron coming from N a may be transmitted as an electron into N b (elastic cotunneling, EC), or it may be transmitted as a hole into N b (crossed Andreev reflection, CAR ). The amplitudes of these two processes decrease exponentially with a characteristic length scale: the coherence length ξ, which is inverse proportional to the energy gap ∆ in the ballistic limit. Therefore, threeterminal nanoscale devices with distance R > ∼ ξ between the contacts are especially interesting: conductance and noise experiments on them probe both the condensate and the quasi-particle states.

Concerning current-current correlations, theorists have envisioned two kinds of experiments for the long term: using entanglement in quantum information devices, and testing entanglement in the electronic Einstein-Podolsky-Rosen (EPR) experiment. 3,25,37 The EPR experiment is not considered here, but instead the basic problem of current-current cross-correlations 38 in NSN structures is addressed. Some experiments based on NISIN structures have been reported recently. 39 Our task here is not to understand the tunnel limit, where an insulating oxide layer I is inserted in between the normal and the superconducting electrodes, but the opposite limit of highly transparent interfaces where Coulomb interactions 16 are not expected to play a predominant role.

Based on the limiting case of tunnel contacts, 6,26 one may erroneously conclude that positive differential current-current cross-correlations dS a,b /dV are equivalent to CAR. However, this is not the case because, as we show, positive dS a,b /dV can well be obtained in the absence of CAR.

Current-current cross-correlations are negative [40][41][42] for non-interacting fermions. A flux of bosons leads to positive cross-correlations 43 and negative cross-correlations are found for bosons impinging one by one onto a beam splitter. 44 Cross-correlations can be positive in interacting fermionic systems, [45][46][47][48][49][50][51] as well as in multiterminal NS structures. [21][22][23][24][25][26][27][28][29][30] The recent experiment 39 and other experiments under way involve normal electrodes separately connected to a superconductor, 28 with a geometry similar to that considered in the following. Not only the noise can be evaluated in various set-ups, but also the full histogram of the charge transmitted in a given time interval. 25,27,30 In addition to these set-ups, relevant information is also obtained from "zero dimensional" chaotic cavities in contact with a superconductor, in connection with the possibility to observe positive current-current crosscorrelations due to CAR. 23 In what follows, attractive interaction binding pairs of electron-like quasi-particles is present everywhere in the superconducting region. Two electron-like quasiparticles of a pair injected into the superconducting region remain glued by the BCS mean field interaction, in contrast with the dissociation of a Cooper pair entering a chaotic cavity (see Ref. 23). The physics behind current-current cross-correlations in the NSN structure with highly transparent contacts considered here was not really elucidated in Ref. 28, in spite of the important observation that dS a,b /dV is positive but the non-local con-ductance is negative at low bias. Unusual properties can be realized with the following experimental conditions: First it is assumed that the same voltage V a = V b ≡ V is applied on the two normal electrodes N a and N b . Second the temperature is very low, and third high values of interface transparencies are used. Direct electron transmission is Pauli blocked at zero temperature because of the same voltage applied on the two normal electrodes. On the other hand, crossed Andreev reflection is strongly reduced at high transmission, regardless of the applied voltages V a and V b . These two features of EC and CAR are shown to be a sufficient condition for deducing dS a,b /dV > 0, not due to CAR, but due to what is called here synchronized Andreev reflection and inverse Andreev reflection (AR-AR). The positive current-current correlations at high transparency are not in disagreement with what is found in Ref. 23 for strong coupling to the superconductor, where a gap is induced in a chaotic cavity. The cross-over between high values of interface transparency and tunnel contacts will also be investigated: CAR has a dominant contribution to dS a,b /dV for tunnel contacts, and dS a,b /dV is negative at the cross-over for intermediate values of interface transparency, where AR-AR is suppressed.

The article is divided into two independent main sections where current-current correlations are evaluated on the basis of (i) the scattering approach for a homogeneous superconducting gap (see Sec. III) and of (ii) microscopic calculations taking into account the strong inverse proximity effect (see Sec. IV). Technically, these two sections rely on different approaches and are based on different assumptions. Overall agreement between the two calculations is obtained. Final remarks are provided in Sec. V. Technical details are as much as possible left for Appendices. The article starts with a preliminary section containing definitions and a summary of the main results.

II. PRELIMINARIES

A. Current, noise and current-current correlations

We start with general definitions of current and current-current cross-correlations. The geometry of the considered set-up is shown in Fig. 1. The central island is superconducting, and it is connected by highly transparent contacts to the two superconducting reservoirs on top and bottom. The superconductor S is made of the central island and of the two reservoirs.

The operator giving the current flowing at time t from the normal electrode N a to the superconducting island S at the N a S interface [see Fig. 1

(b)] takes the form Îa,α (t) = M σ,n=1 t an,αn c + αn,σ (t)c an,σ (t) (1)
+t αn,an c + an,σ (t)c αn,σ (t) ,

Superconducting reservoir

Superconducting reservoir where σ is the projection of the spin on the quantization axis and the sum over n runs over the M tight-binding sites describing the interface. Tight-binding sites on the normal side of the interface N a S are labeled by a n , and their counterparts in the superconducting electrode are labeled by α n . The hopping amplitudes between electrode N a and the superconductor are denoted by t an,αn and t αn,an . One has t an,αn = t αn,an ≡ t a in the absence of a magnetic field. The average current I a ≡ Îa,α (t) is the expectation value of the current operator given by Eq. (1).

Normal electrode N a Normal electrode N b R y = M a 0 V V R x = N a 0 (b) N a R N b S V V I a I b (a)
Current-current auto-correlations in electrode N a are given by

S a,a (t ′ ) = δ Îa (t + t ′ )δ Îa (t) + δ Îa (t)δ Îa (t + t ′ ) , (2)
with δ Îa (t) = Îa (t) -Îa (t) . In the absence of acexcitations, the average current given by Eq. ( 1) is timeindependent and the auto-correlations S a,a (t ′ ) given by Eq. ( 2) depend only on the difference t ′ of the time arguments.

Similarly, current-current cross-correlations between the electrodes N a and N b are given by

S a,b (t ′ ) = δ Îa (t + t ′ )δ Îb (t) + δ Îb (t)δ Îa (t + t ′ ) , (3)
where Îb describes the current at the interface with the

AR R ≫ ξ independent interfaces AR 1 1 2 2 1 1 2 2 (a) AR R ≫ ξ independent interfaces AR 1 1 2 2 1 1 2 2 (c) R ≈ ξ non-local conductance 1 1 2 2 2 2 1 1 (b) R ≈ ξ non-local conductance 1 1 2 2 1 1 2 2 (d)
FIG. 2: (Color online). Two independent local AR processes are shown on panel (a), for two interfaces separated by a distance R much larger than the coherence length ξ. The labels 1 and 2 on the diagram correspond to electrons and holes respectively. For each AR process, an electron impinging from the normal electrode onto the interface is converted into a hole, and a pair is transmitted into the superconductor. If R ∼ ξ, the two AR processes can be coupled coherently by non-local propagation in the superconductor [see panel (b)], with a quartet as an intermediate state and penetration of a charge 4e into the superconductor, which is qualitatively equivalent to double CAR. Panel (c) shows independent AR and AR processes at the two interfaces, supposed to be far apart. Panel (d) shows AR-AR for R ∼ ξ, which is qualitatively equivalent to double EC. normal electrode N b . Zero-frequency auto-correlations (S a,a ) and cross-correlations (S a,b ) are defined as the integral over t ′ of S a,a (t ′ ) and of S a,b (t ′ ). The definition used here for the Fano factor is as follows: F a,a = S a,a /2eI a and F a,b = S a,b /2e √ I a I b , with I a = I b . With this definition, the Schottky formula leads to F = q * /e for a Poisson process transmitting a charge q * .

B. Known results and open questions

A few facts related to the non-local conductance for arbitrary values of interface transparency are already known. Only local Andreev reflection AR and local inverse Andreev reflection AR come into account for contacts separated by a distance much larger than the coherence length [see Figs. 2(a) and(c)]. Local AR means that an electron is converted into a hole and a pair of electron-like quasi-particles is transmitted into the superconductor, and local AR means that a hole is converted into an electron and a pair of hole-like quasi-particles is transmitted into the superconductor, which annihilates a Cooper pair. Local AR and AR contribute to transport if the separation between the interfaces is much larger than ξ, and in two-terminal configurations where the superconductor is not connected to ground.

Other quantum processes appear in a three-terminal configuration if the distance between the contacts is comparable to the coherence length: CAR and EC. However, non-standard types of "non-local" processes can be also obtained by merging AR at the interface N a S to AR at the interface SN b , forming what is called here AR-AR [see Fig. 2(b)]. Physically, AR-AR would correspond to the synchronized transmission of two pairs from electrodes N a and N b into the superconductor, which can be seen as double CAR. However, this process does not contribute to non-local transport at zero temperature. Conversely, AR at interface N a S might be associated to AR at interface SN b , leading to AR-AR. The corresponding non-local resistance is independent on the value of interface transparency in the tunnel limit. 12,13 Qualitatively, AR-AR can also be seen as double EC.

The non-standard non-local process AR-AR involving pairs appears naturally when expanding diagrammatically 13 the non-local conductance to order t 8 , with t the hopping amplitude at the interfaces. As it is shown below, AR-AR plays a central role in understanding the positive 28 current-current cross-correlations for highly transparent contacts, in a regime which is not described by perturbation theory in t.

A very recent preprint 57 points out the possibility of "synchronized Andreev transmission" in the currentvoltage characteristics of a SNS junction array. Synchronization manifests itself in this work as specific features in the current-voltage characteristics of the two-terminal SNS junction array. We arrive here at the conclusion that synchronization of Andreev processes is also possible if the separation between the NS interfaces is comparable to the coherence length. The dominant channel AR-AR is shown to result in positive current-current cross-correlations.

As mentioned in the introduction, experiments on current-current cross-correlations in NSN structures have already started. 39 A few basic questions regarding current-current cross-correlations for highly transparent contacts have not yet received a satisfactory explanation.

