T Jonckheere

J Rech

A Zazunov

R Egger

T Martin

Hanbury Brown and Twiss noise correlations in a topological superconductor beam splitter

Keywords: numbers: 73.23.-b, 72.70.+m, 74.45.+c

We study Hanbury-Brown and Twiss current cross-correlations in a three-terminal junction where a central topological superconductor (TS) nanowire, bearing Majorana bound states at its ends, is connected to two normal leads. Relying on a non-perturbative Green function formalism, our calculations allow us to provide analytical expressions for the currents and their correlations at subgap voltages, while also giving exact numerical results valid for arbitrary external bias. We show that when the normal leads are biased at voltages V1 and V2 smaller than the gap, the sign of the current cross-correlations is given by -sgn(V1 V2). In particular, this leads to positive cross-correlations for opposite voltages, a behavior in stark contrast with the one of a standard superconductor, which provides a direct evidence of the presence of the Majorana zero-mode at the edge of the TS. We further extend our results, varying the length of the TS (leading to an overlap of the Majorana bound states) as well as its chemical potential (driving it away from half-filling), generalizing the boundary TS Green function to those cases. In the case of opposite bias voltages, sgn(V1 V2) = -1, driving the TS wire through the topological transition leads to a sign change of the current cross-correlations, providing yet another signature of the physics of the Majorana bound state.

I. INTRODUCTION

In the last two decades, Majorana fermions, 1 concepts which initially were the strict property of particle physics, found some correspondence in condensed matter physics settings. Instead of looking whether an elementary particle, such as the neutrino, qualifies as a Majorana fermion, nanoscience physicists are now wondering whether a complex many body electronic system with collective excitations could bear such strange objects: a fermion whose annihilation operator is (sometimes trivially) related to its creation counterpart. Indeed, Kitaev 2 showed that a one dimensional wire with tight-binding interactions and p-wave pairing exhibits Majorana fermions at its boundaries. Recent reviews give a broad summary of this work and its consequences in condensed matter physics. [3][4][5] There has been an ongoing effort to study experimentally whether this toy model has some correspondence in physical systems. Among strong candidates are nanowires with Rashba and Zeeman coupling put in proximity to a BCS superconductor, [6][7][8][9][10] and chains of iron atoms deposited on top of a lead surface. 11 In Refs. 9 and 10 the signature for the presence of a Majorana fermion at the edge of a one dimensional nanowire consists of a zero bias anomaly in the current voltage characteristics. These results call for the exploration of more involved transport settings and geometries, where the behavior of the Majorana fermion can be fully investigated and characterized. Among these settings, multi-terminal hybrid devices offer unique perspectives.

Multi-terminal devices have often played an important role for exploring the electronic transport properties of mesoscopic devices. They allow to perform experiments in close analogy with quantum optics scenarios: in the Hanbury-Brown and Twiss 12 (HBT) experiment for instance, photons impinging on a half-silvered mirror are either transmitted or reflected, and the crossed correlations of intensities from these two outputs are measured, yielding a positive signal due to the bunching of photons emitted from a thermal source. Transposed to condensed matter setups, the sign of the HBT crosscorrelations reveals meaningful information concerning the physics at play. For a DC biased three-terminal normal conductor, the electronic analog of the HBT experiment was studied theoretically and experimentally two decades ago. [13][14][15][16][17] This was analyzed in terms of currentcurrent crossed correlations (noise): fermion antibunching (resulting from Pauli principle) leads to a negative HBT noise signal. In the context of conventional BCS superconductivity, three-terminal devices consisting of a superconductor connected to two normal leads were also investigated. In such devices, a Cooper pair can be transferred as a whole in one of the normal leads (via Andreev arXiv:1611.03776v2 [cond-mat.mes-hall] 2 Mar 2017 Reflection or AR), or it can be split into its two constituent electrons in opposite leads (via Crossed Andreev Reflection or CAR). The HBT noise correlations can thus be negative or positive depending on whether AR or CAR is dominant. 18,19 By adding appropriate filters (in energy or spin) to the device in order to rule out AR in each lead, positive noise crossed correlations, due solely to CAR processes, can be guaranteed. 20,21 Experimental evidence for Cooper pair splitting in BCS superconductors has been found both in non-local current measurements [22][23][24] as well as in noise correlation measurements 25 with a device analogous to what was proposed in Refs. 20 and 26. In this work, we study HBT noise correlations for a pair splitter, where a TS nanowire, rather than a standard BCS superconductor, is connected to two biased normal leads. A schematic view of the setup is shown in Fig. 1. Because of the presence of the Majorana zeromode for the TS, the AR and CAR processes are strongly affected. Studying the current and the HBT crossedcorrelations in this setup, one can expect to explore properties and manifestations of the Majorana bound state. This setup was previously studied using scattering theory and tight-binding numerical calculations. [27][28][29] However, these works concentrated on the specific case of equal voltages for the normal leads and focused on voltages below the gap. Our goal is to provide a complete description of the system, exploring the behavior of the current and current correlations, for arbitrary values of the voltages, thus capturing both the effect of the Majorana bound state and the high-energy quasiparticles. Below the gap, we confirm that a TS beam splitter has negative HBT correlations when the two normal leads have equal voltages. 28,29 More importantly, we also predict a reversal of the sign of the current correlations when voltages are changed from equal to opposite, a feature that is directly related to the properties of the Majorana bound state at the end of the TS nanowire.

We perform the calculations within a phenomenological tunnel Hamiltonian approach, using the boundary Keldysh Green function of a semi-infinite TS. Solving non-perturbatively the Dyson equation, this allows us to obtain analytical formulas for the current and current correlations. This approach was introduced in Ref. 30, and can be used to treat a system composed of an arbitrary number of leads. The boundary Keldysh Green function of the TS nanowire encapsulates all the properties of the TS boundary. The corresponding density of states shows a zero-energy peak, associated with the Majorana zero-mode, and a non-zero density of states above the proximity-induced gap without BCS singularity.

Moreover, we can further emphasize the role played by the Majorana bound state by generalizing the TS boundary Green function to the case of a finite length nanowire, or one with an arbitrary chemical potential (adjustable doping). For a TS of finite length L, the two Majorana bound states localized at the opposite ends of the nanowire can overlap, leading to a vanishing of its effect for small enough L, which we confirm by looking at the L-dependence of the HBT correlations. Also, tuning the chemical potential of the TS allows us to drive the transition from a topological regime to a trivial (nontopological) one, which is also manifest in the current correlations.

The structure of the paper is as follow. In Sec. II we introduce the Hamiltonian model and detail the formulas for the transport properties. Sec. III is devoted to our main results. First, analytical expressions and numerical results for the current and differential conductances are presented and discussed. These quantities can readily be measured in experimental setups. We then provide explicit expressions for the current correlations in the subgap regime, along with numerical results at any bias. A detailed qualitative discussion of the noise behavior, in relation to the particular properties of the Majorana bound state is also given. Sec. IV focuses on finite size effects as well as the impact of tuning the chemical potential of the nanowire, driving it away from half-filling. In App. A, the derivation of the boundary Green function for a TS nanowire with a finite bandwidth and arbitrary chemical potential is presented. General analytical expressions for the current and current correlations are provided in App. B. Finally, App. C discusses subtleties of the microscopic tunneling model, and establishes its equivalence with the scattering matrix formalism for subgap voltages.

II. MODEL AND FORMALISM

We consider a three-terminal device in a T-shaped geometry, as illustrated in Fig. 1, where the end of a topological superconductor (TS) nanowire is contacted by two normal-conducting (N) leads. In a general case, two different voltages V 1,2 are applied across the N-TS contacts, while the TS wire is assumed to be grounded. The full Hamiltonian is given by

H = H T S + H N + H t , (1)
where the first two terms describe the TS and two Nleads, respectively, and H t is a tunneling Hamiltonian connecting all three leads to each other (see below for details). We model the TS wire as a semi-infinite 1D spinless p-wave superconductor, corresponding to the continuum version of a Kitaev chain 2,3 in the wide-band limit. The Hamiltonian of the TS wire located at x > 0 reads

H T S = ∞ 0 dx Ψ † T S (x) (-iv F ∂ x σ z + ∆σ y ) Ψ T S (x), ( 2
)
where ∆ is a proximity-induced pairing gap, assumed to be real, the Nambu spinor Ψ T S (x) = (c r , c † l ) T combines right-and left-moving fermions, with annihilation field operators c r (x) and c l (x), respectively, v F is the Fermi velocity and σ x,y,z are the Pauli matrices in Nambu space. In what follows, we use units with

k B = v F = = 1.
In this work, following the approach of Ref. 30, we formulate the transport problem in terms of boundary Keldysh Green functions (GFs) describing the leads which are coupled together by tunneling processes. For such a noninteracting setup with point-like tunneling contacts, the exact boundary GFs can be obtained by solving the Dyson equation to all orders in the tunnel couplings. Below we briefly review the boundary GF approach (see Ref. 30 for details) and summarize relevant formulas needed for the calculation of transport observables in the three-terminal N-TS-N junction.

The boundary Keldysh GF at x = 0 for the TS wire is defined as follows:

ǧT S (t -t ) = -i T C Ψ(t)Ψ † (t ) , (3)
where the Nambu spinor Ψ = (c, c † ) T contains the boundary fermion operator c = [c l + c r ](x = 0), and T C denotes Keldysh time ordering. Explicitly, the Fourier transforms of the retarded and advanced GFs for the uncoupled TS wire in the topologically nontrivial phase derived in Ref. 30 are

g R/A T S (ω) = ∆ 2 -(ω ± i0 + ) 2 σ 0 + ∆σ x ω ± i0 + , (4)
where R/A corresponds to +/-and σ 0 is the unity matrix in Nambu space. Importantly, this simple expression for the retarded/advanced boundary GF of a TS wire captures the zero-energy Majorana bound state as well as continuum quasiparticles, which allows for studying both subgap and above-gap transport on equal footing. The Keldysh component g K T S (ω) is expressed via the retarded and advanced components as

g K T S (ω) =(1 -2n F (ω)) g R T S (ω) -g A T S (ω) , (5)
where n F (ω) = e ω/T + 1 -1 is the Fermi function with temperature T . Throughout the paper we use the chemical potential µ T S of the (grounded) TS wire as a reference energy level and set µ T S = 0. In App. A, we give a derivation of the boundary Green function for a Kitaev chain with the finite bandwidth and arbitrary values for the band filling, while the wide-band expression (4) exhibiting particle-hole symmetry corresponds to half filling.

