Pis'ma v ZhETF, vol.88, iss. 11, pp.780-784
© 2008 December 10
Proximity-induced superconductivity in graphene
M. V. Feigel'man1^, M. A. Skvortsov, K. S. Tikhonov L.D. Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia
Moscow Institute of Physics and Technology, 141700 Moscow, Russia Submitted 27 October 2008
We propose a way of making graphene superconductive by putting on it small superconductive islands which cover a tiny fraction of graphene area. We show that the critical temperature, Tc, can reach several Kelvins at the experimentally accessible range of parameters. At low temperatures, T <C Tc, and zero magnetic field, the density of states is characterized by a small gap Ea < Tc resulting from the collective proximity effect. Transverse magnetic field Hg (T) oc Eg is expected to destroy the spectral gap driving graphene layer to a kind of a superconductive glass state. Melting of the glass state into a metal occurs at a higher field HB2(T).
PACS: 74.20.^z, 74.78.^w, 74.81.-g
Among numerous fascinating properties, graphene [1, 2] provides a unique possibility to study the phenomenon of proximity-induced superconductivity in very favorable conditions. Experimental studies of the Joseph-son current through graphene in standard wide planar SNS junctions [3] have shown that proximity effect in graphene is qualitatively similar to the one known for usual dirty metals. In this Letter, we show that even a tiny amount of graphene area covered by small superconductive islands (with good electric contact to graphene) can lead to a macroscopically superconductive state of the graphene film, with Tc in the Kelvin range.
We consider a system of superconductive (SC) islands of radius a (with the typical value of a few tens of a nanometer) placed approximately uniformly on top of a graphene layer (with the typical distance between the islands b in the sub-micron range) shown in Fig.l. We assume that b is much larger than both a and the
Fig.l. Graphene film covered by superconducting islands
graphene mean free path I. Moreover, present theory will be limited by the case I < a when electron motion in graphene is diffusive at all relevant scales. We will not be particulary interested in phenomena in the
e-mail: feigel@landau.ac.ru
vicinity of the graphene neutral point, assuming relatively large gate potentials > 10 V, and carrier density n > 1012cm2. We assume graphene Fermi energy Ep A0 Tc, where A0 is the island's superconductive gap. Graphene sheet can be either single- or few-layered: the only relevant features are (i) high diffusion constant D > 102 cm2/s, and (ii) very low (in comparison with metals) electron density, which allows to combine moderate values of dimensionless conductance g = (H/e2Rn) > 3 with high Thouless energy Etii = frD/b2. Not very large values of sheet conductance g are practically favorable to avoid suppression of superconductivity in small SC islands due to the inverse proximity effect (the latter can be neglected provided that GtotS < where Gj = G-J + ln(6/a)/27r5 « 1 is the total normal-state resistance from the island to the graphene sheet, Gjnt is the SC-graphene interface conductance, and 5 is level spacing in the island).
Below we treat graphene as a normal diffusive 2D metal within the standard approach based on the Us-adel equation [4]; its applicability to diffusive graphene was proven in Ref. [5, 6]. The intrinsic Cooper channel interaction in graphene can be neglected due to its low DOS [7]. Similarly, phonon-induced attraction is also weak.
Proximity coupling and transition temperature. We start with calculating the Josephson coupling energy between two superconductive islands of radius a separated by distance b a, neglecting the presence of other islands. Such a pair-wise approximation is adequate for determination of Tc, but breaks down at T < Tc/ln(b/a), as shown below. For the single SC island on graphene, the Matsubara-space Usadel equation for the spectral angle 0W and corresponding boundary conditions [8] read as
780 IlHCbMa b ?K3T<D tom 88 btra.11-12 2008
DV29U - 2\w\ sin0w = 0,
двш G,
9■
illt
дг 2жа
cos 9^
= 0.
(1) (2)
The normal (Gw) and anomalous (Fw) components of the matrix Green function in the Nambu-Gorkov space are expressed via the spectral angle 0W and the order parameter phase ip as Gw(r) = cos0w(r) and Fu(t) = = e'^ sin0w(r). The full matrix structure of the anomalous Green function Fw with the valley and spin spaces included (Pauli matrices it and s, respectively) is determined by the usual s-wave pairing in the SC islands: Fw oc ftxSy The interface conductance Gj„t is treated below as a phenomenological parameter which accounts for the Fermi velocity mismatch and a potential barrier on the graphene-metal interface [9].