First what is the physics behind the positive 28 linear differential current-current correlations dS a,b /dV for a highly transparent NSN beam splitter? It is shown that dS a,b /dV > 0 is not an evidence for CAR (which would prevail 26 for tunnel contacts in the absence of Coulomb interactions). An interpretation in terms of AR-AR is proposed.

Second, how do current-current cross-correlations depend on the sample geometry? Current-current crosscorrelations decay with the geometry-dependent coherence length, as it will be obtained from microscopic calculations in Sec. IV.

Third, what is the value of cross-correlations at intermediate transparencies? Experimentalists can realize tunnel or highly transparent contacts, by oxidizing or not the sample during fabrication. Intermediate values of interface transparency are more difficult to control but it is nevertheless useful to quantify how "perfectly transparent" the NS contacts should be in order to obtain AR-AR. A cross-over from positive to negative dS a,b /dV is found in BTK calculations as the normal state transmission coefficient T N is reduced below a value typically of order

T N ∼ 1/2.
Fourth, do the predictions established with a ballistic superconductor hold also for a disordered superconductor? It is shown at the end of Sec. IV that, for strong inverse proximity effect, AR-AR is responsible for positive cross-correlations also in the case of a disordered superconductor in the regime where the elastic mean free path is shorter than the coherence length.

III. HOMOGENEOUS SUPERCONDUCTING GAP: BTK CALCULATION

The BTK approach 58 allows calculations of the current-voltage characteristics of a NS point contact with arbitrary interfacial scattering potential. It was first generalized in Ref. 12 and later in Ref. 17 to the case of non-local transport. Useful physical informations can be obtained even though the gap is not self-consistent in the BTK calculation. We will consider a one-dimensional geometry, as shown in Fig. 1(a).

The current I i at the interface with the normal electrode N i , and the current-current correlations S i,j between the two normal electrodes N i and N j are expressed 21 in terms of the scattering matrix s as

I i = e h dω j,α,β sgn(α) δ i,j δ α,β -s α,β i,j 2 f β (ω) (4) S i,j = 2e 2 h dω k,l,α,β,γ,δ sgn(α)sgn(β) (5)
A kγ,lδ (i, α, ω)A lδ,kγ (j, β, ω)f γ (ω) [1 -f δ (ω)] ,
with

A kγ,lδ (i, α, ω) = δ i,k δ i,l δ α,γ δ α,δ -s α,γ † i,k s α,δ i,l . (6)
Latin labels i, j, k, l run over a,b, referring to the two normal electrodes N a and N b . Greek labels α, β, γ, δ denote electrons or holes in the superconductor. The notation f e (ω) = θ(eVω) stands for the distribution function of electrons at zero temperature, and

f h (ω) = θ(-eV -ω)
is the one of holes, where θ(x) is the Heaviside stepfunction. In Eqs. ( 4) and ( 5), sgn(α) = +1 if α = e, and sgn(α

) = -1 if α = h.
The elements of the s-matrix are evaluated from the BTK approach (see Appendix A) for a one-dimensional N a SN b junction [see Fig. 1(a)].

Step-function variation of the superconducting gap at the interfaces is assumed. A repulsive scattering potential V (x) = H [δ(x) + δ(x -R)] is introduced at the interfaces. The transparency of the interfaces is related to the BTK parameter Z = H/hv F , with the Fermi velocity v F . The interface transparency is characterized by the value of the normal state transmission coefficient

T N = 1/(1 + Z 2 ).
Highly transparent contacts correspond to T N = 1, and tunnel contacts correspond to T N ≪ 1.

In the one-dimensional model considered in this section, current and noise are highly sensitive to the length R of the superconducting region: they oscillate as a function of R with a period equal to the Fermi wavelength λ F ≪ R. These oscillations can be interpreted as Friedel oscillations where the contacts with the normal electrodes play the role of impurities. In a more realistic higher-dimensional model with more than one transmission mode, the oscillations in the different modes are independent and thus are averaged out. In order to simulate qualitatively multi-dimensional behavior with a one-dimensional system, current and noise are averaged over one oscillation period: 6,8,28

I av i (R) = 1 λ F R+λF /2 R-λF /2 dr I i (r) , (7)
S av i,j (R) = 1 λ F R+λF /2 R-λF /2 dr S i,j (r). (8)
In the studied limit of small applied voltage eV ≪ ∆, the energy-dependence of the scattering matrix elements can be neglected and s α,β i,j (ω) ≈ s α,β i,j (ω = 0). Using this approximation, it is possible to perform these integrals analytically.

In this limit, the current I i through the normal lead N i is obtained as

I av i = 2 √ 2e 2 V T 2 N (2 -T N ) cosh(R/ξ) sinh(R/ξ) 2 h [2(2 -T N ) 2 cosh(R/ξ) 2 -8(1 -T N )] 3 2 . (9)
The local current-current correlations S ii at the interface N i S and the current-current cross-correlations S ij

T N R/ξ = 2 R/ξ = 3 R/ξ = 4 R/ξ = 5 0 1 2 3 dI av a /dV (e 2 /h) R/ξ = 0.05 R/ξ = 0.25 R/ξ = 0.5 R/ξ = 1 0 2 4
dS av a,a /dV (e 3 /h) 0 0.25 0.5 0.75 1 between the interface N i S and the interface SN j are

T N R/ξ = 2 R/ξ = 3 R/ξ = 4 R/ξ = 5 0 1 2 3 dS av a,a /dV (e 3 /h) R/ξ = 0.05 R/ξ = 0.25 R/ξ = 0.5 R/ξ = 1 0 0.25 0.5 dS av a,b /dV (e 3 /h) 0 0.25 0.5 0.75 1 T N R/ξ = 2 R/ξ = 3 R/ξ = 4 R/ξ = 5 0 2 0 0.5 1 0 1 2 dS av a,b /dV (e 3 /h) R/ξ = 0.05 R/ξ = 0.25 R/ξ = 0.5 R/ξ = 1
T N R/ξ = 2 R/ξ = 3 R/ξ = 4 R/ξ = 5 0 1 2 F a,a R/ξ = 0.05 R/ξ = 0.25 R/ξ = 0.5 R/ξ = 1 0 0.05 0.15 F a,b 0.01 0.1 1 T N R/ξ = 2 R/ξ = 3 R/ξ = 4 R/ξ = 5 0 1 2 F a,b R/ξ = 0.05 R/ξ = 0.25 R/ξ = 0.5 R/ξ = 1
S av ii = 64 √ 2e 3 V T 2 N sinh(R/ξ) 2 sech(R/ξ) h(2 -T N ) 3 [2(2 -T N ) 2 cosh(R/ξ) 2 -8(1 -T N )] 7 2 -128(1 -T N ) 4 + 2(2 -T N ) 2 (1 -T N )(96 -T N × (192 -T N (116 -T N (20 -3T N )))) cosh(R/ξ) 2 -(2 -T N ) 4 (72 -T N (144 -T N (82 -T N × (10 + T N )))) cosh(R/ξ) 4 + 8(2 -T N ) 6 (1 -T N ) cosh(R/ξ) 6 (10)
and

S av i =j = 64 √ 2e 3 V T 2 N sinh(R/ξ) 2 sech(R/ξ) h(2 -T N ) 3 [2(2 -T N ) 2 cosh(R/ξ) 2 -8(1 -T N )] 7 2 128(1 -T N ) 4 -2(2 -T N ) 2 (1 -T N )(2 + T N ) × (2 -3T N )(8 -(8 -T N )T N ) cosh(R/ξ) 2 + (2 -T N ) 4 (8 -T N (16 + T N (2 -T N (10 + T N )))) × cosh(R/ξ) 4 . ( 11
)
The linear conductance dI av a /dV , the auto-correlations dS av a,a /dV and the cross-correlations dS av a,b /dV given by these formulas are shown in Fig. 3 as a function of the normal transmission coefficient T N , for different values of the distance between the contacts. As it is expected, the conductance increases with interface transparency for R/ξ > ∼ 1. The conductance depends on the ratio R/ξ while R is smaller than the coherence length ξ, but it does almost not vary as R is increased above ξ. For R/ξ < 1, the conductance is non-monotonous when plotted as a function of T N (see Fig. 3 top left). An explanation for the non-monotonous behavior is the enhanced transmission due to the finite size of the superconductor. 54 Starting from tunnel contacts, the linear differential auto-correlation dS av a,a /dV first increases with interface transparency as larger current leads to larger noise. The differential noise reaches a maximum and almost vanishes for perfect transparency if R > ∼ ξ, as it is expected for a single NS junction. 38

different R/ξ > ∼ 1.
As it can be seen, for R/ξ > ∼ 1 the Fano factor is independent of R/ξ after normalizing to its value for T N = 1. The Fano factor F a,b is positive outside the region of the minima. For R/ξ > ∼ 1, F a,a takes the value F a,a ≃ 2 for T N ≪ 1, and the value F a,a ≃ 0 for T N = 1.

We first make some remarks in order to confirm the validity of our calculation. Expected behavior is recovered in some known limiting cases:

(i) For tunnel contacts (T N ≪ 1), the Fano factors are given by F a,a ≈ 2 -1 2 sech(R/ξ) and F a,b ≈ 1 2 sech(R/ξ), which is in agreement with the expected limiting values F a,a = 2 and F a,b = 0 for R/ξ → ∞. This corresponds to the doubling of the effective charge for Andreev reflection at a single NS interface in the tunnel limit. 52,53 (ii) For highly transparent interfaces (T N ≈ 1), the Fano factors vanish exponentially with increasing R/ξ as F a,a = F a,b ≈ sech(R/ξ), as it is expected for a single highly transparent NS interface.

(iii) dS av a,b /dV and F a,b are positive and very small for R/ξ > ∼ 1 and T N ≃ 1, in agreement with Ref. 26. Only CAR contributes to current-current cross-correlations for T N ≪ 1.