In the same manner we construct the Keldysh GFs for the normal leads. Within the wide-band approximation, taking into account that the N-TS tunnel coupling effectively involves only one spin component in the normal conductor, the retarded/advanced GF for the normal electrodes follows from Eq. ( 4) by putting ∆ = 0,

g R/A N (ω) = ∓iσ 0 , (6)
Correspondingly, the Keldysh component g K N (ω) is determined via g R/A N (ω) by a relation similar to Eq. ( 5) but with the respective chemical potential µ N in the Fermi function (matrix in Nambu space),

g K N (ω) = -2i [1 -2n F (ω -µ N σ z )] σ 0 . (7)
In the voltage-biased junction, µ N is shifted with respect to µ T S = 0 by the dc voltage across the N-TS contact.

In terms of the boundary fermions c j representing the three leads, with j = 0 for the TS wire and j = 1, 2 for the normal-conducting electrodes, the tunneling Hamiltonian takes the form 30

H t = 1 2 j,j Ψ † j W jj Ψ j , (8)
with Ψ j = (c j , c † j ) T the boundary Nambu spinor and W = W † is the tunneling matrix in lead and Nambu space. In lead space, we impose that W has vanishing diagonal elements W jj = 0 for all j, while the offdiagonal elements of W are matrices in Nambu space, W jj = λ jj σ z , with a hopping amplitude λ jj . For our setup when two normal electrodes are connected to the central TS lead, the only non-vanishing couplings are λ 0j = λ * j0 = λ j for j = 1, 2. Without loss of generality, λ 1,2 can always be chosen real, and in the case of a single tunnel junction they determine the normal transmission probability of the respective N-TS contact. 30 Once the tunneling matrix W is specified, the full Keldysh GF Ǧ of the system follows from the Dyson equation

Ǧ = (ǧ -1 -W ) -1 , (9)
with the Keldysh matrix W = diag(W, -W ), and the "uncoupled" Keldysh GF ǧ is diagonal in lead space.

From the tunneling Hamiltonian (8), it is straightforward to get the Heisenberg operator for the current flowing from lead j to the contact region,

Îj (t) = i e j =j Ψ † j (t)σ z W jj Ψ j (t), (10)
while the dc current I j = Îj (t) is

I j = 1 2 e ∞ -∞ dω 2π j =j tr N σ z W jj G K j j (ω) , (11)
where tr N is the trace over Nambu space. The Keldysh component of the full GF, G K , is given by 30

G K (ω) =G R (ω)F (ω) -F (ω)G A (ω) + G R (ω) [F (ω)W -W F (ω)] G A (ω), ( 12
)
where

F jk (ω) = δ jk [1 -2n F (ω -µ j σ z )
] contains the distribution functions of the uncoupled leads with the respective chemical potentials µ j . Finally, the HBT correlations are readily obtained through the same formalism by computing the zerofrequency cross-correlations of the above-defined current operator. Quite generally, the current correlations at zero frequency are defined as

S jj = ∞ -∞ dτ δ Îj (τ )δ Îj (0) , (13)
with δ Îj (t) = Îj (t) -I j . In terms of the full GF, these current correlations are given by 30

S jj = ∞ -∞ dω 2π j1 =j j2 =j tr N λ jj1 G -+ j1j2 (ω)λ j2j G +- j j (ω) -G -+ j1j (ω)λ j j2 G +- j2j (ω) , (14)
where

G -s s = (1/2) G K + s G R -G A with s = ±.

III. RESULTS

As the system consists of three electrodes, with 0-1 and 0-2 couplings only, it is possible to solve explicitly the Dyson equation, Eq. ( 9), in order to obtain an analytical expression for the full Green function Ǧ. Using Eqs. ( 11) and ( 13), one can then derive an explicit form for the average current I j , and the current correlations S jj . The expressions we give below depend on the couplings λ 1 , λ 2 between the TS and the normal electrodes 1 and 2, on the voltages V 1 and V 2 of the two normal electrodes, and on the temperature through the Fermi function n F (x). A natural quantity appearing in these formulas is Λ = λ 2 1 + λ 2 2 , which is related to the total transmission probability τ between the TS and the normal leads: τ = 4Λ 2 /(1 + Λ 2 ) 2 . For all the results presented in this work, we focus on the case 0 < Λ < 1, which covers all possible values of the transmission τ ∈ [0, 1]. Taking Λ > 1 would give the same value of τ , but for a different realization of the physical system -see the discussion in App. C for more details.

A. Current and differential conductance

The current flowing through normal lead j = 1, 2 can be written in the simple form:

I j = e h ∞ -∞ dω k=1,2 s=± s n F (ω -s eV k )J jk (ω) (15)
where J jj and J j,k =j determine, respectively, the local and non-local differential conductances at zero temperature.

Focusing e.g. on current I 1 , it can be separated into a "direct" contribution related to the chemical potential of electrode 1 (J 11 ), and a "non-local" contribution related to the chemical potential of electrode 2 (J 12 ). The explicit expressions for the differential conductances J 11 (ω) and J 12 (ω) for |ω| < ∆ are while the expressions for |ω| > ∆ are given in App. B. One can see that in the low-voltage regime, the contribution to I 1 from J 11 is linear in V , while the one from J 12 scales as ∼ V 3 therefore not contributing to the linear conductance in the zero-temperature limit. This means in particular that in the low voltage limit, when coupling to the TS occurs through the Majorana bound state only, the current in one normal electrode is not influenced by the voltage in the other one. 31 One can also check that when setting λ 2 = 0, Eq. ( 16) gives back the known formula for a single N-TS junction. While J 12 trivially vanishes in this case, 30 Fig. 2 shows the local and non-local differential conductances J 11 (ω) and J 12 (ω), for three values of the transmission τ . Focusing for simplicity on the zerotemperature limit, the differential conductance G 1 = dI 1 /dV 1 is given by

J 11 (ω) = 4λ 2 1 Λ 2 (1 -Λ 4 ) 2 ω 2 ∆ 2 + 4Λ 4 -J 12 (ω), J 12 (ω) = 2λ 2 1 λ 2 2 1 -Λ 4 ω 2 ∆ 2 (1 -Λ 4 ) 2 ω 2 ∆ 2 + 4Λ 4 , (16)
J 11 reduces to 1/(1 + ω 2 /Γ 2 ) for |ω| < ∆, with Γ = 2∆Λ 2 /(1 -Λ 4 ).
G 1 (V 1 ) = 2e 2 h J 11 (V 1 ) 2e 2 h λ 2 1 /Λ 2 1 + V 2 1 /Γ 2 . ( 17
)
The factor λ 2 1 /Λ 2 reduces to 1/2 for equal couplings λ 1 = λ 2 , and is otherwise related to the asymmetry of the couplings. The local differential conductance J 11 (ω) has a shape which is similar to the one of a simple N-TS junction, with a peak associated with the Majorana bound state, broadened by the couplings to the normal electrodes. Indeed, the Lorentzian factor (1 + V 2 1 /Γ 2 ) -1 in Eq. ( 17) is reminiscent of the well-known conductance peak of width Γ, and height 2e 2 /h. 32,33 Here, the contribution of this peak is split between the two normal electrodes, resulting in an extra factor 1/2 in the equal coupling case, so that the zero-voltage differential conductance is simply e 2 /h for each electrode. The non-local differential conductance J 12 is shown on the right panel of Fig. 2, for the case of a symmetric junction, and for three different values of the total transmission τ . J 12 is negligible for very small transmission (τ = 0.1). For larger transmissions, it starts from 0 at ω = 0, and is positive for |ω| < ∆. J 12 increases as ω gets closer to

τ = 0 .9 V 1 =V 2 =V V 1 =-V 2 =V
FIG. 3. Current I1 (in units of e∆/h) vs voltage V = V1 in the case of equal (red dashed curve) and opposite (blue full curve) bias voltages V1,2 at zero temperature, λ1 = λ2 and several values of τ . In all figures, voltages are given in units of ∆.

∆, then abruptly changes sign above the gap, becoming negative for |ω| > ∆.

To obtain simple, easily readable formulas for the current, it is useful to take the zero temperature limit, and consider symmetric couplings

λ 1 = λ 2 = Λ/ √ 2.
We then get for the current at voltages below the gap:

I 1 = eΓ 2h 1 + 1 + Γ 2 ∆ 2 tan -1 eV 1 Γ + 1 -1 + Γ 2 ∆ 2 eV 1 -eV 2 Γ + tan -1 eV 2 Γ ( 18
)
which for equal voltages V 1 = V 2 = V reduces to:

I 1 = e h Γ tan -1 eV Γ ( 19
)
Fig. 3 shows numerical results for the current I 1 as a function of V for a symmetric junction in the case of equal

(V 1 = V 2 = V ) and opposite voltages (V 1 = -V 2 = V ).
Here we focused on three different values of the total transmission between the TS and the normal leads (τ = 0.1, 0.5, 0.9) and for simplicity we considered the case of zero temperature. The differences in the current

I 1 between the V 2 = V 1 and V 2 = -V 1 cases
can be understood from the properties of the non-local differential conductance J 12 [see Eq. ( 16) and the right panel of Fig. 2]. For small transparency (τ = 0.1), J 12 is very small, and the two currents cannot be distinguished in the figure. For larger transparency, the effect of J 12 is only noticeable for large enough voltage. As J 12 (ω) > 0 for |ω| < ∆, the current for

V 2 = V 1 is larger than the one for V 2 = -V 1 .
The difference between the two currents is maximal for V close to ∆, when J 12 reaches its maximum, before decreasing at higher voltages since J 12 (ω) becomes negative at high energy.

For arbitrary subgap voltages, the current I 2 is readily obtained from Eq. ( 18) upon exchanging V 1 and V 2 . For symmetric couplings λ 1 = λ 2 , one can easily convince oneself that I 2 = I 1 in the specific case of equal voltages, while in the opposite voltage case, one has I 2 = -I 1 .

B. Hanbury-Brown and Twiss cross-correlations

As we did for the current, we derive here analytical expressions for the current auto-and cross-correlations S jj , and we are particularly interested in the HBT crosscorrelations S 12 between the currents flowing from the central TS to the two normal leads. As the formulas get rapidly long and cumbersome, we give here the zerotemperature expression valid for voltages below the gap ∆, and for equal couplings

λ 1 = λ 2 = λ (thus Λ 2 = λ 2 1 + λ 2 2 = 2λ 2 )
. More general formulas are given in App. B. Introducing as before the broadening Γ = 2∆Λ 2 /(1-Λ 4 ), and taking without loss of generality |V 1 | ≥ |V 2 |, we have:

S 12 (V 1 , V 2 ) = e 2 h Γ 2 4∆ 2 1 + 1 2 Γ 2 Γ 2 + (eV 1 ) 2 |eV 1 | + 1 2 Γ 2 |eV 2 | Γ 2 + (eV 2 ) 2 - 3Γ 2 tan -1 |eV 1 | Γ - Γ 2 tan -1 |eV 2 | Γ -sgn(V 1 V 2 ) |eV 2 | (eV 2 ) 2 + 2Γ 2 + 2∆ 2 (eV 2 ) 2 + Γ 2 -2Γtan -1 |eV 2 | Γ (20)
The last term of this expression is proportional to -sgn(V 1 V 2 ), with a coefficient which is always positive, independently of the voltages. As it turns out, this term is dominant, and gives the sign of S 12 for all voltages below the gap. In the limit of small voltages

|V 1 |, |V 2 | Γ, S 12 becomes S 12 - 1 2 e 2 h sgn(V 1 V 2 )|eV 2 | (21)
Importantly, this means that the HBT cross-correlations are positive when the two voltages V 1 and V 2 have opposite signs.

