PHYSICAL REVIEW B 83, 014510 (2011)

BCS superconductivity in metallic nanograins: Finite-size corrections,low-energy excitations, and robustness of shell effects

Antonio M. Garcıa-GarcıaCFIF, IST, Universidade Tecnica de Lisboa, Av. Rovisco Pais, P-1049-001 Lisbon, Portugal

Juan Diego UrbinaInstitut fur Theoretische Physik, Universitat Regensburg, D-93040 Regensburg, Germany

Emil A. YuzbashyanCenter for Materials Theory, Rutgers University, Piscataway, New Jersey 08854, USA

Klaus Richter1 and Boris L. Altshuler2

1Institut fur Theoretische Physik, Universitat Regensburg, D-93040 Regensburg, Germany2Physics Department, Columbia University, 538 West 120th Street, New York, New York 10027, USA

(Received 15 November 2009; revised manuscript received 23 September 2010; published 20 January 2011)

We combine the BCS self-consistency condition, a semiclassical expansion for the spectral density andinteraction matrix elements to describe analytically how the superconducting gap depends on the size and shapeof a two- and three-dimensional superconducting grain. In chaotic grains mesoscopic fluctuations of the matrixelements lead to a smooth dependence of the order parameter on the excitation energy. In the integrable case wefind shell effects; i.e., for certain values of the electron number N a small change in N leads to large changes inthe energy gap. With regard to possible experimental tests we provide a detailed analysis of the dependence ofthe gap on the coherence length and the robustness of shell effects under small geometrical deformations.

DOI: 10.1103/PhysRevB.83.014510 PACS number(s): 74.20.Fg, 75.10.Jm, 71.10.Li, 73.21.La

Finite-size effects are well documented1 in fermionicinteracting systems such as atomic nuclei and atomic clusters.It is also well established2,3 that the more symmetric the systemis, the stronger are these corrections. For instance, the existenceof magic numbers signaling the presence of a particularlystable nucleus has its origin in the gap between the groundstate and the first excited states caused by the high degree ofsymmetry of the system.

In the field of mesoscopic superconductivity, the study offinite-size effects also has a long history. Already 50 yearsago, Anderson noted4 that superconductivity should breakdown in small metallic grains when the single-particle-levelspacing at the Fermi energy is comparable to the bulksuperconducting gap. In the 1960s the size dependence of thecritical temperature and the superconducting gap were studiedfor a rectangular grain in Ref. 5 and for a nanoslab in Ref. 6.Thermodynamical properties of superconducting grains wereinvestigated in Ref. 7. Results of these papers are restricted torectangular grains, and superconductivity is described by theBardeen, Cooper, and Schriffer (BCS) theory.8

The experiments by Ralph, Black, and Tinkham in the mid-1990s9 on Al nanograins of typical size L ! 3"13 nm showedthat the excitation gap is sensitive to even-odd effects. Morerecently it has been observed10 that the critical temperatureof superconducting ultra-thin lead films oscillates when thefilm thickness is slightly increased. These results have furtherstimulated the interest in ultrasmall superconductors.11–16 Forinstance, pairing, not necessarily BCS, in a harmonic oscillatorpotential was investigated in Ref. 13. The critical temperatureand the superconducting gap for a nanowire were reportedin Ref. 14 by solving numerically the Bogoliubov-de Gennesequations. In Ref. 15 the superconducting gap and low-energyexcitation energies in a rectangular grain were computed

numerically within the Richardson model.16 Shell effects insuperconducting grains with radial symmetry were studied inRefs. 17 and 18. Moreover, recent experiments on Al grainswere interpreted19 as evidence that shell effects can drivecritical temperatures in these grains above 100 K. Mesoscopiccorrections to the BCS energy gap were also considered inRefs. 20 and 21.

We note that if the mean single-particle-level spacingis larger than the bulk superconducting gap, the BCS for-malism breaks down. However, an analytical treatment isstill possible22 with the help of an exactly solvable modelintroduced by Richardson16 in the context of nuclear physics.In particular, finite-size corrections to the predictions of theBCS theory have been recently studied in Refs. 23–27.

Despite this progress, a theory that accounts for all relevantmesoscopic effects in superconducting grains has not emergedso far. The Richardson model alone cannot provide thefoundation for such a theory as it does not allow for mesoscopicspatial fluctuations of the single-particle states. In the presentpaper, for the particular cases of chaotic and rectangularshaped grains, we develop such a theory based on the BCStheory and semiclassical techniques. This formalism permitsa systematic analytical evaluation of the low-energy spectralproperties of superconducting nanograins in terms of theirsize and shape. Leading finite-size corrections to the BCSmean field can also be taken into account in our approach;see Ref. 28 for further details. Results for three-dimensional(3d) grains were also previously published in Ref. 28. Herewe discuss both the two-dimensional (2d) and 3d cases aswell as provide a more detailed account of the techniquesutilized. Moreover, we study the dependence of the mesoscopicBCS order parameter (superconducting gap) on the coherencelength and the robustness of shell effects.

014510-11098-0121/2011/83(1)/014510(14) ©2011 American Physical Society

ANTONIO M. GARCIA-GARCIA et al. PHYSICAL REVIEW B 83, 014510 (2011)

For chaotic grains, we show that the order parameteris a universal function of the single-particle energy; i.e.,it is independent of the particular details of the grain.The mesoscopic fluctuations of the matrix elements of thetwo-body interactions between single-particle eigenstates areresponsible for most of the deviations from the bulk limit.For integrable grains, we find that the superconducting gap isstrongly sensitive to shell effects. Namely, a small modificationof the grain size or number of electrons inside can substantiallyaffect its value. Throughout the paper we study clean (ballistic)grains. The mean field potential is thus an infinite well ofthe form of the grain. We restrict ourselves to system sizessuch that the mean level spacing around the Fermi energyis smaller than the bulk gap, so that the BCS formalismis still a good approximation. For the superconducting Algrains studied by Tinkham and coworkers,9 this correspondsto sizes L > 5 nm.

Our results are therefore valid in the region kF L # 1(limit of validity of the semiclassical approximation),4,23

!/"0 < 1 (limit of validity of the BCS theory), and l # # # L(condition of quantum coherence). Here kF , # = hvF /"0,l, !, "0 are the Fermi wave vector, the superconductingcoherence length, the coherence length of the single-particleproblem, the average single-particle level spacing, and thebulk gap. The Fermi velocity is vF = hkF /m. ConditionskF L # 1 and !/"0 < 1 hold for Al grains of size L ! 5 nm.Further, in Al grains # $ 1600 ! nm and l > 104 ! nmat temperatures T " 4 K.24 Therefore, this region is wellaccessible to experiments.

I. THE SUPERCONDUCTING GAP IN THE BCS THEORY

Throughout the paper pairing between electrons is de-scribed by the BCS Hamiltonian,

H =!

n$

%nc†n$ cn$ "

!

n,n%

In,n%c†n&c

†n'cn%'cn%&,

where cn$ annihilates an electron of spin $ in state n,

In,n% ( I (%n,%n%) = &V !

"'2

n ()r)'2n% ()r) d)r (1)

are matrix elements of a short-range electron-electron interac-tion, & is the BCS coupling constant, and 'n and %n are theeigenstates and eigenvalues of a free particle of effective massm in a clean grain of volume (area) V (A). Eigenvalues %n aremeasured from the actual Fermi energy %F of the system. In thisnotation the mean level spacing is ! = 1/(TF(0), where (TF(0)is the spectral density at the Fermi energy in the Thomas-Fermiapproximation.

The BCS order parameter is defined as

"n ( "(%n) =!

n

In,n% *c†n%&c†n%'+.

Within BCS theory, it is determined by the following self-consistency equation:25

"n = 12

!

|%n% |<%D

"n%In,n%#%2n% + "2

n%

, (2)

where %D is the Debye energy. This result is obtainedin the grand canonical approximation.8 Note that the

BCS order parameter "n is an explicit function of thesingle-particle energy %n since the matrix elements I (%,%%)are energy dependent.