(iv) As it can be seen in Fig. 3(top left), the linear conductance dI av a /dV is suppressed for R/ξ < ∼ 1 because the number of normal states within the gap energy is ∼ R/ξ for this geometry. For a three-dimensional grain it is proportional to ∼ k 2 F R 3 /ξ, which can be much larger than unity even for R/ξ < ∼ 1. The suppression of Andreev processes at high transparency for R/ξ ≪ 1 is thus not expected to occur in the case of a three-dimensional superconducting grain.

The Schottky limit is realized for low values of interface transparencies, which lead to (δ Îaδ Îb ) 2 av = 4eI av a . On the other hand (δ Îa + δ Îb ) 2 av = 4e(I av a + I av b ). One concludes that F a,a ≃ 3/2 and F a,b ≃ 1/2, in agreement with the plateau obtained in Figs. 4(top) for the dependence on T N of the Fano factor. Eqs. ( 9), ( 10) and ( 11) reproduce these values in the limit R/ξ → 0 and T N ≪ 1.

For highly transmitting interfaces T N = 1, we obtain F a,a = F a,b . These values can be confirmed by a more simple calculation in the limit R/ξ ≪ 1 (see Appendix B). The identity

F a,a = F a,b implies that Îav a -Îav b is noiseless for T N = 1, independent of R/ξ: dτ δ Îa (t) -δ Îb (t) δ Îa (t + τ ) -δ Îb (t + τ ) av = 1 2 S av a,a + S av b,b -2S av a,b = I av a 2 (F a,a + F b,b -2F a,b ) = 0.
(12) By comparison, in a fermionic beam splitter with highly transparent contacts, a charge e transmitted from the source to N a means no charge transmitted from the source to N b . Thus, for fermions, it is the sum Îa + Îb that is noiseless.

We can see from Fig. 3 that the linear differential crosscorrelations of current noise dS av a,b /dV are positive for T N ≪ 1 and T N ≃ 1, while they take negative values in between these two limiting cases. For T N ≪ 1, the positive dS av a,b /dV can be explained by the presence of CAR processes. This is in agreement with perturbative calculations carried out in Ref. 26. However, for perfectly transmitting interfaces T N ≃ 1, CAR processes do not occur, as the elements of the scattering matrix describing CAR (e.g. s e,h a,b ) equal zero for T N = 1 (see Appendix B). In this Appendix we obtain in the limit R/ξ ≪ 1

dS av a,b dV = 8 e 3 h    s e,h a,a 2 for R/ξ ≪ 1 s e,e a,b 2
= T EC for R/ξ > ∼ 1 (13)
and

T CAR = s e,h a,b 2 = 0. ( 14
)
In section IV, we will use a microscopic model in order to obtain an understanding of the processes contributing to the noise, and to explain dS av a,b /dV > 0 in the absence of CAR.

To summarize this section, we have obtained analytical expressions for the current and noise in the onedimensional BTK model. It was shown that dS av a,b /dV > 0, and that Eq. ( 13) holds for R/ξ > ∼ 1 and for T N ≃ 1.

Cooper pair splitting dominates for small normal transmission coefficient T N ≪ 1, while what will be interpreted in Sec. IV as AR-AR dominates for T N ≃ 1. These two processes are suppressed for intermediate T N , resulting in negative cross-correlations in this parameter range.

IV. MICROSCOPIC CALCULATIONS

The current-current cross-correlations for highly transparent interfaces can be further investigated in the twodimensional tight-binding set-up shown in Fig. 1(b). First, using analytic calculations, we analyze the different contributions to the noise S a,b and show the absence of contributions due to CAR processes. The positive contributions are attributed to processes which we refer to as AR-AR (this notation stands for Andreev reflection and inverse Andreev reflection), a denomination which is motivated by the microscopic analysis. Second, numerical calculations are performed, which take into account the inverse proximity effect by determining the gap in a self-consistent manner. In addition, disorder will be included in the calculation. As in the last section, we restrict our study to the case where the same voltages V a = V b = V are applied on both normal leads and V is small compared to the gap ∆.

The expression of current-current cross-correlations S a,b can be decomposed as a sum of six contributions according to the types of transmission modes in the superconductor:

S a,b = S CAR + S EC + S AR-AR + S AR-AR + S ′ + S MIXED , (15)
where the expression and the meaning of the different contributions are provided in Appendix D. S CAR contains the noise attributed to crossed Andreev reflections, which contains transmission modes in the electron-hole channels. S EC , the noise due to elastic cotunneling, contains transmission modes in the electron-electron or holehole channels. With AR-AR, we refer to synchronous local Andreev reflections at both interfaces, while AR-AR links a local Andreev process at one interface to a local inverse Andreev process at the other one (see Fig. 2).

The Andreev-reflection is highly local 17 (compare with the BTK calculations in Sec. III, where it vanishes exactly). This motivates the assumption |G 1,2 α,β | ≪ |G 1,1 α,β |, which is confirmed by our numerical calculations also in the presence of the inverse proximity effect (see Fig. 5). Using this simplification, one obtains S CAR = S AR-AR = 0. The terms S ′ and S MIXED vanish after integration over the energy ω. Thus the total current-current crosscorrelation S a,b depends only on the term S AR-AR .

In order to understand what type of microscopic processes are described in AR-AR, we start from the formula giving the current-current cross-correlations in terms of the Keldysh Green functions Ĝ+,and Ĝ-,+ :

S a,b (ω) =Tr Ĝ+,- b,a Ĝ-,+ α,β + Ĝ+,- β,α G -,+ a,b -Ĝ+,- b,α Ĝ-,+ a,β -Ĝ+,- β,a Ĝ-,+ α,b , (16)
where the trace is carried out over the Nambu labels and the different transmission modes at the interfaces. It can be shown that all terms in S AR-AR are obtained from the anomalous contributions of the type G +,-,1,2 G -,+,2,1 and G +,-,2,1 G -,+,1,2 . Let us consider one of these terms as an example (the same conclusions are obtained for all terms) and suppose again that G A,1,2 α,β = 0 if α and β are on different interfaces (that is, crossed Andreev reflection does not contribute), and that a symmetric bias voltage is applied on the two normal electrodes. One has

G +,-,2,1 b,a (t, t ′ )G -,+,1,2 α,β (t ′ , t) = c a,↓ (t ′ )c b,↑ (t) c + β,↑ (t ′ )c + α,↓ (t) . (17)
This equation can be understood as a relation between initial and final states, as it is shown in Fig. 6. In general, these states can be connected by many different processes. However, the microscopic formula for the currentcurrent cross-correlations [see Eq. (D4)] shows that the initial and final states are related by an Andreev process at interface N a S, and, at the same time, by an inverse Andreev process at interface SN b [see Fig. 6(b)]. In an Andreev process, an electron is converted into a hole and a pair of electron-like quasi-particles is transmitted into the superconductor. In an inverse Andreev process, a hole is converted into an electron and a pair of hole-like quasi-particles is transmitted into the superconductor. The pair of electron-like quasi-particles annihilates with the pair of hole-like quasi-particles and the remaining electron and the hole are exchanged between the two interfaces. This results in dS AR-AR /dV > 0.

In addition to the analytic calculation, we performed numerical simulations in order to analyze a more realistic model. Details about the used method 20 are presented in Appendix C. The self-consistent simulations presented below take into account the inverse proximity effect corresponding to the reduction of the superconducting gap within a distance ∼ ξ from the contacts. Self-consistency is equivalent to current conservation for the electrons injected from the normal reservoirs and transmitted into the superconducting ones.

The values for the linear differential cross-correlations dS a,b /dV , and for the EC and CAR transmission coefficients T EC and T CAR are plotted as functions of the length N of the superconductor in Figs. 7 and8. The notations T EC and T CAR refer to the transmission modes in the superconductor (advanced-advanced or retardedretarded modes not exchanging electrons and holes for T EC , and advanced-retarded Green functions exchanging electrons and holes for T CAR ). T EC and T CAR are given by

T EC = W 2 Tr G A,1,1 α,β G R,1,1 β,α + G A,2,2 α,β G R,2,2 β,α (18)
T CAR = W 2 Tr G A,1,2 α,β G R,2,1 β,α + G A,2,1 α,β G R,1,2 β,α , ( 19
)
where W is the hopping amplitude in the bulk and at the interfaces, and α n and β m run respectively over all the sites on the superconducting side of the N a S and SN b interfaces. The notation G already found in a previous work. 20 The differential crosscorrelations dS a,b /dV are positive, which is in agreement with the preceding BTK calculation. Differential crosscorrelations dS a,b /dV show exponential decay (see Fig. 7) as a function of N , because the two normal electrodes N a and N b are coherently coupled by evanescent states in the superconductor. The BCS coherence length as obtained from the fits fulfills R x /ξ > ∼ 1 in a wide range of simulation parameters. This is the range in which the BTK calculation leads to dS av a,b /dV = 8 e 3 h T EC . For a highly transparent NS contact, one has G 2,1 αn,αn G 1,2 βm,βm ≃ 1/4W 2 where W is the hopping amplitude in the normal and superconducting electrodes, and at the interface (see Appendix E). The identity

T A,A AR-AR ≃ T EC holds if |G 1,2 α,β | ≪ |G 1,1 α,β | (see Appendix E), with T A,A AR-AR = -W 2 Tr G A,1,1 α,β G A,2,2 β,α + G A,2,2 α,β G A,1,1 β,α , (20)
where the superscript "A,A" refers to an advancedadvanced transmission mode where electrons and hole are conserved. With these assumptions, the total noise can be written as

dS a,b dV ≃ 8 e 3 h T A,A AR-AR , (21)
T A,A AR-AR ≃ T EC . (22)
Considering the numerical data, plots of dS a,b /dV as functions of T EC and T CAR (see Fig. 9), and a comparison between T A,A AR-AR and T EC confirm Eqs. ( 21) and ( 22) and show in addition

dS a,b dV ≃ 50 e 3 h T CAR . ( 23
)
Comparing with the previous BTK calculation, it is suggested that Eq. ( 23) is model-dependent, because T CAR = 0 for the BTK model while T CAR is finite but small in the self-consistent microscopic calculation. On the other hand, Eq. ( 21), which is obtained also for the BTK model, is expected to be a fundamental relation. Until now, we only studied perfect systems without disorder. The numerical calculations based on microscopic Green functions give us however the additional possibility to add disorder to the model, which is an unavoidable ingredient to describe experiments.