The expression for the correlations S 12 in Eq. ( 20) can be further simplified for some specific choices of the bias voltages. For equal voltages V 1 = V 2 , we have

S 12 (V 1 = V 2 = V ) = - 2e 2 h Γ 2 4 |eV | (eV ) 2 + Γ 2 , ( 22
)
which coincides with existing results. 28 Conversely, for opposite voltages, we have: V τ=0.9

S 12 (V 1 = -V 2 = V ) = 2e 2 h Γ 2 4 |eV | (eV ) 2 + Γ 2 + 2Γ 2 + (eV ) 2 (eV ) 2 + Γ 2 |eV | ∆ 2 - 2Γ ∆ 2 tan -1 |eV | Γ . (23)
V τ=0.1 V 1 =V 2 =V V 1 =-V 2 =V -1.
V τ=0.5 V 1 =V 2 =V V 1 =-V 2 =V -1.
V 1 =V 2 =V V 1 =-V 2 =V -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 V 1 =V 2 =V V 1 =-V 2 =V

C. Discussion of auto-and crossed-correlations

In order to better understand the behavior of the HBT noise correlations S 12 , it is useful to discuss all the noise contributions, in particular the autocorrelations S 00 , S 11 and S 22 are also considered. From the definition of the noise, Eq. ( 13), and using -I 0 = I 1 + I 2 , we have the relation: 14 S 00 = S 11 + S 22 + 2 S 12 (24) so that is is enough to consider S 00 , S 11 and S 12 to achieve a full characterization. For simplicity, we consider in what follows the regime of symmetric couplings (λ 1 = λ 2 ), in the zero-temperature limit.

Equal voltages

For equal voltages V 1 = V 2 = V , the analytical expressions for the various noise correlations (in units of e 2 /h) are

S 00 =2Γ tan -1 |eV | Γ - |eV |/Γ 1 + (eV /Γ) 2 0 ( 25
)
S 11 =Γtan -1 |eV | Γ - 1 2 |V | 1 + (eV /Γ) 2 |eV | 2 (26)
S 12 = - 1 2 |eV | 1 + (eV /Γ) 2 - |eV | 2 ( 27
)
where the final expressions are obtained in the lowvoltage limit eV Γ. In terms of the total coupling Γ, the auto-correlation noise S 00 in the TS lead has the same expression as for a single N-TS junction. 30 Focusing on Eqs. ( 25)-( 27), the behavior of the autoand cross-correlations at low voltage |eV | Γ can be understood from the basic properties of the coupling of the normal electrodes to the Majorana bound state.

From the point of view of the TS, the two normal electrodes are at the same potential and thus act as a single one for the total current I 0 , so that the total conductance has a peak of height 2e 2 /h [see Eq. ( 17) and discussion below]. As a consequence, much like in the single N-TS junction, the total current I 0 is noiseless at low voltage eV Γ, which is confirmed by Eq. ( 25). This total current I 0 = 2(e 2 /h)V is partitioned here with equal prob- ability between the currents I 1 and I 2 (see Fig. 5) :

I 1 = I 2 = (e 2 /h)V. ( 28
)
These two currents are thus equivalent to the transmitted and backscattered current from a quantum point contact with incoming current I 0 and transmission T = 1/2, reflection R = 1 -T = 1/2. This implies that the autocorrelations S 11 and S 22 correspond to the noise associated with currents resulting from random partitioning, 34 leading to (restoring units)

S jj ≡ eI j (1 -T ) = e 2 h |eV | 2 (j = 1, 2) (29)
which coincides with Eq. ( 26) to lowest order in V . Finally, the HBT noise S 12 corresponds to the correlation between the two partitioned currents I 1 and I 2 . Due to the fermionic nature of the electrons, these two currents are totally anti-correlated (see Fig. 5), yielding a negative correlation noise. 13,14 Following Eq. ( 24), and using that I 0 is noiseless, one sees that the HBT correlations and the autocorrelations are simply related as S 12 = -S 11 , which agrees with Eq. ( 27) to lowest order.

Opposite voltages

For opposite voltages V 1 = -V 2 = V the auto-and cross-correlations take the form (in units of e 2 /h):

S 00 =2Γtan -1 |eV | Γ 2|eV | (30)
S 11 =Γtan -1 |eV | Γ - |eV |/2 1 + (eV /Γ) 2 -f (V, Γ) |eV | 2 (31) S 12 = 1 2 |eV | 1 + (eV /Γ) 2 + f (V, Γ) |eV | 2 ( 32
)
where the final expressions correspond to the low-voltage regime eV Γ, and we introduced

f (V, Γ) = Γ 2 2∆ 2 eV 2Γ 2 + (eV ) 2 (eV ) 2 + Γ 2 -2Γtan -1 |eV | Γ ( 33
)
While this opposite voltage case has a behavior strikingly different from its equal voltage counterpart, it can still be understood with the same ingredients, by taking into account that the coupling to the Majorana bound state is perfectly electron-hole symmetric. Indeed, when normal lead 2 is biased at voltage -V rather than V , it can be seen as a reservoir of holes biased at voltage V coupled to the Majorana bound state. The behavior of the system is thus the same as for the equal voltage case, except that electrons are now replaced by holes for the current I 2 . Picturing the total current from the TS as a stream of particles (thus disregarding the charge), this stream is still noiseless, with one particle (electron or hole) emitted during each time interval /eV . The currents I 1 and I 2 still result from the random partitioning of such a noiseless stream of particles, with electrons for I 1 and holes for I 2 , so that

I 1 = -I 2 = (e 2 /h)V (34)
As a consequence the autocorrelation noises S 11 = S 22 = (e 3 /h)|V |/2 are identical to their counterparts in the equal voltage case, to lowest order in eV /Γ [see Eq. ( 31)]. Much like the equal voltage case, the two currents I 1 and I 2 are totally anticorrelated, which leads to the same expression for the HBT correlation noise S 12 , only with the opposite sign, as the carriers in the two leads now bear opposite charges (see Fig. 5).

Finally the total noise S 00 (which accounts for the charge of the carriers) corresponds to the current noise of a noiseless stream of particles (one particle -electron or hole-transmitted at each time interval /eV ) but with particles which can be either electron or holes. According to Eq. ( 24) this creates a total charge noise S 00 = 2(e 3 /h)|V |, which coincides with Eq. (30).

The equal and opposite voltage regimes thus have similar cross-correlation noises but with opposite signs, a direct consequence of the peculiar properties of the Majorana bound state, which by definition does not distinguish electrons from holes.

IV. FINITE LENGTH AND DOPING EFFECTS

The results shown in the previous sections have been obtained for the case of a semi-infinite topological superconductor bearing a (single) Majorana bound state at its end, with the boundary Green function for the TS nanowire computed in the wide-band limit at half-filling, see Eq. ( 4). We wish to address here the case of a finite size TS whose two Majorana bound states (one at each extremity) overlap, leading to radically different behavior for the HBT correlations. Secondly, this TS wire can be doped in such a manner that it becomes a trivial superconductor devoid of topological effects. In this section, we thus briefly discuss how the noise correlations are modified when going beyond the approximations used in the preceding sections.

A. Varying the length of the TS nanowire

The retarded/advanced GF for a TS wire of length L (-L/2 < x < L/2) computed near x = L/2 in the wideband limit is given by

30 g R/A (ω) = ω tanh(ζ ω L) ζ ω σ 0 -tanh(ζ ω L)∆σ x (ω ± i0 + ) 2 -2 ω , (35)
where

ζ ω = √ ∆ 2 -ω 2 and ω = ∆/cosh(ζ ω L) with v F = = 1.
The finite TS has a Majorana bound state localized at each end. When the length of the TS is much larger than the typical scale of the Majorana bound state, which is of the order of the superconducting coherence length ξ 0 = v F /∆, the two end-state wave functions practically do not overlap and one recovers the same result as for the infinite length GF. However, with decreasing L, this overlap becomes important, and we expect to lose the behavior specific to the presence of a Majorana bound state. In Fig. 6, we show the noise correlations S 12 as a function of the voltage V for the opposite voltage case, at transmission τ = 0.5, and for several values of the TS length L (in units of v F /∆). We see that when L 1, the results obtained for V 1 = V = -V 2 are identical (positive HBT correlations) to the ones obtained in the previous section with the infinite length Green function. However, when L is of order 1, the overlap of the two Majorana end states becomes important, and S 12 turns negative around V = 0, over a range of voltage which increases as L decreases.

B. Varying the chemical potential of the TS nanowire

Another important parameter is the intrinsic chemical potential of the topological superconductor, which depends in a real nanowire on the values of the proximity induced coupling, the magnetic field, etc. In our approach, this is modeled by the chemical potential µ of the Kitaev chain. In the calculations presented so far we always set µ = 0, corresponding to half-filling of the chain. By varying this parameter at finite bandwidth, it is possible to go away from half-filling and therefore to drive the system from the topological phase to a trivial one where no Majorana bound state is present. 3 In order to observe this transition, one needs to rederive the Green function for a Kitaev chain beyond the wide-band limit, with arbitrary values of the gap ∆, the hopping parameter t 0 and the chemical potential µ. Explicit formulas for this Green function, and details of the derivation are provided in App. A. Fig. 7 shows the noise correlation S 12 for the opposite (top panel) and equal voltage case (bottom panel), for a chain with a hopping parameter t 0 = 10∆, and chemical potential µ varying from 0 to 13∆. When µ is increased, while still below the bandwidth t 0 , the correlations S 12 are reduced in absolute value, but keep the same qualitative features, with a dip (peak) around V = 0 for opposite (equal) voltages. In the opposite voltage case (top panel), S 12 is always positive for V close to 0, and the decrease at large voltage becomes more pronounced as µ is increased. In the equal voltage case (bottom panel), the correlations S 12 are essentially scaled down when µ is increased, with a notable asymmetry between V > 0 and V < 0 appearing at large µ.