Introducing the exact density of single-particle states((%%) =

$n% !(%% " %n%), one can write Eq. (2) in integral form:

"(%) = 12

" %D

"%D

"(%%)I (%,%%)%%%2 + "2(%%)

((%%)d%%. (3)

The gap equation (3) will be the main subject of our interest.As soon as the order parameter "(%) is known, the low-lying(single-particle) excitation spectrum, E =

%"(%)2 + %2, is

also determined.In the large volume (area) limit, the spectral density, to

leading order, is given by the Thomas-Fermi expression:

(TF(%%) = 2 ,&

V4)2

' 2mh2

(3/2-%% + %F for 3d,

A4)

' 2mh2

(for 2d,

(4)

where the factor 2 in front stands for spin degeneracy. Inaddition, in the bulk limit the matrix elements (1) for chaoticgrains are simply I (%,%%) = &! as a consequence of quantumergodicity. The gap is then energy independent "(%) = "0,and Eq. (2) yields the BCS bulk result:

"0 = 2%De" 1& . (5)

As the volume of the grain decreases, both ((%%) and I (%,%%)deviate from the bulk limit. In this region a more generalapproach to solve Eq. (3) is needed.

Since we are interested in the regime of many particles[(TF(0)%F # 1], an appropriate tool is the semiclassical ap-proximation in general and periodic orbit theory29 in particular(see the Appendix for an introduction). These techniques yieldclosed expressions for ((%%) and I (%,%%) in terms of quantitiesfrom the classical dynamics of the system, which allows us tocalculate analytically the resulting superconducting gap. Suchexplicit expressions for the superconducting gap enable us tostudy deviations from the BCS theory, the spatial dependenceof the gap, and the relevance of shell effects in realistic, notperfectly symmetric grains.

Our general strategy can be summarized as follows:(1) Use semiclassical techniques to compute the spectral

density ((%%) =$

n% !(%% " %n%) and I (%,%%) as series in thesmall parameter 1/kF L, where kF is the Fermi wavevectorand L . V 1/3(.A1/2) is the linear size of the grain (Sec. IIand the Appendix).

(2) Solve the BCS gap equation (2) order by order in 1/kF L(Sec. III).

(3) Study the impact of small deformations of the shape ofa symmetric grain on the gap in realistic models of the grain(Sec. IV).

Finally we stress that all the parameters in our model,&,kF ,%D,%F , are the actual parameters that characterize thematerial at a given grain size and not necessarily the ones atthe bulk limit.

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II. SEMICLASSICAL APPROXIMATION FOR THEDENSITY OF STATES AND INTERACTION MATRIX

ELEMENTS

The first step to solve the gap equation is to find explicitexpressions for the spectral density ((%%) and the interactionmatrix elements I (%,%%) as series in a small parameter1/kF L. While the semiclassical approximation for the spectraldensity has been known for a long time,29 the calculationfor the matrix elements has only recently attracted someattention.28,30 Here we state the results and refer the readerto the Appendix for details.

A. Spectral density

In the semiclassical approximation (see the Appendix), thespectral density is given by

((%%) . (TF(0)[1 + g(0) + gl(%%)], (6)

with a monotonous g(%%) and oscillatory g(%%) (as functionsof system size) parts. The notation g(% = 0) means that g isevaluated at the Fermi energy. This contribution is given bythe Weyl expansion,1

g(0) =&

± S)4kF V

+ 2Ck2F V

3d,

± L2kF A

2d,(7)

for Dirichlet (") or Neumann (+) boundary conditions. InEq. (7), S is the surface area of the 3d cavity and C its meancurvature, while L is the perimeter in the 2d case.

The oscillatory contribution to the density of states is givenby the Gutzwiller trace formula:29

gl(%%) = /

)*

+

2)k2F V

$lp Apei(kF Lp+*p)ei %%

2%FkF Lp 3d,

2kF A

$lp Apei(kF Lp+*p)ei %%

2%FkF Lp 2d.

(8)

The summation over classical periodic orbits (p) with lengthLp includes only orbits shorter than the quantum coherencelength l of the single-particle problem. The semiclassicalamplitude Ap and phase *p in Eq. (8) can also be computedexplicitly using the knowledge of periodic orbits. As wasmentioned previously, the parameters kF and %F in theseexpressions refer to the Fermi wavevector and Fermi energyof the system at a given grain size. Within the free Fermi gasapproximation it is possible to relate the bulk Fermi energywith the one at a given finite size by simply inverting therelation

12N =

" µ

((%)d%, (9)

where ((%) is the spectral density and N is the number ofparticles.

B. Matrix elements

The calculation of the interaction matrix elements I (%,%%)is more complicated as it requires information about classical

dynamics beyond periodic orbits. For a chaotic cavity the finalresult (see the Appendix),

I (%,%%) =

)*

+

&V

,1 + I short

3d (0) " )2S2

16k2F V 2 + I

longdg (0,% " %%)

-3d,

&A

,1 + I short

2d (0,% " %%) + Ilongdg (0,% " %%)

-2d,

(10)

has two types of contributions. Identical pairs of short classicaltrajectories hitting the boundary once give

)..*

..+

I short3d (0) = )S

4kF V3d,

I short2d (0,% " %%) = L

kF A

,C % + Si(4kF L)

)

-2d,

+ L2)kF A

/Ci

, 4(%"%%)kF L%F

-" Ci

, 2(%"%%)%F

-0(11)

with C % = 0.339 . . . , a numerical constant given in theAppendix, and Ci(x) the cosine-integral function.

In the so-called diagonal approximation (see the Appendix)the contribution of longer classical trajectories is

Ilongdg (%F ,% " %%) =

& 1V+l

'%"%%

%F

(3d,

1A+l

'%"%%

%F

(2d,

(12)

where

+l(w) =" l!

,

D2, cos[wkF L, ()r)]d)r (13)

is an integrated sum over trajectories , ()r) starting and endingat position )r . As detailed in the Appendix, due to the ergodicityof the chaotic classical systems, in the limit l # L, Eq. (13)simplifies to

+l#L(w) =

)*

+

4)2

k3F

sin (wkF l)w

3d,

4k2F

sin (wkF l)w

2d.(14)

For integrable grains there is no universal expression forI (%,%%). We restrict ourselves to the rectangular geometrywhere to a good approximation the matrix elements are energyindependent.

Using the knowledge of ((%%) and I (%,%%) as series in 1/kF L,we solve the gap equation (3) in different situations of interest.The resulting gap function, in general, depends the single-particle energy %, the size of the system, and the number ofparticles (or, equivalently, Fermi energy %F ).

III. SOLUTION OF THE GAP EQUATIONIN THE SEMICLASSICAL REGIME

In this section we solve the gap equation (3) for "(%).For a rectangular box in two and three dimensions the gapequation is algebraic, since "(%) = " is energy independent.In the chaotic case, however, we get an integral equationdue to the energy dependence of the interaction matrixelements. As we will see, both cases can be solved analyticallyorder by order in 1/kF L.

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ANTONIO M. GARCIA-GARCIA et al. PHYSICAL REVIEW B 83, 014510 (2011)

A. Rectangular box in two and three dimensions

For the rectangular box the matrix elements are

I (%,%%) =1

i=x,y,z

(1 + !%i ,%%i/2)/V, (15)

where %i 0 k2i , pi = hki is the conserved momentum in the

i = x,y,z direction, and ! stands for Kronecker’s function. Wefirst investigate the role of these matrix elements on the energygap. Qualitatively we expect an enhancement as I (%,%%) >1/V . This enhancement should not be large for !/"0 1 1as the spectrum of a rectangular grain has only accidentaldegeneracy; namely, %i = %%

i typically implies that i = i %. Fora perfectly cubic grain the enhancement is expected to be largerdue to level degeneracy, although they will still relatively smallfor !/"0 1 1. The numerical results of Fig. 1 (upper plot) forthe gap as a function of the grain size confirm this prediction.We compare the cases of trivial matrix elements I (%,%%) $ 1/Vand Eq. (15) (see caption for details). In the region in which ourresults are applicable, ! 1 "0 (L # 6 nm), the enhancementof both the gap average (upper plot) and fluctuations (lowerplot) due to Eq. (15) is small. We note that in the numericalcalculation the chemical potential is not the bulk Fermi energy,but it is computed exactly for each grain size (see caption).This induces an additional enhancement of the average gapwith respect to the bulk limit "0.