Eq. ( 21) remains approximately valid in the presence of weak disorder (see Fig. 10), introduced in the form of a random on-site potential uniformly distributed in the interval [-V, V ], with elastic mean free path l e ∼ (W/V ) 2 a 0 smaller than the ballistic coherence length ∼ (W/∆)a 0 , but not small compared to the Fermi wavelength. The parameter W is the hopping amplitude in the bulk of the superconductor, taking the same value at the interfaces because of highly transparent contacts.

The coherence length is reduced as the strength of disorder increases. The coherence lengths are fitted to ξ = 3.8a 0 , 3.6a 0 for V /W = 1.00, 1.25, as compared to ξ = 8a 0 with V /W = 0 in the ballistic limit. The coherence length in the presence of disorder becomes smaller than its ballistic value, as for a superconductor in the dirty limit. Within error-bars, Eq. ( 21) is fulfilled also in the presence of weak disorder, while the coefficient in Eq. ( 22) is changed, resulting in

dS a,b dV ≃ 40 e 3 h T CAR . (24)
It is concluded that

T CAR ≪ T EC implies that S a,b ≃ S AR-AR a,b
and that Eq. ( 21) is fulfilled. Thus, it was shown that, in this parameter regime, dS a,b /dV > 0 is evidence for AR-AR, not for CAR.

The BTK approach and the microscopic calculations lead to positive dS a,b /dV > 0, which is due to the exchange of fermionic quasi-particles (see Fig. 6). This is the main physical result of our article: dS a,b /dV > 0 at high transparency is not due to CAR. It is due to pairs of electron-like quasi-particles, pairs of hole-like quasiparticles, and exchange of fermions.

V. CONCLUSIONS

The article was already summarized in Sec. II B and thus we conclude with a brief overview and final remarks. We have evaluated current-current cross-correlations in a NSN structure with a homogeneous superconductor (without self-consistency in the order parameter), and with strong inverse proximity effect (with self-consistent microscopic calculations for a two-dimensional threeterminal set-up). For both approaches, the linear differential cross-correlations dS a,b /dV are positive for highly transparent contacts and decay exponentially with a characteristic length set by the coherence length. Positive dS a,b /dV arises in this set-up not because of CAR, but because of what is identified as the correlated penetration of pairs of electron-like quasi-particles and pairs of hole-like quasi-particles into the superconductor in the form of AR-AR. The positive sign of dS a,b /dV is due to the additional exchange of two fermions. It is emphasized that the proposed mechanism does not involve quartets in the superconductor, because AR-AR does not contribute to the current-current cross-correlations. Direct evaluation of dS a,b /dV leads to dS a,b /dV = 4(e 3 /h)T A,A AR-AR , and to T A,A AR-AR ≃ T EC , which holds also for a superconductor with weak disorder and with elastic mean free path shorter than the coherence length.

Finally, correlations between pairs of Andreev pairs were discussed 59,60 in connection with noise in an Andreev interferometer.

Future studies of this class of set-ups might include the effect of arbitrary applied voltages and quantitative effects of three-dimensional geometry. functions in the electrodes N a , S, and N b respectively [see Fig. 1(a)]. The notations q (+) , q (-) , k (+) and k (-) stand for the wave-vectors in the normal and superconducting electrodes:

k (±) = k F ± i/ξ (A4) q (±) = k F , (A5) with k F ξ ≫ 1.
The elements s e,e a,a , s h,e a,a , s h,e b,a and s e,e b,a can be determined using the continuity of the wave-functions at the interfaces [ψ a (0) = ψ S (0) and ψ S (R) = ψ b (R)] and the boundary condition for the derivatives [ψ ′ S (0)ψ ′ a (0) = Hψ a (0) and

ψ ′ b (R) -ψ ′ S (R) = Hψ b (R)]. The BTK pa- rameter Z is defined by Z = H/hv F .
The remaining elements of the scattering matrix can be obtained from the other possible scattering processes (e.g., a hole incoming from electrode N b ) by analogous calculation.

By comparing the different equations, the symmetry

s e,h a,a (ω) = s h,e b,b (-ω) (A6)
of the scattering matrix can be obtained.

The BCS coherence factors u 0 , v 0 appearing in these equations are given by:

u 2 0 = 1 -v 2 0 = 1 2 1 + √ ω 2 -∆ 2 ω . (A7)
The coherence factors u 0 and v 0 are interchanged under complex conjugation and changing sign of the real part of energy (a small imaginary part is supposed to be added to ω).

Appendix B: Current-current cross-correlations for the BTK model in the limit Z ≃ 0

For highly transparent interfaces (Z ≃ 0), the expressions obtained for s α,β i,j simplify considerably, local reflections (s α,α i,i = 0) and non-local Andreev reflections (s α,β i,j = 0, with i = j, α = β) are suppressed. 17 Thus, only local Andreev reflection (s α,β i,i , with α = β) and transmission without branch-crossing (s α,α i,j , with i = j) can occur. The non-zero elements of the scattering matrix are given by s h,e a,a = u 0 v 0 e R/ξe -R/ξ v 2 0 e R/ξu 2 0 e -R/ξ , (B1)

s e,e b,a = e -ikF R u 2 0 -v 2 0 u 2 0 e -R/ξ -v 2 0 e R/ξ . (B2)
Assuming s(ω) to be constant in the range [-eV, eV ] (we study the case eV ≪ ∆), Eqs. ( 4) and ( 5) can be written as

I a = 4e 2 h V s e,
for i = j. In the limit R/ξ ≪ 1, the elements of the scattering matrix (evaluated for ω → 0) are given by

s h,e a,a = s e,h b,b ≃ i R ξ , (B5)
s e,e b,a s h,h a,b ≃ 1, (B6) leading to

S a,a = S a,b = 8e 3 h |V | R ξ 2 ≥ 0 . (B7)
The result for S a,a is obtained by an analogous calculation.

Combining Eq. (B3) for I a with Eq. (B7) for S a,a and S b,b leads to F a,a ≃ F b,b ≃ 1 for eV ≪ ∆ 0 , T N = 1 and R/ξ ≪ 1.

Appendix C: Technical details on microscopic calculations

The microscopic calculations are based on the following tight-binding Hamiltonian on a square lattice: self-consistent calculation gives access to variations of the phase over almost two orders of magnitude. The average current in Eq. ( 1) is evaluated from the Keldysh Green function: the normal lead, and, at zero temperature, n F (x) = θ(-x), with θ(x) being the Heaviside step-function. This gives g +-,11 N,N g -+,22

H S = -W n,m σ c + n,σ c m,σ + c + m,σ c n,σ + n ∆ n c + n,↑ c + n,↓ + c n,↓ c n,↑ - n,σ V n c + n,σ c n,σ , (
N,N = g +-,22 N,N g -+,11

N,N = 0, leading to further simplification of the expression for S a,b .

The contribution S CAR is given by the advanced-advanced or retarded-retarded transmission modes in the electronhole channel, in the form of the combinations of the type G A,12 αβ G R,21 βα . Physically, these microscopic processes can be interpreted as Cooper pair splitting as appearing in CAR. The expression for S CAR is as follows:

S CAR = 2t 4 + g -+,11 bb g +-,22 aa G R,21 αβ G R,12 βα (1 + 2iG A,22 αα πt 2 ρ a )(1 + 2iG A,11 ββ πt 2 ρ b ) + g -+,22 bb g +-,11 aa G R,12 αβ G R,21 βα (1 + 2iG A,11 αα πt 2 ρ a )(1 + 2iG A,22 ββ πt 2 ρ b ) -g -+,22 aa g +-,11 bb G A,21 αβ G A,12 βα (i + 2G R,22 αα πt 2 ρ a )(i + 2G R,11 ββ πt 2 ρ b ) -g -+,11 aa g +-,22 bb G A,12 αβ G A,21 βα (i + 2G R,11 αα πt 2 ρ a )(i + 2G R,22 ββ πt 2 ρ b ) , ( D2
)
where t is the hopping amplitude at the interfaces.

The contribution S EC contains advanced-advanced and retarded-retarded transmission modes in the electronelectron or hole-hole channel, in the form of the combinations of the type G A,12 αβ G A,21 βα . This is the contribution to the noise of normal electron transmission in the form of EC. The expression reads

S EC = 2t 4 + g -+,11 bb g +-,11 aa G R,11 αβ G R,11 βα (i -2G A,11 αα πt 2 ρ a )(i -2G A,11 ββ πt 2 ρ b ) + g -+,22 bb g +-,22 aa G R,22 αβ G R,22 βα (i -2G A,22 αα πt 2 ρ a )(i -2G A,22 ββ πt 2 ρ b ) + g -+,11 aa g +-,11 bb G A,11 αβ G A,11 βα (i + 2G R,11 αα πt 2 ρ a )(i + 2G R,11 ββ πt 2 ρ b ) + g -+,22 aa g +-,22 bb G A,22 αβ G A,22 βα (i + 2G R,22 αα πt 2 ρ a )(i + 2G R,22 ββ πt 2 ρ b ) . (D3)
Now we consider processes that do not appear in lowest order perturbation theory. First the contribution S AR-AR contains advanced-advanced and retarded-retarded transmission modes, in the form of the combinations of the type G A,11 αβ G A,22 βα . Physically, this can be interpreted as the contribution to the noise of synchronized Andreev and inverse Andreev processes (AR-AR), with the exchange of two fermions at the same time, thus leading to dS a,b /dV > 0 (see Fig. 6). This contribution dominates the current-current cross-correlations dS a,b /dV in the considered set-up. This contribution reads