However, the behavior becomes qualitatively different when µ reaches the value of the bandwidth t 0 , as the peak around V = 0 disappears for µ > t 0 . In the opposite voltage case (top panel of Fig. 7), S 12 becomes negative for all V , even close to zero voltage. This is consistent with our previous interpretation: the positive cross-correlations at low voltage are associated with the coupling to the Majorana bound state and this feature 7. Current correlations S12 vs V = V1 in the setup with an infinite TS wire for several values of its chemical potential µ (each curve is labeled by the corresponding value of µ/∆). The top (bottom) panel shows the case of opposite (equal) voltages. We consider a symmetric junction (λ1 = λ2) with a total transparency τ = 0.5, and a hopping strength (TS wire bandwidth) t0 = 10∆. The black curve (labeled "ref") on each plot corresponds to the wide-band limit [using Eq. ( 4)] and serves as a reference for comparison. There is a clear transition when µ reaches the bandwidth t0: the peak behavior around V = 0 disappears when µ > t0, signaling the absence of a Majorana bound state for µ > t0.

S 12 (V) V V 1 =V 2 =V
disappears for µ ≥ t 0 , when we cross into the trivial non-topological phase. In the equal voltage case (bottom panel), the specific behavior which was present for small V also disappears, and the correlations S 12 become fully asymmetric, almost vanishing for V > 0. Note that we avoid on purpose to compute the noise S 12 at the precise value of the transition µ = t 0 . Indeed the spatial extent of the Majorana bound state diverges in this case, 2 and the overlap of the Majorana bound states at the two ends of the system could become important, with a behavior similar to the one presented in Fig. 6.

V. CONCLUSIONS

In this work, we have explored the properties of a topological superconductor nanowire, including a Majorana bound state at its end, by coupling it to two biased normal leads. We computed the currents and the Hanbury-Brown and Twiss current cross-correlations, and showed that the sign of such correlations have a very peculiar dependence on the two voltages V 1 and V 2 of the normal leads: the correlations are negative when the voltages have the same sign, and become positive when the voltages have opposite signs. In addition, for voltages smaller than the coupling between the TS and the normal leads, the correlations for the equal and opposite voltage cases are exactly opposite. This behavior is in stark contrast with the one observed with a conventional BCS superconductor, where typically correlations are positive for voltages of the same sign only. [START_REF] Büttiker | Quantum Noise in Mesoscopic Physics[END_REF] This is directly related to the properties of the Majorana bound state, which by definition makes no difference between electrons and holes. Changing the sign of one of the voltages is equivalent to replacing a reservoir of electrons with a reservoir of holes -the coupling to the Majorana bound state is unaffected, simply leading to a change of sign of the correlations.

The crossed correlations of a TS wire below the gap at equal voltages are similar to the fermionic version of the HBT experiment in normal metals, with differences showing only at V > Γ. [13][14][15][16][17] There, the filled Fermi sea which injects electrons at the two outputs is totally noiseless, and electron partitioning leads to negative crosscorrelations. For the TS beam splitter presented here, it is therefore crucial to also probe the HBT noise at opposite voltages to exhibit its positive sign, in order to rule out a "trivial" interpretation in terms of normal fermionic leads. Positive correlations in this configuration may imply -as for the BCS Cooper pair splitter 20 -that the electron/hole pairs emitted in the two normal metal leads coupled to the TS via the Majorana fermion may form an entangled state.

Calculations were performed using a Keldysh boundary Green function approach, based on a Hamiltonian formalism. 30 Each electrode is then represented by its boundary Green function, and the electrodes are coupled through a tunneling Hamiltonian. Solving the Dyson equation gives exact (non-perturbative) simple formulas for the current and the current correlations in terms of the full Green function of the system. The boundary Green function approach used here is fully equivalent to the scattering matrix approach for voltages below the gap (see App. C), but also allows us to easily access the regime of voltages above the gap.

With this method, we were also able to consider more general situations, simply by adapting the boundary Green function used for the TS nanowire. We considered a TS nanowire of finite length and also studied the effect of varying the chemical potential. Our results show the existence of a crossover when the length L becomes smaller than v F /∆ (the typical length associated with the Majorana bound state), with positive correlations becoming negative for small length. This behavior is due to the hybridization of the two Majoranas at the ends of the TS nanowire, which then behave as a regular fermion. This confirms that the unusual sign observed for the current correlations for a long (or semi-infinite) TS nanowire is specific to the presence of a Majorana bound state. Next, we considered the case where the TS nanowire is represented by a Kitaev chain with a finite bandwidth, and variable chemical potential µ (the corresponding boundary Green function is derived in App. A). By varying µ, one can explore the transition from a topological superconductor to a conventional one. Our results show that above the transition, the cross-correlations become negative at all voltage, and the specific features due to the Majorana bound state (peak at low voltage) disappear.

We believe that these results, obtained by placing the TS nanowire in a three-lead hybrid system, give access to unique properties of the TS nanowire, and of Majorana bound states, which would be more difficult to characterize in a simple two-lead setup. The specific dependence of the sign of the correlations as a function of the voltages could provide an extremely firm experimental proof of the presence of a Majorana bound state in a TS nanowire. Because of its versatility and efficiency, the same boundary Green function approach could be used to consider other multi-terminal setups involving one or several TS nanowires, or more involved effects, such as interactions.

Appendix A: Boundary GF of semi-infinite Kitaev's chain

In this Appendix, we provide a derivation of the retarded/advanced GF used in Sec. IV B, which describes quasiparticle excitations at the boundary of a semiinfinite TS wire. The derivation is performed for arbi-trary values of the chemical potential µ and (normalstate) conduction bandwidth t 0 , so that by varying µ one can drive the wire through a topological transition at µ = ±t 0 . 2,3 This generalizes the derivation outlined in Ref. 30 for the case µ = 0. Following the strategy of Ref. 30, we first compute a "bulk" GF for a homogeneous wire of infinite length, and then the boundary GF is obtained from the Dyson equation of the wire interrupted by a local potential scatterer.

The TS wire is modeled by a Kitaev chain 2,3 representing an effectively spinless single-channel p-wave superconductor. In terms of fermion operators c x on a 1D lattice with site numbers x (we set the lattice constant to unity), the model Hamiltonian reads

H K = 1 2 x -t 0 c † x c x+1 + ∆c x c x+1 + H.c. -µ x c † x c x , (
A1) where t 0 > 0 is the hopping matrix element, ∆ > 0 is the p-wave pairing amplitude and µ is the chemical potential. Imposing periodic boundary conditions c x = c x+N , with the number of lattice sites N → ∞, and passing to momentum space, c x = N -1/2 k e ikx ψ k , the "bulk" Hamiltonian H takes the standard Bogoliubov-de Gennes form

H K = 1 2 π -π dk 2π Ψ † k h k Ψ k , h k = k σ z + ∆ k σ y , (A2)
where

Ψ k = ψ k , ψ † -k T
R/A xx (ω) = π -π dk 2π e ik(x-x ) (ω -h k ) -1 , (A3)
where the frequency ω should be understood as ω + i0 + (ω -i0 + ) for the retarded (advanced) GF. Introducing a new integration variable z = -cos(k), some algebra yields:

g R/A xx (ω) = 1 2π(∆ 2 -t 2 0 ) 1 -1 dz √ 1 -z 2 D(z, ω) s=± ω + (t 0 z -µ)σ z + s∆ 1 -z 2 σ y × -z + is 1 -z 2 x-x , (A4)
with

D(z, ω) = (z -Q + )(z -Q -) Q ± (ω) = -µt 0 ± µ 2 t 2 0 -(∆ 2 -t 2 0 )(ω 2 -∆ 2 -µ 2 ) ∆ 2 -t 2 0 . (A5)
The boundary GF of a semi-infinite Kitaev chain located at x > 0 can be obtained from the "bulk" GF for the translationally invariant model (A1) by adding a local impurity of strength U at site x = 0, which results in the Hamiltonian HK = H K + U c † 0 c 0 . The "full" retarded/advanced GF gR/A xx (ω) then obeys the following Dyson equation:

gν=R/A xx (ω) = g ν xx (ω) + g ν x0 (ω)U σ z gν 0x (ω). (A6)
In the limit U → ∞, i.e., when one effectively cuts the wire into two semi-infinite pieces, Eq. (A6) yields for the boundary GF 30 defined as G(ω) = g11 (ω)

G ν=R/A (ω) = g ν 00 (ω) -g ν 10 (ω) [g ν 00 (ω)] -1 g ν 01 (ω). (A7)
Thus, for computing the boundary GF (A7) one only needs to evaluate Eq. (A4) for x = x and nearestneighbor sites x -x = ±1. Using the following integrals with a complex-valued parameter a, Im a = 0,

1 π 1 -1 dz √ 1 -z 2 (z -a) = -1/a 1 -1/a 2 , ( A8
)
and for n = 1, 2

1 π 1 -1 dz z n √ 1 -z 2 (z -a) = a n-1 1 - 1 1 -1/a 2 ,
(A9) after some algebra we obtain from Eq. (A4):

g ν=R/A 00 (ω) = (ω -µσ z )F -1 (ω) + t 0 σ z F 0 (ω), g ν ±1,0 (ω) = g ν 0,∓1 (ω) = ±i∆F -1 (ω)σ y -(ω -µσ z )F 0 (ω) + (t 0 σ z ± i∆σ y ) [1 -F 1 (ω)] , (A10)
where

F m=0,±1 (ω) = 1 (t 2 0 -∆ 2 )(Q + (ω) -Q -(ω)) × s=± sQ m s (ω) 1 -1/Q 2 s (ω) . (A11)
The boundary GF of the semi-infinite Kitaev chain then follows by inserting Eq. (A10) into Eq. (A7). Eq. ( A10) is an extension of the result of Ref. 30 to the general case of µ = 0. In particular, for µ = 0 and assuming the wide-band limit t 0 max(∆, |ω|) the above expressions in Eq. (A10) simplify to

g ν=R/A 00 (ω) = ω t 0 √ ∆ 2 -ω 2 σ 0 , g ν ±1,0 (ω) = 1 t 0 √ ∆ 2 -ω 2 ∆ 2 -ω 2 σ z ∓ i∆σ y , (A12)
and then using Eq. (A7) one recovers the boundary GF (4) quoted in Sec. II.

Appendix B: Analytical formulas for the current and noise

This Appendix contains more general formulas for the currents and current correlations, which were too lengthy to be shown in the main text.