It is crucial for further analysis that the two-island generalization of the nonlinear problem (1), (2) can be linearized while calculating the Josephson current at inter-island distances b a. Indeed, the total current can be calculated by integrating the current density over the middle line between the islands, on the distance Pi,2 > b/2 from them. This procedure also involves summation over Matsubara energies wn = irT(2n + 1), with the major contribution to the sum coming from wn ~ ETh. At such wn and the spectral angle 9 is small, and linearization of Eqs. (1) and (2) leads to the solution
вш(г) = A(w)K0
L,ь
A(w) =
0(iu
In (Ьш/а)'
(3)
with Lw = yjD/2oj, tw = (Gint/2irg) In(Lw/a), and Q(t) solving the equation ©(f) = tcos@(i). The function A(u>) evolves between the tunnel and diffusive limits as
A(w) =
Gint/(27r5),
n/[2hi(Lu/d)],
Gint -С 2-кд/ ln(Lw/d), Gint > 2-кд/ ln(Lw/d),
(4)
and is always small for In(Lw/a) 1. Thus the Joseph-son current I(ip) = Icsimp between two SC islands with different phases, <pi — <fi2 = <P, can be calculated using the linearized two-island solution for the anomalous Green function: ^(r) = e'^1 sin 0W (|r — — ri |) + e'^2 sin0w(|r — r2|). The standard calculation of the Josephson energy Ej = (h/2e)Ic then leads to
Ej(b,T) = ingT 42K)P(Vi/8^), (5)
w„ >0
where P(z) = z /0°° K0(z cosh t)K1(z cosh t) dt.
A two-dimensional array of SC islands with the coupling energies (5) undergoes the Berezinsky-Kosterlitz-Thouless transition at
Tc = 1Ej(b,Tc),
(6)
where the numerical coefficient 7 depends on the array structure. Below we will assume that the SC islands form a triangular lattice, in which case 7 « 1.47 [10]. For the interface conductance Gjnt comparable with the sheet conductance one finds the transition temperature Tc ~ ETh. In general, Tc can be obtained by numerical solution of Eq. (6) using Eqs. (3) and (5). The result obtained for the ratio Tc/Et\x as a function of Gjnt for g = 6 (Rn « 700 0) and b/a = 10 is presented in Fig.2. With the graphene diffusion constant
^ 0.5
Fig.2. The critical temperature, Tc, and the zero-temperature spectral gap, Eg, vs. the interface conductance Gint (the sheet conductance g = 6, and b/a = 10)
D = 500cm2/s (see, e.g., [11]) and b = 0.5 /tm, one estimates .Etii ~ 1-5 K, leading to Tc in the range 1-f 3 K for 5 < Gint < 20.
Low temperatures: spectral gap and order parameter. Now we switch to the low-temperature range T -C Tc and consider the issue of the spectral gap for the excitation above the fully coherent ground state (with all phases <pt equal). The density of states 1'(E) = i^oRe cos 9(E) is determined then by the periodic solution of Eqs. (1) and (2), analytically continued to real energies: |w| iE. This periodic problem is equivalent to the one defined within the single (hexagonal) elementary cell, supplemented by the additional condition nV0| r = 0, where F is the cell boundary. Solution of the Usadel equations for such a geometry leads to formation of the spectral gap Eg similar to the minigap for one-dimensional SNS junctions [12]. To find it, we write 9(r) = ir/2 + it{i(r) and determine the spectral boundary as the value of E where equation
DV ф + 2E cosh?/' = 0
(7)
г=а
Г
782
M. V. Feigel'man, M. A. Skvortsov, K. S. Tîkhonov
ceases to have solutions with real i/»(r) [13]. At large In(b/d) one may approximate the hexagonal boundary F of the elementary cell by the circle of radius R = 6/2. For the ideally transparent interface, solution of the radially symmetric Eq. (7) gives for the value of the zero-temperature spectral gap:
hD/R2 2.65-Eti,
En
9 1.51n(i?/a) — 1.2 ln(6/4a)'
Decreasing the interface conductance Gjnt leads to the suppression of the minigap, as shown in Fig.2.
In the limit of large In (b/d), the spectral gap Eg -C •C Tc. Smallness of the gap distinguishes the system with superconductive islands from usual dirty superconductors. Roughly speaking, it behaves as a continuous 2D superconductor at the energy/temperature scales smaller than Eg, whereas in the range Eg < (E, T) < Tc it can rather be described as an array of weak Josephson junctions.
The existence of the sharp gap (8) in the electron spectrum looks surprising, as only a tiny fraction (a/b)2 of graphene area is in direct contact with SC islands. The presence of this gap can be traced back to the periodic structure of islands we assumed. Therefore any irregularity in the positions of SC islands will lead to the smearing of the hard gap. Assuming that islands' locations are shifted at random from the sites of the ideal triangular lattice, with the typical shift 5b -C b, one can reduce the problem to the effective one, defined on a scales large than array lattice constant. Random displacements of islands will be seen, in terms of this effective model, as local fluctuations of the superconductive coupling constant [14, 15], leading to the smearing of the gap with the relative width 5Eg ~ (Sb/b)2Eg. The sharp gap will also be smeared by thermal fluctuations of island's phases and finite thermal coherence length Lt■ Thus we expect the spectral gap to be observable at T-C Ea.
Even in the presence of t
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