S AR-AR = -8π 2 t 8 ρ a ρ b + g -+,22 bb g +-,11 aa G A,12 αα G A,21 ββ G R,22 αβ G R,11 βα + g -+,11 bb g +-,22 aa G A,21 αα G A,12 ββ G R,11 αβ G R,22 βα + g -+,11 aa g +-,22 bb G A,11 αβ G A,22 βα G R,21 αα G R,12 ββ + g -+,22 aa g +-,11 bb G A,22 αβ G A,11 βα G R,12 αα G R,21 ββ . (D4)
Another contribution not appearing in lowest order contains processes involving electron-hole conversion both at the interfaces and during propagation in the superconductor. The transmission modes in the superconductor are of the type G A,12 α,β G A,12 β,α . These terms correspond to the synchronization of two Andreev processes (AR-AR). The corresponding expression reads   g A,1,1 α,βg A,2,2 α,β + i g A,1,2 α,β + g A,2,1 α,β i(g A,1,1 α,β + g A,2,2 α,β ) + g A,1,2 α,βg A,2,1 α,β i g A,1,1 α,β + g A,2,2 α,β + g A,2,1 α,βg A,1,2 α,β -g A,1,1 α,β + g A,2,2 α,β + i g A,1,2 α,β + g A, leading to the identification of the advanced-advanced AR-AR transmission coefficient to the opposite of the advancedretarded EC transmission coefficient, for small energy compared to the gap and for N a 0 > ∼ ξ. Eq. (E5) has no imaginary part because g 1,2 α,β g 2,2 α,β av = 0 at small energy compared to the gap. The fully dressed local Green function can be evaluated approximately by inverting the Dyson equation Ĝα,α = ĝα,α + ĝα,α tα,a ĝa,a ta,α Ĝα,α , leading to G A,1,2 α,α = G A,2,1 α,α = 1/2W at energy small compared to the gap.

S AR-AR = 8π 2 t 8 ρ a ρ b + g -+,22 bb g +-,22 aa G A,21 αα G A,21 ββ G R,12 αβ G R,

FIG. 1 :

 1 FIG.1:(Color online). The two models studied in this paper: Panel (a) shows the one-dimensional geometry used in the BTK calculation, which is a simplified description for a NSN structure in three dimensions. The central superconducting electrode has length R and the superconducting gap ∆0 is uniform. Panel (b) shows a representation of a three-terminal device described by a tight-binding Hamiltonian. The two superconducting reservoirs have the same phase and thus constitute a single terminal.

FIG. 3 :

 3 FIG. 3: (Color online). Variations of the linear differential conductance [left panels, in units of e 2 /h], linear differential autocorrelations [center panels, in units of e 3 /h] and linear differential cross-correlations [right panels, in units of e 3 /h] as a function of the normal interface transparency TN , for the values of R/ξ indicated in the figures. A single-channel one-dimensional BTK calculation is used [see Fig. 1(a)]. The lower Panels show that dI av a /dV and dS av a,a /dV are almost independent on R/ξ for R/ξ > ∼ 1, and that dS av a,b /dV becomes very small as R/ξ increases above ∼ 1. The insert of the bottom right panel shows the variations of the normalized current-current cross-correlations [dS av a,b /dV (TN )]/[dS a,b av /dV (TN = 1)]. The data corresponding to R/ξ = 2, 3, 4, 5 superimpose after this rescaling.

FIG. 4 :

 4 FIG. 4: (Color online). Variations of the Fano factors Fa,a = S av a,a /2eI av a [top panels] and F a,b = S av a,b /2eI av a [bottom panels] as a function of the normal interface transparency TN , for the values of R/ξ indicated in the figures. A single-channel one-dimensional BTK calculation is used [see Fig. 1(a)]. The ratio between the gap ∆0 and the Fermi energy ǫF is ∆0/ǫF = 10 -4 in this simulation. The insert of bottom right panel shows the variations of the normalized crossed Fano factor [F a,b (TN )]/[F a,b (TN = 1)]. The data corresponding to R/ξ = 2, 3, 4, 5 superimpose after rescaling.

  Differential current-current crosscorrelations dS av a,b /dV are positive for R/ξ < ∼ 1 in the extreme cases of very high and very low interface transparency, and take negative values in between, as shown in Figs. 3(right), and in the insert of Fig. 3(bottom right). The Fano factors F a,a = S av a,a /2eI av a and F a,b = S av a,b /2eI av a correspond to the noise normalized to the current, which allows to get rid of the trivial effect that, at low transparency, the noise increases when the current increases. The variations of F a,a and F a,b feature local minima at intermediate values of T N . In the insert of Fig. 4(bottom right), the Fano factor F a,b is shown for

FIG. 5 :

 5 FIG. 5: (Color online). The anomalous Green function G 1,2 αn,βm is small compared to the normal Green function G 1,1 αn,βm . As the elements G αn,βm (between site n on the left interface and site m on the right interface) decay exponentially with d = |yn -ym|, we have evaluated the mean values | G αn,βm | n,m of the elements with the same value of d. The main contribution to non-local transport comes from small values of d. While the main frame shows the results obtained for a constant gap ∆, the inset shows the corresponding data with the self-consistent gap.

FIG. 6 :

 6 FIG.6:(Color online). Schematic representation of how AR-AR couples to current-current cross-correlations. The initial state consists of (i) a spin-down electron created at α at time t, on the superconducting side of the NaS interface, and of (ii) a spin-up electron destroyed at time t at b, on the normal side of interface SN b . The final state consists of a spin-down electron destroyed at time t ′ at a, on the normal side of interface NaS, and of a spin-up electron created at time t ′ at β, on the superconducting side of interface SN b . To the AR-AR process shown on Fig.2d, the permutation of two fermions is added. Taking into account the resulting minus sign leads to positive current-current cross-correlations.

25 FIG. 7 :NT 25 FIG. 8 :

 257258 FIG. 7: (Color online). The figure shows dS a,b /dV as a function of N for highly transmitting interfaces [for N see Fig. 1(b)]. In agreement with the BTK calculations dS a,b /dV > 0. The exponential decay is described by the coherence length which increases with M , in agreement with Ref. 20. The values M = 20 (red squares) and M = 25 (blue circle) are used. Strong deviations from the exponential decay (red lines) appear for N ≃ M (see Ref. 20). The data points are obtained from Eq. (C4). The decay is ∝ exp[-2N a0/ξ(M )].

FIG. 9 :

 9 FIG. 9: (Color online). Relation between dS a,b /dV and the transmission coefficients TEC and TCAR: Panel (a) shows the relation between dS a,b /dV and TEC and panel (b) shows the relation between dS a,b /dV and TCAR. The data are the same as in Fig. 7. The different points correspond to different lengths N along Ox axis [see N in Fig. 1(b)]. The blue dashed lines show the linear fits dS a,b /dV = 8TEC, and dS a,b /dV = 50TCAR.

FIG. 11 :

 11 FIG. 11: (Color online). The figure shows the gap profile in panel (a), the phase profile in semi-log scale in panel (b), for a given realization of disorder with V /W = 1. See the text for the meaning of the parameters V and W . The x and y axis are the same as in Fig. 1(b). This figure is obtained with the self-consistent algorithm developed in Ref. 20 for the ballistic case. The phase in panel (b) is small but non-zero because the algorithm takes into account non-equilibrium effects.

22 bb. 8 ĜA

 228 αβ G R,22 βαg -+,11 bb G R,11 αβ G R,12 βα + 4iπt 6 ρ b + g -+,The last term involves advanced-retarded transmission modesS MIXED =8π 2 t 8 ρ a ρ b + g -+,22 aa g +-,11 aa (G A,11 αα G R,12 αα -G A,12 αα G R,22 αα )(G A,21 αβ G R,11 βα -G A,22 αβ G R,21 βα ) + g -+,11 aa g +-,22 aa (G A,21 αα G R,11 αα -G A,22 αα G R,21 αα )(G A,11 αβ G R,12 βα -G A,12 αβ G R,22 βα ) + g -+,22 bb g +-,11 bb (G A,11 βα G R,12 αβ -G A,12 βα G R,22 αβ )(G A,21 ββ G R,11 ββ -G A,22 ββ G R,21 ββ ) + g -+,11 bb g +-,22 bb (G A,21 βα G R,11 αβ -G A,22 βα G R,21 αβ )(G A,11 ββ G R,12 ββ -G A,12 ββ G R,22 ββ ) -4iπt 6 ρ a + g -+,11 bb g +-,22 bb (G A,12 ββ + G R,12 ββ )(G A,21 βα G R,11 αβ -G A,22 βα G R,21αβ ) Evaluation of the Green function Ĝα,β connecting two interfacesIt will be shown that, at small energy compared to the gap, the advanced-advanced AR-AR transmission modeG A,1,1 α,β G A,2,2 β,αav can be replaced by the opposite of the advanced-retarded EC transmission mode -GA,1,1 α,β G R,1,1 β,α av .The Green function is expanded in powers of the exponential coefficient exp(-R x /ξ) appearing in ĝα,β , giving

  α,β = ĝR α,β ≡ ĝα,β for energies within the gap. The off-diagonal Nambu components are vanishingly small if ω ≪ ∆. It is deduced that

  Zero frequency noise (see Sec. II) is obtained from S a,a = 2e 2 t 2 /h dωS a,a (ω) and S a,b = 2e 2 t 2 /h dωS a,b (ω), with

							The trace is evaluated over the Nambu labels. Eq. (C4)
							is a generalization of Ref. 56 to two interfaces with many
							channels. The numerical calculations presented in the
			55				main body of the article are based on Eqs. (C2), (C3)
	I a,α =	h m 2e	dω tam,αm	Ĝ+,-αm,am (ω)		(C2)	and (C4).
			-tαm,am	Ĝ+,-am,αm (ω)	1,1	.
	S a,a (ω) =	Tr Ĝ+,-am,an (ω) Ĝ-,+ αn,αm (ω)		(C3)
		n,m	+ Ĝ+,-αm,αn (ω) Ĝ-,+ an,am (ω)
			-Ĝ+,-am,αn (ω) Ĝ-,+ an,αm (ω)
			-Ĝ+,-αm,an (ω) Ĝ-,+ αn,am (ω) ,
	S a,b (ω) =	Tr Ĝ+,-bm,an (ω) Ĝ-,+ αn,βm (ω)		(C4)
		n,m	+ Ĝ+,-βm,αn (ω) Ĝ-,+ an,bm (ω)
			-Ĝ+,-bm,αn (ω) Ĝ-,+ an,βm (ω)
			-Ĝ+,-βm,an (ω) Ĝ-,+ αn,bm (ω) .