The expressions (16) for the local and non-local differential conductances can be extended for energies above the gap, leading to the following general forms

J 11 (ω) =          2λ 2 1 λ 2 2 (Λ 4 -1) ω 2 ∆ 2 +4λ 2 1 Λ 2 (1-Λ 4 ) 2 ω 2 ∆ 2 +4Λ 4 |ω| < ∆ -2λ 4 1 Λ 4 +2Λ 2 1-∆ 2 ω 2 -2∆ 2 ω 2 +1 +2λ 2 1 (3Λ 4 +1) 1-∆ 2 ω 2 +Λ 2 (Λ 4 +3)-2∆ 2 Λ 2 ω 2 Λ 4 +2Λ 2 1-∆ 2 ω 2 +1 2 |ω| > ∆ (B1) J 12 (ω) =          -2λ 2 1 λ 2 2 (Λ 4 -1) ω 2 ∆ 2 (1-Λ 4 ) 2 ω 2 ∆ 2 +4Λ 4 |ω| < ∆ -2λ 2 1 λ 2 2 Λ 4 +2Λ 2 1-∆ 2 ω 2 -2∆ 2 ω 2 +1 Λ 4 +2Λ 2 1-∆ 2 ω 2 +1 2 |ω| > ∆ (B2)
Similarly, one can obtain closed-form expressions for the noise cross-correlations S 12 at zero temperature for generic values of the coupling constants λ 1 , λ 2 and voltages V 1 , V 2 , thus generalizing Eq. ( 20). In the subgap regime |V 1 |, |V 2 | ≤ This relation generalizes the equivalent one for a simple junction composed of two leads. 30,[START_REF] Cuevas | [END_REF] We see that when Λ goes from 0 to 1, τ also varies from 0 to 1, so the whole range of transparencies is covered by taking Λ in [0, 1]. From Eq. (C2), one also sees that for a given Λ in [0, 1], the value 1/Λ gives the same value of the transparency τ . However, as we show below, taking Λ > 1 leads to a different physical realization of the system. Indeed, choosing the value τ of the total transparency between the TS and the normal leads, even for a symmetric system, does not totally specify the system. This can be understood simply from the scattering matrix formalism, as noted by Valentini and co-workers 29 . Writing the scattering matrix for a three-lead system, where the two lateral leads (lines/columns 1-2) are symmetrically connected to a central one (line/column 3) with a transparency τ = √ 1 -r 2 , we have

S =   • • t • • t t t r   (C3)
with r 2 + 2t 2 = 1. Imposing the unitarity of the S matrix gives the values of the coefficients in the 1-2 block s 12 [written as dots in Eq. (C3)], which represent direct reflection/transmission in the subsystem of the two lateral leads. There are two possible solutions (written here for simplicity with real coefficients)

s 12,+ = 1 2 (1 -r) -(1 + r) -(1 + r) (1 -r) (C4)
and

s 12,-= 1 2 -(1 + r) (1 -r) (1 -r) -(1 + r) (C5)
and we note S + and S -the complete scattering matrix corresponding to the choice of s 12,+ and s 12,-respectively. The difference between the two choices can be understood for example by taking r close to 1 (very poor transmission to the central lead). Then the amplitude of direct transmission between 1 and 2 [elements (1,2) and (2,1)] are very different for S + and S -: for S -, it is (1 -r)/2, which is close to 0; while for S + it is -(1 + r)/2 which is close to one in absolute value. This means that for a given transmission to the central lead (which fixes r), the currents I 1 and I 2 have totally different values for the two choices S + or S -as soon as the voltages V 1 and V 2 are not equal (if V 1 = V 2 , then the direct transmission between leads 1 and 2 has no consequence). The impact of the choice of S + or S -is illustrated in Fig. 8, with a plot showing the current I 1 (V ) in the opposite voltage configuration (V 1 = -V 2 = V ), for Λ = 1/4 and Λ = 4. The two values of Λ correspond to the same transparency τ 0.22 between the TS and the two normal leads, but the currents I 1 and I 2 = -I 1 are very different for the two values of Λ. For Λ = 4 the current I 1 is much larger as V increases, because of the direct current going from normal lead 1 to normal lead 2.

The two choices for S correspond precisely to the choice Λ < 1 (→ S -) or Λ > 1 (→ S + ) in the microscopic calculation. The relation between Λ and r is:

Λ = 1 -r 1 + r (Λ < 1) or Λ = 1 + r 1 -r (Λ > 1) (C6) or equivalently r = 1 -Λ 2 1 + Λ 2 (Λ < 1) or r = Λ 2 -1 Λ 2 + 1 (Λ > 1)
(C7) with 0 < r < 1, and Λ 2 = λ 2 1 +λ 2 2 is given from the Hamiltonian. As a proof of these relations, we show below that the currents obtained from the scattering matrices S + and S -, for voltages smaller than the gap, coincide with the expressions obtained from our Hamiltonian calculation in terms of Λ [see Eq. ( 16)]. The details of the scattering matrix calculation follows Ref. 28. Channels 1 and 2 correspond to the two normal leads, while channel 3 is related to the superconductor. We first construct the s ee and s he 2x2 matrices describing the scattering between the normal leads, of an electron into an electron (ee) or a hole (he). These are 28 : s ee = s 12 -a(ω) 2 r 1 + r 2 a(ω) 2 t 2 t 2 t 2 t 2 (C8)

s he = a(ω) 2 1 + r 2 a(ω) 2 t 2 t 2 t 2 t 2 (C9)
where a(ω) = exp[-i arccos(ω/∆)] is the amplitude for Andreev reflection at energy ω, s 12 is given by Eq. (C4) or (C5), and r = √ 1 -2t 2 is the reflection amplitude from the superconductor. For simplicity, we consider a symmetric system, and we take r and t real.

The expression for the current I 1 in terms of the scattering matrix elements is 28 where the ± sign refers to the choice of (s 12,+ ) or (s 12,-). One can show that these expressions are equal to 2 J 11 (ω) and 2 J 12 (ω) from Eq. ( 16) with λ 1 = λ 2 = Λ/ √ 2 if the relation between r and Λ is r = 1 -Λ 2 1 + Λ 2 (for s 12,-) or r = Λ 2 -1 Λ 2 + 1 (for s 12,+ ) (C13) For a given transparency τ , taking Λ > 1 thus represents a system where there is a strong, direct link between the lateral leads 1 and 2, which is not the system we are studying here. This explains why, in all the results presented in this work, we consider 0 < Λ < 1 only.

FIG. 1 .

 1 FIG. 1. Schematic view of the setup: a grounded TS nanowire is tunnel coupled (with hopping amplitudes λ1, λ2) to two normal-conducting (N1, N2) leads which are biased at voltages V1 and V2, respectively.

1 FIG. 2 .

 12 FIG. 2. Zero-temperature local J11(ω) (left panel) and nonlocal J12(ω) (right panel) differential conductances (in units of 2e 2 /h) vs ω/∆ for λ1 = λ2 and several values of the total transmission probability τ .

9 FIG. 4 .

 94 FIG.4. HBT cross-correlations S12 (in units of e 2 /h) vs V = V1 in the case of equal (red dashed curve) and opposite (blue full curve) voltages V1,2 for λ1 = λ2 and several transparencies τ (as noted on each panel), at zero temperature. The bottom right panel combines all curves to show the overall scale. S12 is negative (positive) for equal (opposite) bias voltages, and the values of S12 are simply opposite in sign in the two cases for |V | Γ.

Fig. 4

 4 Fig. 4 shows the HBT cross-correlation noise S 12 for equal and opposite voltages, computed for three different values of the transmission probability τ . One can see that the cross-correlations in these two cases are simply opposite as long as eV Γ, with negative (positive) values of S 12 for the equal (opposite) voltage case. For |eV | larger than the gap ∆, S 12 is always a decreasing function of |V |, which, for the opposite voltage case, eventually becomes negative for |eV | ∆ (not shown), as the TS behaves essentially as a normal electrode at such high voltages.

FIG. 5 .

 5 FIG. 5. Schematic picture of the current partitioning between the TS and two normal leads (N1,2) at low voltage |eV |Γ. Electrons (holes) are shown as full (empty) circles. Left panel: the case of equal voltages, where a noiseless stream of electrons from the Majorana bound state is partitioned between the two normal leads, with perfect anti-correlations of the two electron streams. Right panel: the case of opposite voltages where lead N2 is biased at potential -V , so that electrons are emitted into N1 while holes are emitted into N2. The two fermion streams are perfectly anticorrelated, which leads to positive cross-correlation noise. The arrows indicate directions of quasiparticle motion.

5 FIG. 6 . 1 ,

 561 FIG.6. Cross-correlations S12 vs V = V1 in the case of opposite bias voltages (V1 = -V2) for a symmetric junction (λ1 = λ2) with total transmission τ = 0.5 and several values of the TS wire length L. Each curve is labeled by the corresponding value of L in units of ξ0 = vF /∆. For L 1, S12 is identical to the infinite TS case, while for L ∼ 1 the cross-correlations S12 become negative over the range |V |/∆ (ξ0/L)2

  is a Nambu spinor subject to the reality constraint Ψ k = σ x Ψ * -k , k = -t 0 cos(k) -µ is the kinetic energy, ∆ k = ∆ sin(k) is the Fouriertransformed pairing, and Pauli matrices σ x,y,z act in Nambu space. Correspondingly, in coordinate space the retarded/advanced Nambu GF of Ψ(x) = c x , c † x T for the Kitaev model (A1) is given by g

V 1 =-V 2 =VFIG. 8 .

 128 FIG.8. Panels (a) and (b) : schematic illustration of the behavior of the system for the choice of Λ < 1 and Λ > 1 , in the case of a strong reflection (r close to 1). While Λ = Λ0 < 1 and Λ = 1/Λ0 > 1 correspond to the same transparency between the TS and the normal leads, the transparency of the direct channel between the two normal leads is totally different in the two cases. Panel (c) : plots of the current I1(V ) in the V1 = -V2 = V configuration, for Λ = 4 and Λ = 1/4.

s eh 11 2 -s hh 11 2 2 + s he 12 2 +n F,2h (ω) -s eh 12 2 + s hh 12 2 ( 2 + s he 11 2 = 2 + s he 12 2 = 2 + s he 11 2 = 1 -r 2 2 ± ( 1 -r) 2 rω 2 4r 2 ω 2 + ( 1 -+ 1 ) 2 ω 2 4r 2 ω 2 + ( 1 -

 1111221222222221212211221 +n F,2e (ω) -|s ee 12 | C10) where n F,1e (ω) is the Fermi function of electrode 1, with n F,1h (ω) = 1 -n F,1e (-ω), and s hh (ω) = [s ee (-ω)] * , s eh (ω) = [s he (-ω)] * . Comparing with Eq. (15), we see that the two expressions of the current are identical if we have 1 -|s ee 11 | 2J 11 (ω), and -|s ee 12 | 2J 12 (ω). Using Eqs. (C8)-(C9), we get after some algebra (recalling that r 2 + 2t 2 = 1): 1 -|s ee 11 |

ACKNOWLEDGMENTS

We acknowledge discussions with A. Levy Yeyati. The authors also acknowledge the support of Grant No. ANR-2014-BLANC ("one shot reloaded"). This work was granted access to the HPC resources of Aix-Marseille Université financed by the project Equip@Meso (Grant No. ANR-10-EQPX-29-01) and has been carried out in the framework of the Labex ARCHIMEDE (Grant No. ANR-11-LABX-0033) and of the AMIDEX project (Grant No. ANR-11-IDEX-0001-02), all funded by the "investissements d'avenir" French Government program managed by the French National Research Agency (ANR). This work has also been supported by Deutsche Forschungsgemeinschaft (Bonn) Grant No. EG 96/11-1. A. Z. is grateful to the CPT for hospitality during his visit to Marseille.