Since we are mainly interested in the study of gap fluctu-ations (discussed later in this paper), we neglect in the restof this section the nontrivial part of Eq. (15) [I (%,%%) $ 1/V ].Therefore to a good approximation the gap does not dependon energy, "(%) = ", and satisfies the equation

2&

=" %D

"%D

1 + g(0) + gl(%%)-%%2 + "2

d%%, (16)

where g(0) for a 3d rectangular box is given by Eq. (7) withoutthe curvature term.

Using Eq. (7) for g(0) and Eq. (8) for gl(%%) (henceforthwe drop the subscript l to simplify the notation), and takinginto account the scaling of each contribution with 1/kF L asdescribed in the Appendix, we look for a solution of the gapequation (16) for the 3d case in the following form:

" = "0(1 + f (1) + f (3/2) + f (2)), (17)

where f (n) 0 1/(kF L)n. Substituting " into Eq. (16), expand-ing in powers of 1/kF L, and equating the coefficients at eachpower, we obtain an explicit expression for f (i):

&f (1) =2g(0) + &

2

" %D

"%D

g(3)(%%)#%%2 + "2

0

d%%3,

(18)

&f (3/2) =3!

i,j 2=i

&

2

" %D

"%D

g(2)i,j (%%)

#%%2 + "2

0

d%%,

10 15 20L(nm)

0

0.5

1

1.5

2

!(L

)/! 0

10 15 20L(nm)

0.1

0.2

"(!(

L))

/!av

e

FIG. 1. (Color online) Upper figure: The energy gap " in unitsof the bulk gap "0 for a cubic grain of side L with & = 0.3, %D =32 meV, %F $ 11.65 eV, kF = 17.5 nm"1 as a function of the grainsize L. The chemical potential was computed exactly as a functionof N by inverting the relation 1

2 N =4 µ

((%) d% where ((%) is thespectral density. Similar results (not shown) are obtained for othervalues of &. Red circles stand for the exact numerical solution ofthe gap equation (2) with matrix elements (15). The black curve isits average value "ave. Blue squares are the numerical solution ofEq. (2) for trivial matrix elements I (%,% %) = 1/V . The green curveis its average value. Lower figure: The standard deviation of thegap $ (L) in units of the average gap "ave, a typical estimation ofthe average fluctuation, as a function of the grain size. The black(green) curve is the typical deviation for the case of nontrivial matrixelements given by Eq. (15) [I (%,% %) = 1/V ]. As can be observed,in the region !/"0 # 1 (Al L # 6 nm), in which our semiclassicalformalism is applicable, the nontrivial matrix element (15) does notmodify substantially the average gap or the typical fluctuation. Wenote that the average fluctuation (see also Fig. 2) is in reasonable

agreement with the theoretical prediction, ""0

$#

)!4"0

.20

&f (2) =3!

i

&

2

" %D

"%D

g(1)i (%%)

#%%2 + "2

0

d%% + f (1)[f (1) " g(0)]

" f (1)!

i

"20

2

" %D

"%D

g(1)i (%%)

'%%2 + "2

0

(3/2 d%%, (19)

where g(k) 0 (kF L)"k denotes the oscillating part of thespectral density. Explicit expressions for g(k), g

(k)i , and g

(k)i,j

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BCS SUPERCONDUCTIVITY IN METALLIC NANOGRAINS: . . . PHYSICAL REVIEW B 83, 014510 (2011)

for a rectangular box in terms of periodic orbits can be foundin the Appendix and in Ref. 1.

Equations (18) and (19) can be further simplified by thefollowing argument. After we express g(3), g(2), and g(1) interms of a sum over periodic orbits, the integration over %% canbe explicitly performed. The resulting expression is again anexpansion in terms of periodic orbits with two peculiarities:(1) the spectral density is evaluated at the Fermi energy, and(2) in the limit %D # "0 the contribution of an orbit of periodLp is weighted with the function

W (Lp/# ) = &

2

" 3

"3

cos(Lpt/# )-

1 + t2dt. (20)

This cutoff function is characteristic of the BCS theory asopposed to the smoothing due to temperature or inelasticscattering (recall that in this paper we assume that the single-particle coherence length l is much larger than superconduct-ing coherence length # ). In a similar fashion, the last term inf (2) is weighted with

W3/2(Lp/# ) ="2

0

2

" 3

"3

cos(Lpt/# )(1 + t2)3/2

dt.

The effect of W3/2(Lp/# ) is, again, to exponentially suppressthe contribution of periodic orbits longer than # . Therefore thesum over periodic orbits in the definition of the spectral densityis effectively restricted to orbits with lengths of the order orsmaller than the superconducting coherence length # .

Following standard semiclassical approximations, we intro-duce g# (0) as a spectral density evaluated at the Fermi energywith a cutoff function that suppresses the contribution of orbitsof length Lp > # . With these definitions, we get

&f (1) =,g(0) + g

(3)# (0)

-, &f (3/2) =

3!

i,j 2=i

g(2)i,j# (0), (21)

&f (2) =3!

i

g(1)i# (0) + f (1)

5

f (1) " g(0) "3!

i

g(1)i# (0)

6

.

Equation (21) is our final result for the finite-size correctionsto the gap function for a 3d rectangular box. As expected, it isexpressed in terms of classical quantities such as the volume,surface, and periodic orbits of the grain.

In Fig. 2 we compare the analytical expression for the gap(17) and (21) (solid blue line) to the numerical solution ofthe gap equation using the exact one-body spectrum (circles)and the semiclassical prediction for the spectral density (redsquares). It is observed that the analytical expression for thegap is in fair agreement with the exact numerical results.Moreover it is also clear from the figure that the semiclassicalformalism provides an excellent description of the numericalresults. We note that the small differences observed for smallvalues of the gap are a consequence of the finite l ! 50Rsingle-particle coherence length entering into the semiclassicalexpression of the spectral density (6). Since our motivationhere is to test the validity of the semiclassical formalism, weare assuming for simplicity that the chemical potential is fixedat the bulk Fermi energy.

The following argument can shed light on our results. Thedensity of states cannot be pulled out of the energy integration

4.22#105

4.24#105

4.26#105

4.28#105

N

0

0.5

1

1.5

!(N

)/! 0

FIG. 2. (Color online) The energy gap " in units of the bulk gap"0 for a cubic grain with & = 0.3, %D = 32 meV, %F $ 11.65 eV,kF = 17.5 nm"1 as a function of the number of particles N (L $13.23–13.32 nm) inside the grain. The solid line is the analyticalprediction from (17) and (21). Black circles (red squares) are resultsfrom a numerical evaluation of the gap equation using the exact[semiclassical Eq. (6)] spectral density. The semiclassical formalismprovides an excellent description of the exact numerical results forthe gap. We stress that, for the sake of simplicity, it has been assumedthat I = 1/V .

in the gap equation (16) unless it is smoothed. However, this isexactly what our result (21) means, since truncating the sums isequivalent to smoothing the energy dependence. We concludethat our result (16) should be similar to the standard BCSsolution in the bulk,"0 = 2%De"1/&, with the substitution& 4&[1 + g(0) + g# (0)]. Indeed, an expansion of this expressionin 1/kF L gives exactly Eq. (21).

In order to simplify notation henceforth we will drop thesubscript # in the spectral density g# smoothed by the cutofffunction W (Lp/# ). In two dimensions we find

" = "0(1 + f (1/2) + f (1)), (22)

with

&f (1/2) = g(2)1,2(0),

(23)&f (1) = g(0) +

!

i=1,2

g(1)i (0) + 1 " &

&

,g

(2)1,2(0)

-2.

The sums implicit in gi ,gi,j are smoothly truncated by the sameweight function W (Lp/# ). Similar to the 3d case, this resultcan also be obtained by expanding the bulk expression for thegap with the full density of states in 1/(kF L). We note that,contrary to the 3d case, in 2d grains, oscillatory contributionsto the density of states are of leading order.