  Another contribution contains transmission modes of the type G A,11 α,β G A,12 β,α . These terms read S ′ =8π 2 t 8 ρ a ρ b + g +-,11

	aa	G A,11 αα	g -+,22 bb	G A,21 ββ G R,12 αβ G R,11 βα -g -+,11 bb	G A,12 ββ G R,11 αβ G R,21 βα
	+ g +-,11 aa	G A,12 αα	g -+,22 bb	G A,22 ββ G R,22 αβ G R,21 βα -g -+,11 bb	G A,11 ββ G R,21 αβ G R,11 βα
	+ g +-,22 aa	G A,21 αα	g -+,11 bb	G A,11 ββ G R,11 αβ G R,12 βα -g -+,22 bb	G A,22 ββ G R,12 αβ G R,22 βα
	+ g +-,22
	+ g -+,22 aa	g +-,22 bb	12 βα + g -+,11 bb αα G R,12 βα G R,12 αβ G A,21 G A,21 ββ + g -+,11 aa	g +-,11 aa g +-,11 bb	G A,12 αα G A,12 ββ G R,21 αβ G R,21 βα G A,12 αβ G A,12 βα G R,21 ββ αα G R,21	.	(D5)

A(R),ni,nj i,j

Acknowledgments

The authors acknowledge fruitful discussions with B. Douçot, D. Feinberg, M. Houzet, F. Lefloch, P. Samuelsson, R. Whitney. R. M. is grateful to S. Bergeret and A. Levy Yeyati for a previous collaboration, 20 and thanks the latter for useful remarks on a preliminary version of the manuscript. We thank one of the referees for pointing out to us the possibility to perform the integrals within the BTK model analytically. Support from contract ANR P-NANO ELEC-EPR is acknowledged.

stands for the Nambu component (n i , n j ) of the advanced (retarded) Green function connecting i to j.

The dependence on M of the coherence length was The elements s α,β i,j of the scattering matrix are calculated within the BTK approach 58 using two-component wave-functions describing electrons and holes respectively. The indices i, j refer to the normal electrodes N a and N b , while α, β run over the two components describing the electrons and holes.

For example, the wave-functions for an electron incoming from electrode N a take the form ) x + s e,e a,a e -iq (+) x + s h,e a,a 0 1 e iq (-) x , (A1)

where ψ a (x), ψ S (x) and ψ b (x) are the parts of the wave-

Appendix D: Microscopic calculations for the noise formula

In this Appendix we provide the complete formula for the noise S a,b (ω) given by Eq. (C4) for sub-gap voltage (|ω| < eV < ∆). This allows considerable simplification of the expression of S a,b because the Keldysh Green function g +-= 0 in the isolated superconductor, as there exist no single-electron states. The total noise can be decomposed into different terms according to the types of transmission modes in the superconductor as:


Mimoza ; Gjikaj
email: mimoza.gjikaj@tu-clausthal.de

Wolfgang Brockner

Tao

Dr * Pd

Keywords: Synthesis, crystal structure and vibrational spectrum 2, 3, 5-Triphenyltetrazolium hexachloroniobate(V), 2, 3, 5-Triphenyltetrazolium oxotetrachloroniobate

    Synthesis, crystal structure and vibrational spectrum of 2,3,5-triphenyltetrazolium hexachloroniobate(V) and oxotetrachloroniobate(V) acetonitrile, [TPT] [NbCl6] and [TPT] [NbOCl4(CH3CN)]

D calculated /g cm - 3 1.738 Abstract. Single crystals of 2,3,5-triphenyltetrazolium hexachloroniobate(V), [TPT] [NbCl 6 ], (1) and of 2,3,5-triphenyltetrazolium oxotetrachloroniobate(V), [TPT] [NbOCl 4 (CH 3 CN)], (2) could be obtained and their crystal structures determined. The yellow compound 1 crystallizes in the monoclinic space group P2 1 /c (C 2h 5 , no. -units [1][2][3]. In case of Na[NbCl 6 ] and of the low temperature K[NbCl 6 ] phase [2], another still unknown crystal structure is formed according to X-ray Guinier and Raman spectroscopic investigations [2,3]. This is in accordance to the respective phase diagrams of the NbCl 5 -MCl systems [3,4]. Secondly, the existence of from [9]. Another way to oxotetra-chloroniobates(V) runs via a synproportionation of niob(V)oxides with NbCl 5 in the presence of CsCl in molten salt state [10]. The all together rare oxotetrachloroniobate(V) unit is found furthermore in the uncommon compounds [(η 5 -C 5 H 5 )Nb(CNCMe 3 ) 4 Cl] [NbOCl 4 (THF)] ⋅ (THF)] [11] and in Te 6 (NbOCl 4 ) 2 [12]. With the obtaining of single crystals of the title compounds the opportunity is used to investigate and characterize the new compounds in more detail.

Results and Discussion

Crystal Structure of [TPT][NbCl 6 ] (1)

Treatment of niobium pentachloride, NbCl 5 , with one equivalent of anhydrous 2,3,5triphenyltetrazolium chloride in acetonitrile afforded [TPT] [NbCl 6 ] (1) as yellow crystals. The product shows rapid decomposition on the slightest exposure to the atmosphere, but the crystals were manipulated under argon.

The complex salt 1 crystallizes in the monoclinic space group P2 1 /c (no. 14) with four formula units per unit cell. The asymmetric unit contains isolated [TPT] + cations and [NbCl 6 ] -anions and its structure is isotypic with [TPT] [SbCl 6 ] [13] (Fig. 1 and Tab. 1).

Figure 1

Table 1 The packing diagram of 1 shows the face-to-face stacked orientation between the phenyl rings of the two neighboring molecules, which is caused by intermolecular π⋅⋅⋅π ). These values agree with those of reported hexachloroniobate(V) salts [14][15][16][17].

Table 2

Crystal Structure of [TPT] [NbOCl 4 (CH 3 CN)] (2)

Reaction of 2,3,5-triphenyltetrazolium chloride monohydrate with NbCl 5 in 1:1 ratio led to precipitation of yellow crystals of title compound 2. cations and one one [NbOCl 4 (CH 3 CN)] -anion [22]. The whole structure can also be seen as being from alternating layers along b axes, in which the [TPT] + cations and the [NbOCl 4 (CH 3 CN)] -anions are linked by hydrogen bonds (Fig. 3).

Figure 3

Vibrational spectra and assignment

2,3,5-Triphenyltetrazolium hexachloroniobate(V), [TPT] [NbCl 6 ], (1)

According to the crystal structure determination the title compound 1 is built up by

[TPT] + cations and [NbCl 6 ] -anions. These units can be clearly detected in the vibrational spectra of 1 (Fig. S1 / Supporting information).

The relevant vibrational modes of [NbCl 6 ] -are summarized in Table 3 along with the proposed assignment and in comparison to literature data [2,23].

Table 3

The Octahedral [NbCl 6 ] -units (O h symmetry) are here the more interesting inorganic components of the crystal structure of 1. A vibrational analysis for such a unit results in [START_REF] Weidlein | Schwingungsspektroskopi[END_REF]: In detail, the very strong Raman mode at 370 cm -1 and the medium FIR band at 372 cm -1 can be assigned to ν 1 in A 1g (Table 3). The very strong and remarkable broad FIR bands at 341/331 cm -1 show a Raman counterpart at 339 cm -1 and they belong to ν 3 / F 1u . ν 2 / E g is split and existent not only in Raman but also in FIR spectrum. The bending mode ν 5 /F 2g , in Raman at 173 cm -1 , is present in FIR spectrum and it is split.

Γ vib ([NbCl 6 ] -/ O h ) = A 1g (RE) + E g (RE) + 2 F 1u (IR) + F 2g (RE) + F 2u (i
Only ν 4 / F 1u appears in the FIR spectrum at 161 cm -1 as a very strong band. Altogether, our presented vibrational data are in agreement with these given in [2,23], as well as with those of [Et 4 N] + [NbCl 6 ] - [START_REF] Beattie | [END_REF]. Further details may be obtained from Tab. 3. Raman and IR bands in this region are split due to Fermi resonance [27] and shifted by about 20 cm -1 to higher wave numbers in comparison to these ones of liquid acetonitrile [28][29][START_REF] Smith | Infrared Spectral Interpretation[END_REF] and in accordance to other weak bounded acetonitrile adducts [19,[START_REF] Kawano | [END_REF].

2,3,5-Triphenyltetrazolium oxotetrachloroniobate(V) acetonitrile adduct,[TPT] [NbOCl 4 (CH 3 CN)], (2)

Title
The Raman and IR/FIR spectra of crystalline compound 2 are shown in Fig. S2 / Supporting information. In this contribution, the main interest is directed on the inorganic [NbOCl 4 ] -respective [NbOCl 4 (CH 3 CN)] -group. Therefore, the relevant [NbOCl 4 ] -vibrational modes of crystalline 2 in the spectral region below 1000 cm -1 are presented in Table 4 along with their estimated intensities and the proposed assignment.

All the vibrational frequencies of compound 2 with marked peculiarities (in parenthesis) are given as footnote 2 below Table 4 as well as the values of acetonitrile for completeness reasons.

Table 4

As frequencies and their PED (Potential Energy distribution) values were done with help of the programs GAMFORCE [START_REF] Bache | [END_REF] and MOLVIB [START_REF] Sundius | [END_REF].