∆, this reads

where we introduced δ = λ 2 1 -λ 2 2 and focused on voltages

Appendix C: Discussion of the case Λ > 1 vs. Λ < 1

Our microscopic model contains the two parameters λ 1 , λ 2 describing the tunneling from the TS to normal electrodes 1 and 2. As shown in the formulas for the currents and current correlations, a natural parameter is

which is related to the total transmission probability τ between the TS and the normal electrodes


Jorge Trasobares

Jérôme Rech

Thibaut Jonckheere

Thierry Martin

Olivier Alévêque

E Levillain

Valentin Diez-Cabanes

Yoann Olivier

Jérôme Cornil

Jean-Philippe P Nys

O Aleveque

R Sivakumarasamy

K Smaali

P Leclere

A Fujiwara

D Théron

D Vuillaume

& N Clément

Estimation of π-π electronic couplings from current measurements

Keywords: Cooperative effect, -interaction, transfer integral, molecular electronics, nanoelectrochemistry, coupled quantum dot

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Interactions between -systems 1 , 2 are involved in diverse and important phenomena, such as the stabilization of the double helical structure of DNA, 3 protein folding, 4 molecular recognition, 5 drug design, 6 and crystal engineering. 7 These interactions are of fundamental technological importance for the development of organic-based devices, 8 in particular for organic light-emitting diodes, 9 field-effect transistors 10 , or (bio-) molecular devices. [11][12][13][14][15][16] A key parameter in these interactions is the transfer integral (or electronic coupling energy) parameter t, which is included as t 2 in simple semiclassical formulations of charge carrier mobility. 17 In symmetric dimers, t is directly related to energy-level splitting of the highest occupied/lowest unoccupied molecular orbital (HOMO/LUMO) due to intermolecular interactions for hole and electron transport, respectively. 8 The parameter t has mainly been discussed by using photoelectron spectroscopy and quantum-chemical calculations. 18 -20 In the ideal scenario for (opto-)electronic applications, t should be deduced directly from electronic measurements in a device configuration and related to the molecular structure. Such knowledge of t would help us to understand and optimize charge transport through molecular systems. For example, cooperative effects, induced by molecule-molecule and molecule/electrode electronic couplings, are attracting substantial theoretical attention. 21 , 22 The distribution or fluctuation of t plays a key role in the charge transport through organic semiconductors or biomolecules by inducing charge localization or conformational gating effects. [23][24][25] A Gaussian distribution of t with a standard deviation (SD) in the range of the mean t is usually assumed from thermal molecular motions, 25 but remains to be confirmed experimentally. The experimental measurement of t could potentially be used as an ultrasensitive chemical characterization technique because t is expected to be more sensitive to molecular structural order than other physical constants such aselectrostatic interactions (φ) measured by Cyclic Voltammetry (CV) (Figure 1a). However, recent efforts to establish correlations between electrochemical and molecular electronics results [26][27][28][29][30][31] have neglected -intermolecular interactions.

To reach these goals, two main issues need to be addressed. A first issue is related to disorder. Structural variability makes it difficult to extract t from electronic measurements because t is extremely sensitive to order at the angstrom level 8 . One recently implemented and elegant way to measure charge transport at the local scale is through photoinduced time-resolved microwave conductivity (TRMC), 32 but this contactless approach differs from the measurement of charge transport in a device configuration. The alternative approach is to reduce electrodes and organic layer dimensions.

A second issue is that comparisons between experimental and theoretical charge transport data are usually qualitative. Even without molecular organization disorder, many parameters influence the measured current including molecule/molecule or molecule/electrode coupling and electron-vibration (phonon) interactions 8,33 . A recent theoretical proposal suggested additional degrees of freedom. Reuter et al. found that quantitative information on cooperative effects may be assessed by statistical analysis of conductance traces. 21,34 This approach is based on the Landauer Buttiker Imry formalism that typically is used in mesoscopic physics for the study of electron transport through quantum dots in the coherent regime. The related experimental model system is a single layer of -conjugated molecules (quantum dots), which is sandwiched between two electrodes. Thousands of molecular junctions are required for statistical analysis. The authors suggested that cooperative effects between molecules should provide asymmetrical conductance histogram spectra (Figure 1b). Histogram fitting may be achieved by considering the mean and SD of molecule site energies ( , δ ), molecule-electrode coupling (V, δV) and transfer integrals (<t>, δt). 21 This fitting differs from the usual experimental log-normal conductance histogram shape (normal distribution when conductance G is plotted in log scale) reported in single moleculebased molecular electronics (Figure 1b) (see Supplementary Note S1 in Supporting Information [SI] for a detailed history of conductance histograms in molecular electronics). 11, [35][36][37][38][39][40][41] Here, we explore -intermolecular interaction energies from the electrochemical perspective (coupling between charge distributions) and molecular electronic perspective (coupling between orbitals) using a large array of ferrocene (Fc)-thiolated gold nanocrystals. First, we show that the two peaks observed in voltammograms on these systems can be controlled by the nanocrystal diameter. Each peak corresponds to a dense or dilute molecular organization structure located at the top or side facets of the nanocrystals, respectively. Second, the dense molecular organization structure is resolved by Ultra-High-Vacuum Scanning Tunneling Microscopy (UHV-STM). This structure is used as a reference for estimating t from quantum chemical calculations at the Density Functional Theory (DFT) level. Based on current measurement statistics for ~3000 molecular junctions between the top of the nanocrystals and a conducting atomic force microscope (C-AFM) tip, we confirm the theoretical prediction of histograms that shape is affected by cooperative effects. 21 Furthermore, we extend the previously proposed tight-binding formalism to fit the histograms 21,34 . The estimated electronic coupling energy distribution for t is quantitatively compared with quantumchemical calculations. The φ and t obtained from CV traces and current histograms, respectively, are discussed on the basis of intermolecular distance fluctuations. Finally, we highlight the implications and perspectives of this study to molecular electronics, organic electronics and electrochemistry.

Results

Electrochemical characterization of Fc-thiolated gold nanocrystals

We selected ferrocenylalkylthiol (FcC 11 SH) as an archetype molecule with a πconjugated head for electrochemistry 29,30,42,43 and molecular electronics. 12,14,[29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46] CV is a powerful tool to gain insights into the molecular organization, extract surface coverage Γ, and evaluate the energy level of the HOMO (E HOMO ± δE HOMO ). In particular, as different molecular organization structures usually lead to multiple CV peaks, 47,48 the aim of this section is to demonstrate that molecules located at the top of the Fc-thiolated gold nanocrystals correspond to a single CV peak, from which φ can be extracted. 49 We have previously demonstrated the possibility of performing CV on Fc-thiolated gold nanocrystal surfaces, 14 although we studied only one dot diameter and did not investigated cooperative effects. CV cannot be performed at the single-dot level with these molecules because the currents are too weak (≤ fA range). 14,50,51 First, we assess E HOMO and dot-to-dot dispersion in E HOMO (δE HOMO ) by CV. The voltammogram is averaged over millions of nanocrystals with a few Fc molecules per nanocrystal (diluted in a C 12 SH matrix) to avoid cooperative effects (Figure 2d). The peak energy position is in the expected range for Fc molecules (0.41 eV vs Ag/AgCl). 12,14,47,42 Furthermore, the voltammogram width at half maximum (FWHM) is close to 90 mV for the main peak, i.e. the theoretical value in the absence of interaction between redox moieties. 27 This result suggests that δE HOMO is less than 45 meV (Figure S3). S6). Peak splitting can be observed [42][43][44] . The peak area is related to the total faradic charge and, therefore, to the number of molecules per dot. Only the number of molecules per nanocrystal related to peak 1 significantly varies with nanocrystal diameter D (Figure 2h). Based on a simple model with a truncated conical shape for dots (Figure 2c), we suggest that peak 1 corresponds to molecules at the top of the dot, whereas peak 2 corresponds to molecules on the side of the dot (Figure 2h, inset). Thus, the density of molecules is smaller on the sides (Γ~2 nm 2 /molecule) than on the top (Γ~0.39 nm 2 /molecule) of the nanocrystals (see Figure 2h for fits and Methods for details). In other words, a highly ordered structure corresponding to a single peak in the voltammogram can be successfully formed on the top of the gold nanocrystals. This hypothesis is consistent with FWHM ≥90 mV for peak 1 (global repulsion between Fc moieties in the electrolytic media used 56 ) and FWHM ≤90 mV for peak 2. The position and shape of the second peak can be explained by a local change of the environment (presence of Na + counter ions from negatively charged silica at the dot borders and pH>2; Figure 2i) and a modification of ion-pairing equilibrium 47,52 (fewer ClO 4 -ions at dot borders due to SiO -surface sites). CV on the smallest dots results in a single peak whose width is smaller than the width expected at room temperature without molecular interactions. This result could be technologically useful for improving the sensitivity of electrochemical biosensors beyond the thermal Nernst limit. [53][54][55] The strength of electrostatic interactions for molecules located at the top of the gold nanocrystals can be quantitatively assessed by the extended Laviron model 27,56,57 (see Supplementary Methods). Coulomb interactions (φ when Fc moieties are fully oxidized) tune the FWHMs of the voltammograms because they are modulated by the fraction of oxidized species. Reasonable fits can be obtained with φ = 4.5 meV for all dot diameters (see Table S1 in SI for fit parameters). The φ obtained from CV will be linked to t from the current measurements in the Discussion section.