B. 3d chaotic cavity

The energy dependence of the interaction matrix elements,I (%,%%), in this case is given by Eqs. (10)-(14), i.e.,

I (%,%%) = &

V

21 + )S

4kF V" )2S2

16k2F V 2

+ 1V+l

7% " %%

%F

83,

where

+l(w) = 4)2

k3F

sin(kF l-)-

. (24)

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ANTONIO M. GARCIA-GARCIA et al. PHYSICAL REVIEW B 83, 014510 (2011)

The details of the calculation based on the semiclassicalapproximation for Green’s functions can be found in theAppendix.

The expression for I (%,%%) together with the semiclassicalexpression for the spectral density (8) are the starting pointfor the calculation of the superconducting order parameter.The energy dependence of the matrix elements implies a gapequation of integral type and, most importantly, that the orderparameter itself depends on the energy. Based on the 1/kF Ldependence of the different contributions to I (%,%%), we write

"(%) = "0[1 + f (1) + f (2) + f (3)(%)] (25)

for a 3d chaotic grain. Substituting this expression into the gapequation (3) and comparing powers of 1/kF L, we get a simplealgebraic equation for f (1) with the solution

&f (1) = (1 ± 1)S)

4kF V. (26)

It shows that for Dirichlet (") boundary conditions, thesuperconducting order parameter for a chaotic 3d cavitydoes not have mesoscopic deviations of order 1/kF L. Thissuppression is a hallmark of the chaotic case and appearsdue to the fluctuations of the interaction matrix elements.It can be also found by substituting & 4 &(1 + S)/4kF V )into Eq. (5), which accounts only for the surface contributionto the density of states, and expanding the modified "0 to firstorder in 1/kF L.21

The second-order correction reads as

&f (2) = 2Ck2F V

+ 27

51 + 1 ± 1&

87)S

4kF V

82

+ g(0), (27)

with

g(0) = 2)k2F V

!

p

ApW (Lp/# ) cos(kF Lp + *p), (28)

where the contribution of periodic orbits Lp longer than thecoherence length # is exponentially suppressed.

Equating terms of order (kF L)"3, we obtain forf (3)(%) an integral equation of the form f (3)(%) = h(%) +4

K(%%)f (3)(%%) d%%, which is solved with the ansatz f (3)(%) =h(%) + c, where c is a constant. We obtain

f (3)(%) = )&!

"0

7"0#

%2 + "20

+ )

4

8. (29)

Note that (1) since !/"0 1 1 is an additional small parameterthe contribution (29) can be comparable to lower orders inthe expansion in 1/kF L and (2) the order parameter "(%) hasa maximum at the Fermi energy (% = 0) and decreases onan energy scale % ! "0 as one moves away from the Fermilevel. One can also show that mesoscopic corrections given byEqs. (26) and (27) always enhance "(0) as compared to thebulk value "0. A couple remarks are in order: (1) the energydependence of the gap is universal in the sense that it does notdepend on specific grain details and (2) the matrix elementsI (%,%%) play a crucial role; e.g., they are responsible for mostof the deviation from the bulk limit. Finally we briefly addressthe interplay of mesoscopic fluctuations and parity effects (seeRef. 28 for a more detailed account). The Matveev-Larkin(ML) parity parameter "p,23 a experimentally accessible

observable, accounts for even-odd asymmetries in ultrasmallsuperconductors. While the ML parameter coincides with thestandard superconducting gap in the bulk limit, in Ref. 23 itwas found that its leading finite-size correction is given by

"p ( E2N+1 " 12

(E2N + E2N+2) = "(0) " !

2, (30)

where EN is the ground-state energy for a superconductinggrain with N electrons.

We see that these corrections to the BCS mean fieldapproximation are comparable to mesoscopic fluctuationsbut have an opposite sign. For Al it seems that mesoscopiccorrections are larger than those coming from Eq. (30).

C. 2d chaotic cavities

In this section we study a 2d superconducting chaotic grainof area A, perimeter L, and linear size L =

-A. Our starting

point is the gap equation (3) together with the semiclassicalexpressions for the spectral density [Eqs. (7) and (8)] and thematrix elements, I (%,%%) [Eqs. (10)–(14)], namely,

I (%,%%) = &

A

71 + L

kF A

2C % + Si(4kF L)

)

3

+ L2)kF A

9Ci

24(% " %%)kF L

%F

3" Ci

22(% " %%)

%F

3:

++l

7% " %%

%F

88, (31)

where C % $ 0.339 . . . and Si(x),Ci(x) are the sine and cosineintegral functions, respectively. For l # L, the chaotic classi-cal dynamics leads to a universal form for the function +l(w):

+l(w) = 4k2F

sin(kF l-)-

. (32)

As in the 3d case, the energy dependence of matrix elementsimplies that the equations to be solved for the gap are of integraltype and that the gap itself is energy dependent. However,unlike the 3d case, we have logarithmic corrections comingfrom the contribution of the matrix elements. Based on theexpansion in powers of 1/kF L of the spectral density andI (%,%%) [see also Eqs. (A34) and (A44)] we propose for a 2dchaotic grain the expansion

"(%) = "0[1 + f (log) + f (1) + )"1f (2)(%)]. (33)

Following the same steps to solve the gap equation as in the3d case, we get to leading order:

&f (log) = L log 2kF L

2)kF A. (34)

Similar logarithmic corrections to residual interactions in 2dchaotic quantum dots in the Coulomb blockade regime werereported in Ref. 30.

The next order correction is given by

&f (1) = (C % ± 1)L

2kFA+ g(0), (35)

with (") for Dirichlet and (+) for Neumann boundaryconditions, respectively. The truncated spectral density g(0)

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BCS SUPERCONDUCTIVITY IN METALLIC NANOGRAINS: . . . PHYSICAL REVIEW B 83, 014510 (2011)

-4 -2 0 2 4$/!0

1

1.1

1.2

1.3!(

$)/!

0

FIG. 3. (Color online) Superconducting order parameter "(%),Eq. (33), in units of the bulk gap "0 for 2d chaotic Al grains(kF = 17.5 nm"1, ! = 7279/N, "0 $ 0.24 meV) as a functionof the energy % with respect to the Fermi level, % = 0. Differentcurves correspond to grain sizes (top to bottom) and boundaryconditions: L = 6 nm, kF L = 105, !/"0 = 0.77) (Dirichlet andNeumann boundary conditions), L = 8 nm, kF L = 140, !/"0 =0.32 (Dirichlet), and L = 10 nm, kF L = 175, !/"0 = 0.08 (Dirich-let). The leading contribution comes from the energy-dependentmatrix elements I (%,% %).

is defined as in the 3d case, with semiclassical amplitudescorresponding to 2d systems.

Finally, the energy-dependent correction to the gap in 2dchaotic grains f (2)(%) is given by the same function (29) as in3d grains.

We note that (1) in two dimensions the leading finite-sizecontribution comes from the interaction matrix elements, notfrom the spectral density, (2) finite-size effects are strongerthan in three dimensions and the leading correction does notvanish for any boundary condition, and (3) since effectivelythere are two expansion parameters 1/kF L 1 1 (ensuring thevalidity of the semiclassical approximation) and !/"0 < 1(in order to apply the BCS formalism) it can happen thatin a certain range of parameters the contribution f (2)(%) isdominant.

In Fig. 3 we plot the gap as a function of the energy in unitsof the bulk gap "0 for Al grains (kF $ 17.5 nm"1, & $ 0.18,and ! $ 7279/N meV, where N is the number of particles),of different sizes L. Note the single peak at the Fermi energy.For the smallest grains the leading contribution is f (2)(%).This is yet another indication that the matrix elements playa dominant role in the finite-size effects in superconductingmetallic grains.

IV. ENHANCEMENT OF SUPERCONDUCTIVITY INNANOGRAINS: IDEAL VERSUS REAL GRAINS

According to the findings of previous sections the super-conducting gap is an oscillating function of the system sizeand the number of electrons inside the grain. Even for grainswith N ! 104 " 105 electrons considerable deviations fromthe bulk limit are observed. For a fixed grain size, the deviationsfrom the bulk limit are larger the more symmetric the grainis. This is a typical shell effect similar to that found in otherfermionic systems, such as nuclei and atomic clusters.1 These

shell effects have their origin in the geometrical symmetriesof the grain. Symmetries induce degeneracies in the spectrumand, consequently, stronger fluctuations in the spectral density.The superconducting gap is enhanced if the Fermi energy is ina region of level bunching (large spectral density). Likewise,if the Fermi energy is close to a shell closure (small spectraldensity) the superconducting gap will be much smaller than inthe bulk limit.