In the Table 4 our [NbOCl 4 ] -vibrational modes of crystalline 2 are presented along their proposed assignment based on the mentioned PED data and in comparison to the vibrational frequencies of [NbOCl 4 ] -given in [10]. According to that (Fig. S2 and Table 4) ν(Nb=O) is proved by the strong band at 945 and 955 cm -1 , respectively. The bands at 341/330 and 244/243 cm -1 belong to NbCl 4 stretchings whereas these ones at 168/175 cm -1 are NbCl 4 bendings. The Raman band at 140 cm -1 is attributed to a ONbCl 4 bending. Bands below 100 cm -1 may be ascribed to so-called lattice vibrations.

Herewith our proposed assignment of the anionic unit differs considerable from this one of the [Ph 3 MeP] + salt [9] especially due to the differences in the FIR spectra.

Concluding remarks

Two new niobate compounds, 2,3,5-triphenyltetrazolium hexachloroniobate(V) (1) and oxo-tetrachloroniobate(V) acetonitrile ( 2), have been synthesized. The identity and purity of the synthesized compounds were confirmed by elemental analysis, infrared and Raman spectroscopy, and single X-ray diffraction.

In compound 1, no hydrogen bonds are found between the octahedral [NbCl 6 ] -anions and 2,3,5-triphenyltetrazolium cations.

Experimental Section

Materials and Methods: All manipulations and chemical reactions were carried out in a dry argon atmosphere using standard Schlenk, vacuum-line and glove-box techniques.

NbCl 5 was sublimated for purification, and 2,3,5-triphenyltetrazolium chloride monohydrate (Fluka) was used without additional purification. CH 3 CN was distilled and dried by P 4 O 10 . Obviously, the commercial anhydrous [TPT]Cl contained water, therefore the "anhydrous" 2,3,5-triphenyltetrazolium chloride was dried under reduced

X-Ray diffraction analysis

Since both compounds are sensitive towards hydrolysis by moist air and the selection of crystals was performed in an argon filled glove box. Crystals were transferred into glass capillaries which were sealed. The crystals selected had the size 0.

1 and 2 Keywords: 2 , 3 , 5 -

 12235 14) with a = 1321.6(3), b = 1080.7(2), c = 1740.2(4) pm, β = 111.54(2)°, and Z = 4. Its structure is isotypic to that of the recently characterized [TPT] [SbCl 6 ] and built up by [TPT] + cations and octahedral [NbCl 6 ] -anions. The light yellow compound 2 crystallizes in the orthorhombic space group Pbcm (D 2h 11 , no. 57) with a = 1574.4(1), b = 1047.7(1), c = 1723.4(1) pm, and Z = 4. The ionic compound contains [TPT] + cations and [NbOCl 4 (CH 3 CN)] -anions. The nitrogen atom of the acetonitrile solvate molecule is coordinated to the niobium ion in the trans-position to the oxo ligand. The Nb-O bond length of 169.6 pm corresponds to a double bond. The [TPT] [NbOCl 4 (CH 3 CN)] formation proceeds with the hydrate [TPT]Cl ⋅ H 2 O obvious via NbOCl 3 . The FT-Raman and FT-IR/FIR spectra of both compounds have been recorded and interpreted, especially with respect to the inorganic structure parts. The melting temperatures of are given. Triphenyltetrazolium hexachloroniobate(V); 2,3,5-Triphenyltetrazolium oxotetrachloroniobate(V) acetonitrile; Crystal structure; Raman; IR/FIR Introduction Crystalline niobium pentachloride, consisting of dimeric Nb 2 Cl 10 units, forms with the heavier alkali chlorides MCl (M = K, Rb and Cs) M[NbCl 6 ] compounds with octahedral [NbCl 6 ]

  interactions. The closest contact between the neighboring phenyl rings is 369 pm, and it is similar to that found in [TPT] [SbCl 6 ] [13]. In the cationic unit the five atoms in the tetrazolium ring (N(1), N(2), N(3), C(4) and N(5)) are coplanar. N(1) -N(5) and N(2) -N(3) are equivalent (131 pm). The N(1) -N(2) bond is much longer than the formers two bonds (2.0 pm). The elongation of this bond may be ascribed to the repulsion between the relevant phenyl groups. The deviations of C(6), C(12) and C(18) atoms from the tetrazolium plane are 5.4, -35.4 and 19.9 pm, respectively. The torsion angles of C(6), C(12) and C(18) phenyl rings are 94.31, -36.81 and 22.05°, respectively. These values agree well with those reported for [TPT] [SbCl 6 ] [13]. The hexachloroniobate anion is approximately octahedral coordinated by six chlorine atoms. The metal-chlorine bond distances range from 232.1(2) to 239.4(2) pm (mean value of 235.0 pm). The cis-Cl -Nb -Cl angles range from 89.7 to 92.4°, and trans-Cl -Nb -Cl angles of 176.9 and 179.4° in the equatorial positions, and 177.9° in the axial positions (Tab. 2

  [TPT] [NbOCl 4 (CH 3 CN)] crystallizes in the orthorhombic space group Pbcm (no. 57) with four formula units per unit cell. The crystal structure of 2 consists of discrete [TPT] cations and [NbOCl 4 (CH 3 CN)] anions (Fig. 2).

Figure 2 The[

 2 Figure 2The [NbOC1 4 (CH 3 CN)] -ion has C 4v symmetry long known for oxotetrahalometallate anions of the early transition metals[18,19]. The niobium atom is displaced by 33 pm toward the oxo ligand from the best mean plane through the four chlorine atoms. The short Nb-O bond of 169.6 pm is comparable to those found for [(η5 -C 5 H 5 )Nb(CNCMe 3 ) 4 Cl][NbOCl 4 (THF)] • THF [11] and Tl[NbOCl 4 ] [20] and indicates multiple bond character while in the trans-position the long Nb-N(3) bond of 245.8 pm indicates only weak bonding (Fig. 3 and Tab. 2). The coordination of the acetonitrile molecule is linear with Nb-N(3)-C(14) of 178.9 and N(3)-C(14)-C(15) of 179.1°. The Nb-Cl bond lengths range from 239.1 to 239.8 pm. The cis-Cl -Nb -Cl angles range from 86.7(1) to 90.8°, and the trans-Cl -Nb -Cl angle is 164.2° in the equatorial positions and they compare well with those of known hexachloro-niobate(V) salts [9, 21].

  [TPT] cation exhibits approximately C 2 symmetry, therefore all its vibrations are Raman active and infrared allowed, what is also verified by its whole vibrational spectrum. The [TPT] + vibrations result in their characteristic modes for the aromatic -C-H stretchings and bendings as well as for the phenyl ring and the tetrazolium heterocyclus modes. The corresponding Raman as well as IR bands are very similar to those of [TPT] + [Cl] -[24] and [TPT] [SbCl 6 ] [13]. The relevant [TPT] + Raman and IR modes may be taken from footnote 1.

  .a.), with RE = Raman, IR = Infrared and i.a. = inactive, and with A 1g + E g + F 1u as stretchings and F 1u + F 2g as bendings.The Raman as well as the IR/FIR spectra of 1 do not exhibit the expected pattern of a [MCl 6 ] octahedron, but they show much more vibrational modes. It is also noteworthy that the selection rules and the rule of mutual exclusion do not have an effect, so that the [NbCl 6 ] -fundamentals appear in the Raman as well as FIR spectrum. According to the crystal structure determination, the [NbCl 6 ] -ion is built up octahedrally, so that the sketched peculiarities are caused by symmetry lowering of the anion by the crystal lattice of 1 (removal of degeneracy). (In the monoclinic [TPT] [NbCl 6 ]: P2 1 /c (C 2h 5 ), Nb is on the general position of 4e, what means site symmetry C 1 for Nb respective [NbCl 6 ] -).

  compound 2 is built up by [TPT] cations, [NbOCl 4 (CH 3 CN)] anions. Additional acetonitrile solvate molecules are included in the crystals, likely as fluid inclusions. In the vibrational spectra of 2 the mentioned building components can be clearly recognized. The acetonitrile is weakly coordinated to Nb in the complex anion confirmed by the Nb-N distance 245.8 pm. The two different acetonitrile components can be clearly distinguished by their CN-stretchings in the range 2320-2250 cm -1 .

in title compound 1 ,

 1 the [TPT] + vibrations of 2 are Raman active and infrared allowed and they are found in the characteristic group frequency ranges. The [TPT] + Raman as well as IR bands are very similar to those in [TPT] [NbCl 6 ] (c.f. footnote 1), [TPT] + [Cl] -[24] and in [TPT][SbCl 6 ] [13]. All the relevant [TPT] + Raman and IR modes are summarized in footnote 2. weak acetonitrile coordination the anionic spectral parts are best interpreted by [NbOCl 4 ] -units rather than by [NbOCl 4 (CH 3 CN)] -ions as in the first structurally characterized saltlike complex triphenylmethylphosphonium oxotetrachloroniobate(V) acetonitrile, [Ph 3 MeP] + [NbOCl 4 (CH 3 CN)] -[9]. The square pyramidal [NbOCl 4 ] -group exhibits C 4v symmetry. A vibrational analysis for this unit results in [32]: Γ vib ([ONbCl 4 ] -/ C 4v ) = 3 A 1 (IR, RE) + 2 B 1 (RE) + B 2 (RE) + 3 E (IR, RE), with IR = Infrared and RE = Raman, and with 2 A 1 + B 1 + E as stretchings and the remaining A 1 + B 1 + B 2 + 2 E as bendings. The Raman and IR/FIR frequencies for [NbOCl 4 (CH 3 CN)] -in crystalline [Ph 3 MeP] [NbOCl 4 (CH 3 CN)] are given and assigned in part only [9]. Furthermore, NbOCl 3 -NbCl 5 -CsCl melts have been investigated by Raman spectroscopy [10] and the characteristic [NbOCl 4 ] -frequencies were determined therefrom. The given interpretation of the Raman data has been assisted by quantum mechanical ab initio SCF (Self Consistent Field) calculations. The included calculations of the [NbOCl 4 ] -

Title compound 2

 2 could be obtained by the reaction of [TPT]Cl ⋅ H 2 O with niobpentachloride in acetonitrile according to [TPT]Cl ⋅ H 2 O + NbCl 5 (in CH 3 CN) → [TPT] + [NbOCl 4 (CH 3 CN)] -+ 2 HCl As expected, [NbOCl 4 (CH 3 CN)] -exhibits C 4v symmetry and Nb 5+ is coordinated by one oxygen, four chloride ions and one acetonitrile molecule. The niobium is displaced by 33 pm toward the oxo ligand from the best mean plane through the four chlorine atoms. The short Nb-O bond of 169.6 pm indicates multiple bond character while in the transposition the long Nb-N bond of 245.8 pm indicates only weak bonding. The [NbOCl 4 (CH 3 CN)] -anion is hydrogen bonded to four 2,3,5-triphenyltetrazolium cations.