Estimation of t from quantum-chemical calculations

The self-assembled monolayer (SAM) structure on a gold substrate has been resolved by UHV-STM (Figure 3a) and used as a reference for DFT calculations. The STM image shows a regular structure of elongated shapes corresponding to groups. The extracted average area per molecule (0.40 nm 2 ) is in agreement with our CV results and is slightly larger than the 0.36 nm 2 considered for a hexagonal structure with a diameter of 0.66 nm per Fc 58,59 . The area corresponds to a configuration in which Fc units are at the same level in the vertical position. Each Fc unit forms a tilt angle of 56° ± 15° with respect to the surface normal (Figure S7), consistent with estimates obtained by obtained by Near-edge X-Ray absorption fine structure spectroscopy (60° ± 5°) 60 and by molecular dynamics simulations (54° ± 22°). 60 When molecules are organized as in Figure 3b, t can be calculated by DFT for two neighboring Fc units (fragments). This simulation is only based on the Fc units and not on the full FcC 11 SH molecule because the contribution of the saturated part of the molecule to t is negligible. As structural fluctuations in monolayer organization are expected experimentally, we compute t a and t b between fragments of molecules 1 and 3 and molecules 1 and 2 at different positions along the X and Y axes. Figure 3c shows t b when molecule 2 moves along the X axis in a collinear geometry. t b strongly depends on displacements of molecule 2 at the angstrom level because t is related to the electronic (rather than spatial) overlaps between orbitals. [61][62][63][64] Maxima are in the 20-30 meV range. Figure 3d shows the evolution of t b as a function of the variation of the intermolecular distance d (δd) around the equilibrium position, without lateral displacement. The decay ratio b =1.94/ Å is close to the tunnel decay ratio in molecular electronics. Similar results are obtained for t a (cofacial geometry; see Figure S8). For consistency with our previous studies, the B3LYP functional (see Methods) has been chosen due to the good agreement with mobility values extracted from the TRMC technique. 32 A recent theoretical study illustrated that B3LYP behaves very similarly to long-range corrected functionals and that the size of the basis set has a weak impact on the calculated transfer integrals. 65 Overall, the results indicate that the -conjugated Fc molecules are electronically coupled and suggest that a signature of cooperative effects should be observed on current measurements.

Cooperative effects on current histograms

We have conducted the statistical study proposed in ref. 21 (i.e., current histograms).

"Nano-SAMs"(i.e., SAMs with diameters of a few tens of nanometers) are ideal for this experiment. Use of nano-SAMs enables us to obtain sufficient molecules for cooperative effects, but limits the number of molecules to avoid averaging over many molecular structures, grain boundaries, and defects. The C-AFM, as the top electrode, 66 is swept over thousands of nanocrystals. 39 We previously showed that log-normal histograms are systematically obtained when such a statistical study is performed with nano-SAMs composed of alkyl chains without groups. 39 In contrast, as predicted in ref. 21, we find that the presence of cooperativity between π-conjugated orbitals (in the head group) affects the line shape of histograms. Figure 4a S2 for fitting parameters). In the case of 15-nm-diameter nanocrystals, a second peak, corresponding to another molecular organization structure 39 appears at a lower current in the histograms (Figure 4b). We suggest that this peak, which is barely seen in the histograms, is averaged on larger dots. Fitting parameters for the main current peak are almost unchanged (see Table S2). When FcC 11 molecules are diluted 1:1 with dodecanethiol molecules (C 12 SH) to reduce coupling between molecules, the lognormal histogram is recovered (Figure 4c), similarly to alkyl-chain-coated nanocrystals. 39 We tried to fit the current histograms using a coherent scattering formalism similar to the one proposed in ref. 21 the external potential, depends on these parameters accordingly (see SI Methods). To generate current histograms, V t ,V b ,t, are chosen from Gaussian distributions with predefined means and SDs (e.g. eq.1a) for each individual molecular junction. For t, we additionally considered eq 1b to explicitly consider the fluctuation of the intermolecular distance (see Figure 3d). meV, and δ =30 meV). Considering eq 1b for t gives an even better fit to experimental data with t 0 =0.34 eV, β = 1.96/Å, and SD(δd) = 0.8Å. When intermolecular coupling is suppressed (t = 0) while keeping other parameters constant to mimic the diluted monolayer (Figure 5c), the resulting log-normal histogram reproduces the experimental results (Figure 5d).

Discussion

In molecular or organic electronics, comparisons of experimental and theoretical charge transport data are usually qualitative. Therefore, any step towards a more quantitative analysis is important to the field.

A strong coupling asymmetry of α=V t 2 /(V t 2 +V b 2 )≈0.9 was required to fit histograms (see Figure S10 in SI), as expected from the structure of the molecule and previous studies 12, 14 . The "large" values of V t and V b (molecular orbital energy broadening amounts of 100 meV and 15 meV, respectively) confirm our expectation of strong molecule/electrode couplings, which we previously exploited to obtain a high-frequency molecular diode 14 .

Extracted distributions of t corresponding to best fits in Figure 5d are shown in Figure 6a. We have explored two t distributions corresponding to eq 1a and eq 1b. In both cases, maxima are found at t ≈ 35 meV, which is in the expected range from our DFT calculations. However, both deviate quantitatively from the theoretical distribution prediction for t based on thermal molecular motions (SD(t) ≈ <t> in eq 1a). 25 Using eq 1a, we find SD(t) ≈ 140 meV, suggesting that the structural fluctuations are larger than those generated from solely thermal motions (phonons). Structural fluctuations are explicitly considered with parameter δd in eq 1b. The extracted SD(δd) = 0.8Å is reasonable given that a more packed configuration for these monolayers is possible. 12 Based on these results, we suggest that Van der Waals interactions between alkyl chains, which compete with the -interactions in the molecular organization of such monolayers, 12 could play a role in the distribution of t.

We stress that the number of molecules NxN, considered for current histograms generation, affects the quantitative extraction of t. An approximately 150 molecules are used in the experiment. A large enough N was required in the model to avoid overestimating the extracted value of t (Figure S11 in SI). At N=9, the extracted t depends to a lesser extent on the molecule/electrode coupling parameters, which reduces the error on the estimated t (Figure S11 in SI). We suggest that t ≈ 35±20 meV is extracted from the present model based on both t distributions and the possible error on V t and V b .

From this quantitative analysis on t, we can discuss the results in the general contexts of charge transport in organic semiconductors 32,33,68 and chemical characterization tools.

As high-mobility organic semiconductors are often composed of a -conjugated backbone substituted by one or more alkyl side chains, 32 as in the present study, a t distribution following eq 1b may be considered in charge transport models.

Semiclassical theories of charge transport in organic semiconductors show that the electron transfer (hopping) rates along the -conjugated molecular planes scale as t 2 .

Figure 6b represents such probability distributions for t 2 corresponding to the two t distributions shown in Figure 6a (related to eq 1a and 1b). Distributions have similar shapes in both cases, but the tail is narrower for the Gaussian distribution of t. In both cases, the broadened distribution of t 2 would open new hopping pathways.

The exploration of -intermolecular interaction energies from both CV and current histograms using the same samples composed of a large array of Fc-thiolated gold nanocrystals enables a direct comparison of both techniques as chemical characterization tools. Parameters φ and t are different in nature, but both are related to the molecular organization. As for t with eq 1b, φ can be related to δd from a simple electrostatic model (Figure 6c, inset):

φ=q.[1-(1+(r a /(d+δd)) 2 ) -0.5 ]/[4 ε 0 ε r (d+δd)] (2)
r a is the counter-ion pairing distance, q is the elementary charge, ε 0 and ε r are the dielectric permittivity of vacuum and the relative permittivity of water, respectively.

φ=4.5 meV, for d=7 Å and δd=0 Å, corresponds to an Fc-ClO 4 -ion pairing distance of 4.9 Å (5.5Å is expected from molecular dynamics simulations 69 ). With eq 2, a Gaussian distribution for δd implies a non-Gaussian distribution for φ (Figure 6c). Combining eq 2 and the extended Laviron model (see SI Methods), we see that such a distribution should induce a broadening of the CV peak, but only when d+δd approaches the ionpairing distance (Figure 6d). Therefore, CV would not be sufficiently sensitive to assess information on the small molecular organization fluctuations expected here (e.g. SD(δd) = 0.8Å from parameter t analysis). This feature illustrates the potential of using t as an ultra-sensitive chemical characterization parameter.

In summary, we have investigated the possibility of assessing the -electronic couplings from charge transport measurements in a connected device, using a statistical analysis of current from a large array of Fc-thiolated gold nanocrystals. The results have been quantitatively compared to DFT calculations. Extracted parameters, including a molecule/electrode coupling asymmetry of 0.9 and t of 35 meV, were in the range of expectations. However, the distribution of t was broader than expected from the solely thermal fluctuations. This observation is attributed to structural fluctuations and to a variation of the intermolecular distance of 0.8 Å in the model. The results confirm the need for charge transport model to consider small structural fluctuations, even on the order of 1Å; however, CV does not have sufficient sensitivity to reveal such small fluctuations. This limitation may be overcome by measuring extremely small CV currents (on the single-dot level), and performing statistical analyses on φ (as predicted in Figure 6c). The origin of these structural fluctuations remains unclear, but could be related to the competitiveand -interactions due to the presence of alkyl chains.

Overall, the present study provides insights into understanding -intermolecular interactions in organic and (bio-) molecular devices. The findings confirm that Landauer-type coherent-scattering models, which are usually dedicated to lowtemperature mesoscopic physics, are relevant at room temperature for molecular electronics, even in the presence of cooperative effects. Statistical current analysis could be applied to various systems, because current histograms represent a common approach in molecular junctions. The study of -electronic couplings is a unique opportunity to link quantum chemistry, mesoscopic physics, organic electronics, and electrochemistry, indicating the importance of each subfield in the development of organic electronics.

Methods

Additional Methodological information related to STM, gold nanodot fabrication, monolayer self-assembly, experimental conditions for CV and related fits, image treatment DFT calculations and theoretical histogram generation is available in the SI Methods.

UHV STM

The high resolution image was performed at room temperature with a substrate biased at 2V and at a constant current of 1 pA.

Areas of top and side facets of the nanocrystals

To estimate the number of molecules per peak, the area was considered from the following formula, based on Figure 2c:

/4*(D-2*h/tanξ) 2 + *(D-h/tanξ)*h/sinξ (3)
where the first term corresponds to the area on the top and the second term to the area on the sides of the nanocrystal. Reasonable fits are obtained with h=2.7 nm and ξ=30°, as expected from the nanocrystal structure.

C-AFM

We measured current voltage by using C-AFM (Dimension 3100, Veeco) with a PtIrcoated tip on molecules 66 in N 2 atmosphere. Each count in the statistical analysis corresponds to a single and independent gold nanodot. The tip curvature radius is about 40 nm (estimated by SEM), and the force constant is in the range of 0.17-0.2 N/m. C-AFM measurements were taken at loading forces of 15 and 30 nN for the smallest and largest dots, respectively to keep a similar force per surface unit. As shown in ref. 14, a weak effect of the force is observed for these molecules in the range of 10-30 nN. In scanning mode, the bias is fixed and the tip sweep frequency is set at 0.5 Hz. With our experimental setup being limited to 512 pixels/image, the parameters lead to a typical number of 3000 counts for a 6×6 µm C-AFM image. In the presence of -electronic couplings, current histogram peaks are well-fitted with an asymmetric double sigmoidal function ( ) given by

( ) = + !("!" # $% & /()/% ( )1 - !("!" # !% & /()/% , - (4)
where the values of the various parameters are presented in SI Table 2.