Therefore, thanks to shell effects, one can adjust thegap value by adding or removing few electrons in such away that the Fermi energy moves into a region of high orlow spectral density. In fact, shell effects in metallic grainsof different geometries have recently attracted considerableattention.14,15,17–19,32,33 A superconducting spherical shell anda rectangular grain were studied numerically in Ref. 15, asimilar analysis was carried out in Ref. 14 for a nanowire,and a qualitative analysis of a spherical superconductor wasreported in Ref. 17.

Discrepancies with experiments are expected becausefactors such as decoherence, deformations of the shape of thegrain, and surface vibrational modes are not taken into accountin the theoretical analysis. In this section we discuss the impactof small deformations of the grain and of decoherence effectsthat shorten the coherence length. We will see that weaklydeformed grains can be modeled as symmetric ones but withan effective coherence length that incorporates the details ofthe deformation. The semiclassical formalism utilized in thispaper is especially suited to tackle this problem.

A. Superconductivity and shell effects

We study the dependence of the gap on the number ofelectrons N inside the grain and compare the gap between twograins with a slightly different degree of symmetry. We focuson 3d rectangular grains where deviations from the bulk resultsare expected to be larger. In this case the chemical potentialcan be computed exactly as a function of N by inverting therelation

12N =

" µ

((%)d%, (36)

where ((%) is the spectral density.As is shown in Fig. 1 matrix elements do not

affect the gap oscillations. Therefore we can solvethe gap equation (3) following the steps of Sec. III withthe spectral density given by Eqs. (A7), (A8), and (A10)and I $ 1/V . The spectral density depends on the cutoff,namely, on the number of periodic orbits taken into account.This cutoff is set by the single-particle coherence length l.Here we take l ! 12L where L, is the length of the longestside of the parallelepiped, and study the differences between acubic and a rectangular grain. The cutoff is chosen to be muchlarger than the system size in order to observe fluctuationsbut considerably smaller than the superconducting coherencelength # in order to accommodate other effects (discussedlater in this paper) that might reduce the typical single-particlecoherence length in realistic nanograins. We study a range ofN such that the BCS theory is still applicable but deviationsfrom the bulk limit are still important.

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ANTONIO M. GARCIA-GARCIA et al. PHYSICAL REVIEW B 83, 014510 (2011)

3.1#104

3.2#104

3.3#104

3.4#104

N

0.5

1

1.5!/

! 0

FIG. 4. (Color online) The superconducting gap " in units of"0 $ 2.286 meV, as a function of the particle number N for a cubic(circles), of side L, and a parallelepiped-shaped (1.0288 : 0.8909 :1.0911) (squares) grain. Fluctuations are on average stronger in thecubic grain due to its larger symmetry. The parameters utilized are& =0.3, %D = 32 meV, %F $ 11.85 eV, kF = 26 nm"1. The energy gapwas obtained by solving Eq. (16) with the semiclassical expressionof the spectral density given by by Eqs. (A7), (A8), and (A10) and asingle-particle coherence length l ! 12L.

In Fig. 4 we plot ", from Eq. (3), as a function of N for acube an a parallelepiped with aspect ratio 1.028:0.89:1.091.For both settings we observe strong fluctuations with respectto the bulk value. The fluctuations are clearly stronger in thecubic case since the grain symmetry is larger. We also observethat a slight modification of the grain size (or equivalentlyN ) can result in substantial changes of the gap. The observeddifferences between the cube and the parallelepiped are dueto the different symmetry of these grains. In the cube theoverall symmetry factor in the spectral density is 0N1/2.The parallelepiped has only two symmetry axis and thereforethe symmetry factor !N1/3.

In addition to the fluctuations due to periodic orbits, we alsoexpect smooth corrections to the bulk limit Sdue to the surfaceand perimeter term of the spectral density. These correctionswill be clearly observed as the coherence length is shortened,and the contribution of periodic orbits is therefore suppressed.

B. Finite-size effects in real small grains

Highly symmetric shapes are hard to produce in thelaboratory. It is thus natural to investigate to what extent smalldeformations from a perfect cubic shape weaken the finite-sizeeffects described in previous sections. For applications it isalso important to understand the dependence of the results onthe single-particle coherence length l. In order to study thisdependence, we assume that the superconducting coherencelength # is the largest length scale in the system. This is themost interesting region because in the opposite case l # # theresults for the gap (21) are to a great extent independent of l.By contrast, in the limit # # l, the cutoff (20) induced by #has little effect as the contribution of periodic orbits Lp # #is already strongly suppressed by the cutoff induced by l. Ifl ! # both cutoffs must be taken into account.

2.9#104

3.0#104

3.1#104

N

0.8

0.9

!/! 0

l = 10 Ll = 6 Ll = 2.25 L

FIG. 5. (Color online) Superconducting gap " for a cubic grain(volume N/181 nm3) for different single-particle coherence lengthsl = 2.25L, l = 6L, l = 10L in units of"0 $ 0.228meV as a functionof the number of particles N . The parameters utilized are & = 0.3,%D = 32 meV, %F $ 5.05 eV, kF = 18 nm"1. The energy gap wasobtained by solving Eq. (16) with the semiclassical expression ofthe spectral density given by Eq. (6). As the coherence length isreduced, less periodic orbits contribute to the spectral density, andfluctuations are smaller. Fluctuations are strongly suppressed forcoherence lengths l " 2L. In this limit the gap is still smaller than"0 as a consequence of the surface and curvature terms in Eq. (16).

We now address these two related issues. We note thatnot only the effect of a finite coherence length l but alsosmall deviations from symmetric shapes can be included inour analytical expressions for the gap by adding an additionalcutoff D [besides Eq. (20)], which suppresses the contributionof periodic orbits longer than D. The details of D dependstrongly on the source of decoherence or the type of weakdeformation. Indeed, in certain cases D may modify not onlythe amplitude but also the phase of the contribution of theperiodic orbit to the trace formula used to compute the spectraldensity. For instance, the effect of small multipolar correctionsto an otherwise spherical grain34 is modeled by adding anadditional D cutoff in term of a Fresnel integral that smoothlymodulates the amplitude and phase of the periodic orbits ofthe ideal spherical grain.

If the deformation is in the form of small, nonoverlappingbumps,35 the cutoff is exponential and affects only theamplitude. The numerical value of the cutoff depends on theoriginal grain and is directly related to the typical size ofthe bump. If the source of decoherence is due to finite temper-ature effects,36 D = Lp/l

sinh(Lp/l) with l inversely proportional tothe temperature.

In Fig. 5 we show the effect of a finite coherence length l inthe superconducting cubic grain investigated previously. Thegap equation (3) was solved exactly with the semiclassicalspectral density given by Eqs. (A7), (A12), and (A10) andI = 1/V . For simplicity we use D = Lp/l

sinh(Lp/l) as a cutoff withl now the single-particle coherence length. This is enough fora qualitative description of the suppression of shell effects asa consequence of decoherence or geometrical deformations.

The cutoff (20), related to the superconducting coherencelength, does not affect the calculations as it is much longer(!1600 nm) than the ones employed in Fig. 5. Similar resultsare obtained if the analytical result (21) is utilized.

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BCS SUPERCONDUCTIVITY IN METALLIC NANOGRAINS: . . . PHYSICAL REVIEW B 83, 014510 (2011)

As expected, the amplitude is reduced, and the fine structureof the fluctuations is washed out as the coherence length isshortened. We did not observe any gap oscillations with Nfor l # 2.5L. This can be regarded as an effective thresholdfor a future experimental verification of shell effects in super-conductivity. Smooth nonoscillatory corrections depending onthe S (or perimeter L in two dimentions) term in the spectraldensity are not affected by the coherence length and shouldbe clearly observed in experiments. Note that " in Fig. 5 is,on average, below "0 even for the maximum N investigated.This is a direct consequence of the negative sign of the surfaceterm in the spectral density for Dirichlet boundary conditionsused in the numerical calculations [f (1) in Eq. (17)].