Figure 1

 1 Figure headings

Figure 2

 2 Figure 2 View of the crystal structure of [TPT] [NbOCl 4 (CH 3 CN)] along the a-axis.

Figure 3

 3 Figure 3 Interaction (dotted lines) between two [TPT] cations and one [NbOCl 4 (CH 3 CN)] anion in compound 2. (The rest of hydrogen atoms are not displayed for clarity).

crystal structure and vibrational spectrum of 2,3,5-triphenyltetrazolium hexachloroniobate(V) and oxotetrachloroniobate(V) acetonitrile, [TPT] [NbCl 6 ] and [TPT] [NbOCl 4 (CH 3 CN)]

	Synthesis, Mimoza Gjikaj*, Tao Xie and Wolfgang Brockner
	Clausthal-Zellerfeld, Institut für Anorganische und Analytische Chemie der Technischen Universität
	Clausthal
	Received, 2009
			1.374
	µ Mo-Kα /mm -1	1.229	0.819
	F(000)	1200	1172
	Temperature /K	223(2)	223(2)
	Diffractometer	Stoe IPDS 2	Stoe IPDS 2
	Monochromator	Graphite	Graphite
	Wavelength /pm	71.073	71.073
	2θ range /°	1.00 -25.02	1.00 -25.03
	Index ranges	-15 ≤ h ≤ 15	-18 ≤ h ≤ 18
		-12 ≤ k ≤ 12	-11 ≤ k ≤ 12
		-20 ≤ l ≤ 20	-19 ≤ l ≤ 20
	Unique reflections	4087	2612
	Data / refined parameters	4087 / 331	2612 / 180
	Goodness-of-fit on F 2	0.984	1.159
	R 1 (I ≥ 2σ(I))	0.0543	0.0631
	wR 2 (I ≥ 2σ(I))	0.0973	0.1070
	R 1 (all data)	0.0812	0.0731
	wR 2 (all data)	0.1194	0.1226
	Largest diff. peak and hole Peak / e • pm -3	0.880 / -1.139	0.698 / -1.252

  [NbCl 6 ] - deviating species as e.g. [Nb 2 Cl 11 ] -in dependency of the NbCl 5 : Cl stoichiometry is still open [5]. The formation of such "acidic" [Nb 2 Cl 11 ] -species out of Nb 2 Cl 10 + Cl -

	triphenylmethylphosphonium	oxotetrachloroniobate(V),	[Ph 3 MeP] +	[NbOCl 4
	(CH 3 CN)] -, characterized until now only, was synthesized by an intricate route via
	NbCl 5 with CCl 3 NO 2

would be analogous to the formation of well known [Al 2 Cl 7 ] -ions

[6,7]

. The large [TPT] + ion could influence the kind of the chloroniobate(V) species in the reaction of NbCl 5 with [TPT] + [Cl] -. Moreover, [TPT]Cl is hygroscopic and reacts with moisture to a triphenyltetrazolium chloride hydrate, ([TPT]Cl ⋅ H 2 O) [8]. Besides its chloride ion donor properties this hydrate can form oxohalide compounds with water sensitive halides. In such a reaction fortunately and more fortuitously single crystals of [TPT] [NbOCl 4 (CH 3 CN)] could be obtained. Obviously, the formation reaction proceeds via NbOCl 3 . The crystalline

Page 16 of 23 Wiley-VCH ZAAC pressure

  in Schlenk tubes (120-130 o C, 2 h). The melting points of 1 and 2 were measured on a BÜCHI 520 apparatus and they are uncorrected values. NbCl 5 were mixed in a Schlenk tube. The mixture was kept for at 140 °C for 3 h. Upon cooling 15 mL CH 3 CN was added forming a yellow clear solution. After some days the yellow, block-shaped crystals were formed. C 19 H 15 Cl 6 N 4 Nb: calc. C 37.72, H 2.50, N 9.26; found C 37.32, H 2.53, N 9.10. The melting temperature was found to be 128 o C. was synthesized by using the same method as that for compound 1, except for the replacement of [TPT]Cl ⋅ H 2 O by dried [TPT]Cl. Upon cooling to room temperature, yellow, block-shaped crystals formed. C 21 H 18 Cl 4 N 5 NbO: calc. C 42.67, H 3.07, N 11.85; found C 41.99, H 3.11, N 11.21. The melting temperature was found to be 154 o C.

	2,3,5-Triphenyltetrazolium hexachloroniobate(V) (1): 352.8 mg (1 mmol) [TPT]Cl ⋅
	H 2 O and 270.4 mg (1 mmol) 2,3,5-Triphenyltetrazolium oxotetrachloroniobate(V) acetonitrile (2): Compound 2

Table headings Table 1

 headings1 Crystal data and structure refinement of [TPT] [NbCl 6 ] (1) and [TPT]

	[NbOCl 4 ] (2)
	Table 2 Selected internuclear distances / pm and angles / ° for [TPT] [NbCl 6 ] (1) and
	[TPT] [NbOCl 4 ] (2)

Table 3

 3 Raman and FIR frequencies (cm -1 ) of crystalline [TPT] [NbCl 6 ] in the [NbCl 6 ] -

Table 3

 3 Raman and FIR frequencies (cm -1 ) of crystalline [TPT] [NbCl 6 ] in the [NbCl 6 ] -relevant spectral region below 500 cm-1 along with their estimated intensities and proposed assignment Raman (cm -1 / intensity) of crystalline [TPT] [NbCl 6 ] : 3161 vw, 3080 sh, 3068 s, 3045 sh, 1607 vvs, 1583 s, 1527 vs, 1485 m, 1457 s, 1410 sh, 1394 w, 1354 vvs, 1336 vw, 1306 w, 1285 vw, 1220 w-m, 1181 w, 1074 w, 1042 sh, 1005 vs, 993 vs, 830 vw, 729 vw, 684 vw, 624 vw, 608 vvw, 523 vvw, 406 vw, 370 vvs, 339 w, 292 vw, 260 w, 246 w, 173 s, 137 sh, 118 sh, 104 s. IR/FIR (cm -1 / intensity) of crystalline [TPT] [NbCl 6 ]: 3175 vw, 3087 w, 3064 m, 3047 sh, 1608 s-vs, 1575 sh, 1528 vs, 1487 vs, 1470 sh, 1458 vvs, 1357 w-m, 1300 s, 1273 w, 1172 s, 1158 vs, 1145 m, 1105 m, 1075 m, 1030 m, 998 vs, 918 m, 788 s, 764 vvs, 755 vvs, 730 sh, 726 vvs, 687 vvs, 670 sh, 618 w, 607 vs, 520 m, 508 m-s, 490 s, 440 vvw, 423 vvw, 403 vvw, 372 m, 341 vs, br, 331 vs, br, 275 m, 268 sh, 188 vw, 174 sh, 161 vvs, 144 vvw.

	Page 22 of 23
	Raman	FIR	[NbCl 6 ] -	Assignment / O h
			[2, 23] a	mode description
	370 vvs	372 m	375 vs, br	ν 1 / A 1g : ν s
	339 w	341vs, br	333 vs, br	ν 3 / F 1u : ν as
		331 vs, br
	292 vw			[TPT] +
	260 w	275 m	287 w	ν 2 / E g : ν
	246 w	268 sh
	173 s	188 vw	182 m, sh	ν 5 / F 2g : δ
		174 sh	177 s
		161 vvs	176 s	ν 4 / F 1u : δ
			163 s
	137 sh	144 vvw		(ν 6 / F 2u : δ)?
	118 sh			Lattice vibration
	104 s			Lattice vibration

Estimated intensities: s: strong, m: medium, w: weak, v: very, sh: shoulder, br: broad. ν: stretching, δ: bending. a) [2]: Raman for [TlNbCl 6 ]; [23]: FIR for [KNbCl 6 ]. Footnote 1.

Table 4

 4 Vibrational frequencies (cm -1 ) of [NbOCl 4 ] -in title compound 2 in the relevant spectral region below 1000 cm -1 along with their estimated intensities and proposed assignment and in

	comparison to literature data
	Raman	IR / FIR	[NbOCl 4 ] -[10]	Assignment / C 4v , mode description
	945 s	955 vs, 945 vs	942 -962	ν 1 / A 1 : ν Nb = O
	341 m-s	330 vvs, br	312 -333	ν 2 / A 1 : ν s NbCl 4
	168 w	175 w	150	ν 3 / A 1 : δ NbCl 4
			268	ν 4 / B 1 : ν NbCl 4
			(141) calc.	ν 5 / B 1 : δ NbCl 4
	140 w-m		165	ν 6 / B 2 : δ ONbCl 4
	337 sh	330 vvs, br	313 -315	ν 7 / E : ν as NbCl 4
	244 s	243 s	232 -237	ν 8 / E : δ NbCl 4
			(47) calc	ν 9 / E : δ ONbCl 4 / or lattice vibration

Estimated intensities: s: strong, m: medium, w: weak, v: very, sh: shoulder, br: broad, ν: stretching, δ: bending.

Acknowledgements. The authors are indebted to Prof. Dr. Arnold Adam for his support.

We thank K. Bode for recording the Raman spectra and Dr. V. Zapol'skii for the determination of the melting points of compounds 1 and 2.