Landauer Imry Buttiker Formalism and histograms fits

The model has been adapted from ref. 21 to account for a large number of molecules and asymmetrical contacts (detail in SI Methods). It provides a good description of the fundamental aspects of electron transport via the computation of the energy-dependent transmission through the device. The formalism described in ref. 21 focused on the zerobias conductance (and at low temperature), a result which can be extended to the evaluation of the current at low bias, provided that one integrates the transmission over a range of energy given by the external potential. Here we have used this model with conditions of relatively high bias and at room temperature. The assumption that the Landauer approach remains applicable under such conditions is often made in the field of molecular electronics with a single-level model. 31 We believe that these assumptions are further justified due to the strong coupling of the Fc molecules, as evidenced by the level broadening estimated in the range of 100 meV. The generation of current histograms (instead of conductance histograms in ref 21) required an additional assumption (midpoint rule) to efficiency compute the 10 6 realizations (see SI Methods).

The validity of this approximation has been confirmed for the present study (Figure S14 in SI).

The process of fitting the line shape of the experimental histograms relies on a relatively large number of variables, which can be defined by a step by step procedure.

We considered a site energy = 0.2 eV versus Fermi level at V bias =0 V, given the CV results and related energy band diagram proposed in ref.

14. An upper limit of . <45

meV was considered based on CV analysis (Figure S3d). First, we optimized the parameters for 3×3 molecules due to computational time. Because . does not significantly affect the extracted value of t (Figure S10), we considered δ =40 meV (similar to the value considered in ref. 21. V t and V b were adjusted to get a good current level and to reproduce the histogram shape. An optimal asymmetry factor of =0.9 was considered (Figure S10), in agreement with refs. 12 and 14. δV was tuned to fit the histogram line shape when t = 0 (mixed monolayer). δt was tuned with t while fitting asymmetric histograms line shapes. Current histograms were generated based on 10 6 realizations. When the number of molecules in the matrix is large (e.g. 9x9 molecules), histogram fitting takes several days. Efficient hardwares (i.e. Ising machine) is being developed to solve such problems efficiently. 70,71 Figure S2.

(h) Graphs showing number of molecules per dot (obtained from (e-g)) averaged with data from reduction peak (Figure S6). Error bars are based on dispersion between CV (oxidation/reduction peak and various speeds). Data are fitted with eq 3 (truncated cone approximation). Inset: Schematic representation of molecular organization. Peak 1 (purple) and peak 2 (green) correspond to molecules on top and sides, respectively. (i) Schematic representation of Fc-thiolated gold nanocrystals in NaClO 4 electrolyte when all Fc are oxidized. S2.

Figure

  Figure 2a-c (and Figure S1,S2 in SI) show the experimental setup with NaClO 4 electrolyte (0.1 M) facing Fc molecules and the Au nanocrystal electrodes.49 We have

Figure

  Figure 2e-g show conventional CV results for nanocrystals (of different diameters)

  , with the additional consideration of asymmetrical coupling to the electrodes and the possibility of simulating up to 9×9 molecules (only two molecules were considered in ref.21). Figure5aillustrates the modeled system. Each Fc molecule is considered as a single-level quantum dot coupled to both electrodes. Dots are coupled together with coupling term t in a tight binding model. This coupling term is equivalent to the transfer integral in DFT. For simplicity, t is considered to be identical along both axes in the plane. Each molecule within a molecular junction composed of N×N molecules has the same parameters (molecule orbital energy), V t , V b , (molecules coupling to top and bottom electrodes, respectively) and t. Cooperative effects, whose strengths are controlled by parameters V t ,V b ,t,N, cause a smearing out of the energy-dependent transmission coefficient shape, with a peak transmission being less than one21 (see FiguresS13-S15in SI for additional illustrations). The current, obtained from the integral of the transmission coefficient over a range of energy set by

  δd in eq 1b is chosen from a Gaussian distribution. A step-by-step fitting protocol is detailed in the Methods section and Figures S10, and S11 in SI. Figure 5b illustrates the possibility of generating histograms that reproduce the experiments. Optimized parameters for 9×9 molecules using eq 1a for t (t = 0.04 eV, V t = 0.401 eV, V b = 0.144 eV, δ = 40 meV, δt = 0.14 eV, and δV=22 meV) are in the range of those considered in ref.21 based on ref. 67 where t was 0.1 eV, V t = V l = 0.6 eV, δt = 75 meV, δV = 37

1

  Figure 1 Signatures of cooperative effects with the introduction of parameters φ and t. (a) Schematic representation of CV results in the absence (black curve) and presence (orange curve) of Coulomb interactions between Fc molecules according to the Laviron model 27 based on the Frumkin isotherm. Inset: Schematic representation of the microscopic process. When Fc is oxidized (green cloud), it shifts the energy level of the neighboring molecule by φ. (b) Schematic representation of a theoretically predicted 21 conductance histogram in the absence (black curve) and presence (orange curve) of coupling between two molecules (tight binding model). Inset: Schematic representation of intermolecular coupling (t) related to charge transfer between adjacent molecules.

Figure 2

 2 Figure 2 Quantitative analysis of intermolecular interaction energy φ from CV (a) Schematic representation of intermolecular interaction in an electrochemical setup. Electrolyte is NaClO 4 (0.1 M). φ, Coulomb repulsion between adjacent molecules (see SI Methods); CE counter electrode; RE Ag/AgCl reference electrode; WE working electrode (gold nanocrystals). (b) SEM image of gold nanocrystal array on highly doped silicon. Scale bar = 200 nm. (c) Schematic representation of a nanocrystal cross-section based on ref.49. Parameters of interest are the nanocrystal diameter D, height h≈3 nm and angle ξ≈30°. (d) Square wave voltammogram (SwV) for (1:10) FcC 11 SH SAM diluted with C 12 molecules on an array of 15 nm diameter dots. Integration of the main peak area corresponds, after normalization (see SI Methods), to 5 to 15 FcC 11 SH molecules per dot. The curve is fitted with 2 peaks. The main peak has a FWHM~90 mV (φ≈0). (e-g) CV results (anodic peak) for arrays of FcC 11 SH-coated gold nanoelectrodes of different diameters (as indicated). FWHM correspond to sweep rate of 1V/s (oxidation peak). Fitting parameters are indicated in TableS2.

Figure 3

 3 Figure 3 Estimation of cooperative effects from supramolecular organization and DFT calculations (a) UHV-STM image of a SAM of FcC 11 SH molecules grafted on gold. Molecular structure is resolved and used as reference for full DFT calculations. Periodic black lines, with cell delimited by pink clouds, indicate positions of Fc molecules. (b) Cell composed of four FcC 11 SH molecules based on (a,b). A number is attributed to each molecule due structural anisotropy. (φ a ,t a ) and (φ b ,t b ) refer to interactions between molecules 1 and 3 and molecules 1 and 2, respectively. X and Y axes are aligned along molecules 3 and 1 and molecules 2 and 1, respectively. (c) Full DFT calculation of parameter t b between molecules 1 and 2. Position of molecule 2 is translated along the X axis to mimic disorder. Inset: Molecular configuration at each maximum for t b . (d) Evolution of t b as a function of the variation of intermolecular separation δd modulated from the initial geometry (normal displacement = 0 Å). Decay ratios b is indicated.

Figure 4

 4 Figure 4 Current histograms used to evaluate -intermolecular interaction energy (~ 3000 counts per histogram) Current histogram obtained at a tip voltage of -0.6 V for (a) 40 and (b) 15-nm diameter dots (with 5-nm diameter on top). Plain curve is the fit with asymmetric double sigmoidal function (eq 4). Dashed curve is the lognormal fit. Inset: Schematic view of the setup. (c) Same as (b) but with a (1:1) FcC 11 SH:C 12 SH-diluted SAM. Fitting parameters are shown in TableS2.

Figure 5

 5 Figure 5 Histograms fits with Landauer Buttiker Imry formalism. (a) Schematic representation of the model. Each molecule (quantum dot) is coupled to other molecules with coupling term t and coupled to top/bottom electrodes with coupling energies V t and V b , respectively. , molecule orbital energy. SDs of these parameters are used to generate histograms. Related experimental setup shown for clarity. (b) Experimental (V tip = -0.6 V) and simulated histograms (V t = 0.401eV, V b = 0.144eV, .V = 22meV, = 0.2eV, t = 0.04 eV, .t = 140 meV, t 0 = 0.34eV, β = 1.96/Å SD(δd)=0.8Å) considering 81 molecules. (c) Schematic representation of model with fewer molecular interactions. Parameters are same as in (a) except t = 0. Related experimental setup (diluted monolayer) is shown for clarity. (d) Experimental (V tip = -0.6 V) and simulated histograms (t = 0).

Figure 6

 6 Figure 6 Extracted distributions of t and implications for organic electronic and electrochemical field (a) Distribution of t obtained from best fits in Figure 5d with eqs 1a and 1b. Expected (Gaussian) distribution from solely thermal motions shown in grey. Each electronic level has an associated t that can be positive or negative, so the sign is of little importance. (b) t 2 distribution obtained from (a). (c) Estimated distribution of φ from eq 2 given an intermolecular distance fluctuation of δd=0.8Å. Inset: Schematic representation of the electrostatic model (equation 2). (d) Experimental and theoretical CV results (coupled eq 2 and eq S1) with δd=0, 0.8 and 3 Å. Energy level of Fc vs Ag/AgCl (E p ) is used to center the CV peak at 0.

Acknowledgments

The authors thank C.A. Nijhuis from NTU Singapore for discussions, C. Wahl from CPT for beginning simulations on current histograms, D. Guerin and A. Vlandas from IEMN for discussions related to electrochemical measurements, and T. Hayashi, K. Chida and T.Goto from NTT Basic Research Labs for fruitful discussions. J.T. thanks PhD funding from Marie Curie ITN grants and the EU-FP7 Nanomicrowave project and J.C thanks the iSwitch (GA No. 642196) project. We acknowledge support from Renatech (the French national nanofabrication network) and Equipex Excelsior. Work

Associated content:

Supporting information

The Supporting Information is available free of charge on the ACS Publications website at DOI:10.1021/acs.nano-lett.xxx

Corresponding Author:

e-mail : nicolas.clement@lab.ntt.co.jp

ToC Image

The -electronic coupling energy parameter t is assessed from a statistical analysis of current histograms. (1) Hunter, C.A.; Sanders, J.K.M. J.Am.Chem.Soc. 1990, 112, 5525-5534.

(2) Herman, J.; Alfe, D.; Tkatchenko, A.; Nat. Commun