V. CONCLUSIONS

We have determined the low-energy excitation spectrum,E =

%"(%)2 + %2, of small superconducting grains as a

function of their size and shape by combining the BCS meanfield approach and semiclassical techniques. For chaotic grainsthe nontrivial mesoscopic corrections to the interaction matrixelements make them energy dependent, which, in turn, leadsto a universal smooth energy dependence (29) of the orderparameter "(%); see Fig. 3. In the integrable (symmetric)case we found that small changes in the number of electronscan substantially modify the superconducting gap; see, e.g.,Fig. 4. Due to its potential relevance for experiments, we haveinvestigated how these shell effects decrease (Fig. 5) when thegrain symmetry and/or the single-particle coherence length isreduced.

ACKNOWLEDGMENTS

AMG thanks Jorge Dukelsky for fruitful conversationsand acknowledges financial support from FEDER and theSpanish DGI through Project No. FIS2007-62238. KR andJDU acknowledge useful conversations with Jens Siewert andfinancial support from the Deutsche Forschungsgemeinschaft(GRK 638). EAY’s research was in part supported by the Davidand Lucille Packard Foundation and by the National ScienceFoundation under Award No. NSF-DMR-0547769.

APPENDIX: SEMICLASSICAL APPROXIMATION FORTHE DENSITY OF STATES AND THE INTERACTION

MATRIX ELEMENTS

Semiclassical techniques such us periodic orbit theory1 arenot a common tool in the study of superconductivity; however,they are a key ingredient in our analytical treatment. In orderto solve the gap equation (3) we first need a closed expressionfor the spectral density and the interaction matrix elementsI (%,%%). In this Appendix we describe in detail how thesequantities are computed using a semiclassical approximationfor 1/kF L 1 1, where kF = k(%F ) =

-2m%F

his the momentum

at the Fermi energy %F and L is the linear system size. Theresulting semiclassical expansion will be organized in powers(possibly fractional) of the small parameter 1/kF L.

In order to observe deviations from the bulk limit, thesingle-particle coherence length must be larger than the systemsize, l # L. The time scale, . $ l/vF , associated with l has a

meaning of the lifetime of states near the Fermi energy. Thecondition l # L means that the Cooper pairs are composedof quasiparticles with a lifetime longer than the flight timethrough the system.

A. Density of states

We start with the analysis of the density of states. Thesemiclassical expression for ((%) for a given grain geometry isalready known in the literature:1

((%%) . (TF(0)[1 + g(%) + gl(%%)]. (A1)

The spectral density gets both monotonous g(%) and oscillatingg(%) corrections. The monotonous correction at the Fermienergy is given by the Weyl expansion:

g(0) =&

± S)4kF V

+ 2Ck2F V

3d,

± L2kF A

2d,(A2)

for Dirichlet (") or Neumann (+) boundary conditions. InEq. (A2), S is the surface area of the 3d cavity, C is its meancurvature, while L is the perimeter in the 2d case.

The oscillatory contribution to the density of states issensitive to the nature of the classical motion. For a systemwhose classical counterpart is fully chaotic it is given to theleading order by the Gutzwiller trace formula:29

gl(%%) = /

)*

+

2)k2F V

$lp Apei[kF Lp+*p]ei %%

2%FkF Lp 3d,

2kF A

$lp Apei[kF Lp+*p]ei %%

2%FkF Lp 2d,

(A3)

where we used k(%%) . kF + e%kF /2%F . The summation isover a set of classical periodic orbits (p) of lengths Lp < l.Only orbits shorter than the quantum coherence length lof the single-particle problem are included. The amplitudeAp increases with the degree of symmetry of the cavity1

(discussed later in this Appendix). In the chaotic caseAp = Ap(%F ) is given by

Ap(%F ) = Lp

|det(Mp " I)|1/2, (A4)

with the monodromy matrix Mp taking into account thelinearized classical dynamics around the periodic orbit. Theclassical flow also determines1 the topological index *p inEq. (A3).

Note that Eqs. (A3) and (A4) indicate that the scaling of gin terms of the small parameter

/ = 1/kF L (A5)

is

gl(%%) 09/ 2 3d,

/ 2d.(A6)

1. Rectangular grain

Consider a rectangular box of sides ai with i = 1, . . . ,d ind dimensions. For these systems the sum over periodic orbits

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ANTONIO M. GARCIA-GARCIA et al. PHYSICAL REVIEW B 83, 014510 (2011)

is exact and given by1,3

g(%%) =

)*

+g(3)(%%) " 1

2

$i

$j 2=i g

(2)i,j (%%) + 1

4

$i g

(1)i (%%) 3d,

g(2)1,2(%%) " 1

2

$i g

(1)i (%%) 2d.

(A7)

Here g(3) is a sum over families of periodic orbits. Each familyis parametrized by three (nonsimultaneously zero) integers)n = (n1,n2,n3):

g(3)(%%) =l!

L)n 2=0

j0

7kF L)n + e%

2%F

kF L)n

8, (A8)

where L)n = 2#

a21n

21 + a2

2n22 + a2

3n23 is the length of an orbit in

the family and j0(x) = sin x/x is the spherical Bessel function.We see that

g(3) 0 /. (A9)

In the same spirit, g(2)i,j is written as a sum over families of

periodic orbits parallel to the plane defined by sides ai,aj . Inthis case the families are labeled by two integers )n = (n1,n2)and

g(2)i,j (%%) =

& aiaj)

kF V

$lL)n2=0 J0

'kF L

i,j)n + e%

2%FkF L

i,j)n

(3d,

aiaj

A

$lL)n 2=0 J0

'kF L

i,j)n + e%

2%FkF L

i,j)n

(2d,

(A10)

where Li,j)n = 2

#a2

i n21 + a2

j n22 is the length of the orbit (n1,n2)

and J0 is a Bessel function. Using the asymptotic expressionfor J0, we find that this contribution scales with / as

g(2)i,j (%%) 0

&/ 3/2 3d,

/ 1/2 2d.(A11)

Finally, for g(1)i we have periodic orbits labeled by a single

integer n:

g(1)i (%%) =

& 4)ai

k2F V

$lLi

ncos

'kF Li

n + e%

2%FkF Li

n

(3d,

4ai

kF A

$n cos

'kF Li

n + e%

2%FkF Li

n

(2d,

(A12)

with lengths Lin = 2nai . The dependence on / in this case is

g(1)i (%%) 0

&/ 2 3d,

/ 2d.(A13)

It is important to note that depending on the classical dynamicsand the spatial dimensionality there are different types ofscaling with / . The amplitude of the spectral fluctuationsincreases with the degree of symmetry of the cavity. It ismaximal in spherical cavities and minimal in cavities withno symmetry axis.1 The latter typically includes chaoticcavities, namely, cavities such that the motion of the classicalcounterpart is chaotic.

This relation between symmetry and fluctuations can beunderstood as follows. In grains with one or several symmetryaxis there exist periodic orbits of the same length. As a result oftaking all these degenerate orbits into account, the amplitudeof the spectral density is enhanced by a factor /"1/2 for eachsymmetry axis.37,38 For instance, a spherical cavity has three

symmetry axes, so the symmetry factor is proportional to/"3/2 # 1. Periodic orbits in chaotic cavities are not in generaldegenerate, and the symmetry factor is therefore equal to one.For the range of sizes L ! 5–10 nm studied in this paper thedifference between a chaotic and an integrable grain can beorders of magnitude.

B. Interaction matrix elements

1. Semiclassical approximation to the average density

Unlike the case of the density of states, there is no generalsemiclassical theory for quantities, such as the interactionmatrix element I (%,%%), involving the spatial integration ofmore than two eigenfunctions in clean systems. For integrablesystems the ergodic condition

I (%,%%) = &

0(A14)

with 0 = V or A in three and two dimensions, respectively, istypically not met due to the existence of constants of motion.The constraints imposed by conservation laws effectivelylocalize the eigenfunctions in a smaller region of the availablephase space.

On the other hand, for chaotic systems Eq. (A14) is welljustified as a result of the quantum ergodicity theorem.39 Thevast majority of the eigenfunctions spread almost uniformlyover the whole volume (area) due to the lack of constants ofmotion besides the energy. If the position )r is far enough fromthe boundaries, we have

|'2n ()r)|2 = 1

0[1 + O(/ )] (A15)

for almost all states close to the Fermi energy. In order toevaluate explicitly deviations from Eq. (A14), we propose thereplacement

|'2n ()r)|2 4 *|'()r)|2+%n

. (A16)

The average is over a small window of states around %n.The width of this window is controlled by an energy scaleh/. related to the single-particle coherence length l $ vF . .This averaging procedure is justified since eigenfunctionsof classically chaotic systems have well-defined statisticalproperties.41

This average is exactly given by

*|'()r)|2+% = 1g(%)

!

%n

w(% " %n)|'en()r)|2

= 1)g(%)

"w(%%)6G()r,)r,%% " % + i0+) d%%,

(A17)

where G()r,)r %,z) is the Green function of the noninteractingsystem at complex energy z, w(x) is a normalized windowfunction of width h/. centered around x = 0, and g(%) is thedensity of states smoothed by w(x).

Next, we express the Green function as

G = G0 + G, (A18)

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where G0 is given by the free propagator

G0()r,)r %,% + i0+) =&

" m2)h2

eik(%)|)r")r% |

|)r")r %| 3d,

" im2h2 H

+0 [k(%)|)r " )r %|] 2d,

(A19)

and H+0 is the Hankel function. The corresponding contri-

bution to the average intensity, obtained by taking the limit)r 4 )r % of the imaginary part of G0 on Eq. (A19), is thenspatially uniform and given by

*|'()r)|2+0% = 1

0. (A20)

The effect of such so-called zero-length paths joining )r with)r in zero time is then to produce a constant backgroundindependent of the position (see, for example, Ref. 42). Thisresult should not come as a surprise, as zero-length paths areresponsible for the leading-order terms in the Weyl expansionof the density of states.

In the semiclassical approach29 the other part of the Greenfunction G is expressed in terms of nonzero paths , goingfrom )r to )r in a finite time ., as

G()r,)r,%) =!

,

D, e i(kF L, + %2%F

kF L, +*, ). (A21)

This contribution is responsible of the typical spatial oscil-lations of the average intensity. The classical properties ofeach trajectory are encoded in its topological phase *, (equalto)/4 times the number of conjugate points reached by the tra-jectory) and the smooth function D, = D, ()r,)r %,%F )|)r=)r % :1,29

D, ()r,)r %,%F ) =

).*

.+

1kF

;;det 12L, ()r,)r %)1qi1q

%j

;;1/2 3d,#

2)kF

;; 12L, ()r,)r %)1q1q %

;;1/2 2d.(A22)

Here qi and q %j are local coordinates transverse to the trajectory

, at points )r , respectively, and )r %, and L, ()r,)r %) is its length.In three dimensions we have two perpendicular components,while in two dimensions there is only one.

After substitution of Eq. (A21) into Eq. (A17), the inte-gration over energies can be carried out explicitly providedthat, consistently with the stationary phase approximationused to derive the semiclassical Green function, all smoothfunctions of the energy are evaluated at %F . The resultingFourier transform of the window function acts as a cutoff forthe sum. We finally obtain, after using the expression for thedensity of states and factorizing the Thomas-Fermi density:

*|'(r)|2+% = 10

1 + R()r,%)1 + g(%F ) + g(%)

. (A23)

In both three dimensions and two dimensions, R()r,%) is simplyobtained form the Green function as a sum over classical paths, ()r) = , starting and ending at point )r with finite lengthsL, ()r) = L, < l and actions S, ()r) = hk(%)L, :44

R()r,%) =l!

,

D, cos7

kF L, + %

2%F

kF L, + *,

8. (A24)

Inspection of Eq. (A22) shows that R scales as

D, 0&/ 3d

/ 1/2 2d. (A25)

Furthermore, the normalization condition implies

10

"R()r,%)d)r = g(%) + g(%). (A26)

Equation (A26) can also be used as the definition of the densityof states without the Thomas-Fermi contribution.

The separation between smooth, g(%) . g(0), and os-cillatory terms g(%) in Eq. (A26) is as follows. Smoothcontributions come from trajectories starting and endingat )r after hitting the boundary only once, L, < L. Onthe other hand, trajectories hitting the boundary more thanonce will have in general Lp > L, and their contribution tothe spatial integral can be evaluated using the stationary phaseapproximation to give g(%).

Using Eqs. (A2), (A6), (A25), and (A26) the interactionmatrix elements have the following semiclassical expansion:

I (%,%%) =

)*

+

&V

,1 + I (%F ,%,%%) " S2)2

16k2F V 2

-3d,

&A

,1 + I (%F ,%,%%) " L

2kF A+ gl(%F )

-2d,

(A27)

where

I (%,%%) = 10

"R()r,%)R()r,%%) d)r. (A28)

2. Evaluation of I(!,!!)

As we will see, I (%,%%) is a smooth function of both % and %%;it does not oscillate as rapidly as eiS/h where S is the classicalaction. The key point in carrying out the spatial integrationin Eq. (A28) is the separation of R = Rshort + Rlong into shortand long classical trajectories. A similar separation leads to thesmooth and oscillatory contributions to the density of statesdiscussed in the previous section. In other words, our approachto evaluating I (%,%%) is similar to the Weyl expansion for thedensity of states.

To calculate Rshort, we note that in the regime l # L, theshort trajectories of length L, < L are insensitive to thesmoothing, and hence their contribution to the imaginarypart of the Green function in (A17) can be pulled out fromthe energy integration. This means that Rshort is simplyproportional to the imaginary part of the Green functionassociated with the short paths.

Following Balian and Bloch37 the basic idea of thesubsequent calculation is that the boundary of the grain canbe locally approximated as a plane in three dimensions anda straight line in two provided that the observation point isclose enough to it. Within this approximation, the exact Greenfunction representing a single reflection off an infinite wallcan be calculated using the method of images. For the 3d caseEq. (A22) is quantum mechanically exact and gives the sameresult.

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ANTONIO M. GARCIA-GARCIA et al. PHYSICAL REVIEW B 83, 014510 (2011)

Following this idea, we construct the Green function for aninfinite straight boundary by means of the method of imagesto obtain

Gshort()r,)r %,% + i0+) = ±G0[)r,T ()r %),% + i0+], (A29)

where the action of the linear operator T is to map the position)r % into its image point on the other side of the boundary. Theplus and minus sign give Neumann and Dirichlet boundaryconditions, respectively.

It is easy to see that when )r 4 )r %, the function Gshort

depends only on the distance between )r and the boundary,which we denote by x. Since this distance is still of theorder of the system linear size, it is possible to perform theenergy average in Eq. (A17) with the Green function given byEq. (A29). As a result,

Rshort()r,%) = ±&

sin 2k(%)x2k(%)x 3d,

J0[2k(%)x] 2d.(A30)

After Rshort is inserted into Eq. (A28), the integral alongdirections parallel to the plane simply yields a factor of Sin three dimensions and L in two. The integration in theperpendicular direction is naturally truncated at the systemlinear size L.In three dimensions, using

4 L

0 =4 3

0 "4 3L

, weobtain

I short3d (%,%%) = " S

8k2F LV

+ )S4V

Min[k(%),k(%%)]k(%)k(%%)

,

which, as expected, is a smooth function of % and %%. The secondterm in this expression was previously obtained in Ref. 21 viaa slightly different method which misses the first term of theright-hand side of Eq. (A31).

A similar analysis in two dimensions is more subtle due todivergence of the integration in the direction perpendicularto the boundary. However, there is a natural upper limitfor this integration given by the linear system size L.Upon using k(%) . kF (1 + %/2%F ) and introducing the scaledperpendicular distance to the boundary y = 2kF x:

I short2d (%,%%)

= L2kF A

" 2kF L

0J0

271 + %

2%F

8y

3J0

271 + %%

2%F

8y

3dy.

(A31)

Employing the asymptotic expression for the Bessel functions,we find

I short2d (%,%%)

= L2kF A

2C + 1

)

" 2kF L

1

sin 2y + cos 2(% " %%)y/%F

ydy

3

(A32)

valid for kF L # 1. In Eq. (A32) the constant C =4 10 J 2

0 (y) dy . 0.850 . . . . We see that, contrary to the 3d case,I short

2d depends on energy (through the difference % " %%). Thisimplies that in 2d chaotic systems the superconducting gap isenergy dependent even to leading order in / .

The integrals in Eq. (A32) can be expressed in terms of thesine-integral (Si) and cosine-integral (Ci) functions. Our finalresult is

)...*

...+

I short3d (%F ) = )S

4kF V3d,

I short2d (%F ,% " %%) = L

kF A

,C % + Si(4kF L)

)

-2d,

+ L2)kF A

,Ci

' 4(%"%%)kF L%F

(" Ci( 2(%"%%)

%F)-,

(A33)

with C % = C " Si(2)/) = 0.339 . . . . Thus for fixed % and %%,I short scales with / = 1/kF L 1 1 as follows:

&I short

3d (%F ) 0 / + b/ 2 3d,

I short2d (%F ,% " %%) 0 / + b%/ log / 2d,

(A34)

where b and b% are constants independent of the system size.Note the nonalgebraic dependence on / in the 2d case. Theconstant b turns out to be much smaller than all other second-order contributions to the gap and will be dropped henceforth.

Now we focus on the contribution of long paths, Rlong, tothe spatial integral (A28). We use the expression for R as asum over classical closed paths , ()r) starting and ending at )rwith length L, ()r). Now we impose the condition

L, ()r) # L, (A35)

expressing the fact that the paths are long; namely, they hitthe boundary several times. As is standard in these cases, weevaluate the smooth functions D, in Rlong at the Fermi energyand expand k(%) . kF + k(%)2/2kF to get

I long(%,%%) = /" l!

, ,, %

D,D, %e i2+

, ,, % + e i2", ,, %

4d)r. (A36)

The phases involved in the spatial integration are (we do notinclude topological indexes for simplicity)

2±, ,, % (%,%%,)r) = kF (L, ± L, %) + kF

2%F

(L, % ± L, %%%), (A37)

where L, = L, ()r) is the length of the trajectory , .In chaotic systems different trajectories, in general, will

have lengths differing by at least L (see, however, Ref. 45).This means that since the first term in Eq. (A37) scales as1// # 1, the integral over r in 2+

, ,, % (%,%%,)r) and 2±, ,, (%,%%,)r)

can be evaluated by the stationary phase method. Within thisapproximation, oscillatory integrals of the form

"f (x)e i&h(x) dx

are given to leading order in 1/& by

"f (x)e i&h(x) dx . f (x7)

e i&h(x7)

-2)&h%%(x7)

,

where h%(x7) = 0, and therefore each spatial integration inEq. (A36) yields an extra factor 0 1// 1/2. Combining thiswith the prefactors (A25), we find that the contribution of

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BCS SUPERCONDUCTIVITY IN METALLIC NANOGRAINS: . . . PHYSICAL REVIEW B 83, 014510 (2011)

pairs , 2= , % (the so-called nondiagonal contribution) Ilongndg ) is

of order

Ilongndg (%,%%) 0

&/ 5/2 3d,

/ 2 2d.(A38)

On the other hand, terms that involve 2", ,, (%,%%,)r) do not

oscillate rapidly, because in this case the highly oscillatoryterms in the phase cancel each other, leaving the second termin Eq. (A37), which scales as / and not as 1// :

2", ,, (%,%%,)r) = kFL, ()r)

2%F(% " %%). (A39)

This contribution involves coherent double sums over classicaltrajectories and is usually referred to as the diagonal contribu-tion I

longdg . Taking , = , % in Eq. (A36), we easily find

Ilongdg (%F ,% " %%) =

" l!

,

D2, cos2"

, ,, (%,%%,)r) d)r, (A40)

which can be cast in a very compact form by introducing thepurely classical function

+l(w) =" l!

,

D2, cos wkF L, ()r) d)r (A41)

as follows:

Ilongdg (%F ,% " %%) =

& 1V+l

'%"%%

%F

(3d,

1A+l

'%"%%

%F

(2d.

(A42)

Keeping also in mind the / dependence of the coefficientsD, , we have

Ilongdg (%F ,% " %%) 0

&/ 2 3d,

/ 2d.(A43)

Equations (A33), (A38), and (A42) complete the evaluationof I . Restricting ourselves to the first two orders in / (/ and/ log / in the 2d case), we finally obtain

I (%,%%) =

)*

+

&V

,1 + I short

3d (%F ) " )2S2

16k2F V 2 + I

longdg (%F ,% " %%)

-3d,

&A

,1 + I short

2d (%F ,% " %%) + Ilongdg (%F ,% " %%)

-2d.

(A44)

Equations (A44) together with the definitions (A33) and(A42) allow for the calculation of interaction matrix elementsin 3d and 2d chaotic grains. In general, the explicit evaluationof +l(w) requires the precise knowledge of all classical pathsup to lengths of the order of the single-particle coherencelength l that have a crossing at )r for every point inside thecavity. However, if l is large enough compared to L (in practicel . 5L suffices), ergodic arguments can be invoked, and aclosed expression for the interaction matrix elements can befound. In situations when l . L one must carry out the explicitsystem-dependent calculation.

Classical ergodicity of chaotic systems can be formulatedin various ways,46 and we are going to give only a briefsketch of its consequences here. The main mechanism behinduniversality in the quantum mechanical description of classi-cally chaotic systems resides in the behavior of typical (in the

sense of measure theory) classical trajectories. By definition, atypical trajectory of a chaotic system will explore in an uniformway the available phase space, thus implying the equivalencebetween temporal and microcanonical averages.

This uniformity extends, in a nontrivial way, to the periodicorbits as well. The key concept here is the classical probabilityof return, defined as

P ()x0,t,e) = 1Z(e,t)

![)x0 " )x()x0,t)]![H ()x0) " e], (A45)

where )x0 = ()r0, )p0) is a point in phase space mapped at time tinto )x()x0,t) by the solution of the classical equations of motion.Clearly, the function ![)x0 " )x()x0,t)] is nonzero only when theclassical flow maps an initial point into itself after a time t andplays the role of a probability of classical return. Moreover,in case we want to select a fixed energy, we use an extracondition given by the value of the Hamiltonian function alongthe trajectory. Finally, the probability must be normalized suchthat

"P ()x0,t,e) d )x0 = 1, (A46)

thus fixing Z(t,e). The key observation here is that, bydefinition, the set of points where P ()x0,t,e) is different fromzero belongs to periodic orbits with period t . Although theoriginal ergodicity criteria were given in terms of typicaltrajectories, the theory of dynamical systems provides a strictlyequivalent definition of ergodicity in terms of periodic orbits:

P ()x0,t,e) 4 const. for t 4 3. (A47)

That means that not only typical trajectories but also periodicorbits uniformly fill the available phase space. We remarkthat the left-hand side of this equation, a set of delta peaksat the periods of the classical periodic orbits, must beunderstood in the sense of distributions; namely, both sides areassumed to be integrated over a smooth function of time andphase-space position.

In order to make contact with the coordinate representationused so far, we use the uniformity of periodic orbits in phasespace expressed by Eq. (A47) and integrate out the momentum.This integral can be exactly calculated.47 It involves a Jacobianof the form 1 )p(t)1()r0), which is indeed proportional to thesemiclassical prefactors D, . In summary, in the present contextclassical ergodicity leads to the following sum rule47 forclassical closed orbits:

l#L!

,

D2, ![l " L, ()r)] =

& 4)2

k2F V

3d,

4kF A

2d.(A48)

As was mentioned previously, integration over lengths up to lon both sides with a smooth weight function is also assumed.Using this result and noting that the right-hand side of Eq.(A48) is independent of the position )r , we get

+l#L(w) =

)*

+

4)2

k3F

sin wkF lw

3d,

4k2F

sin wkF lw

2d.(A49)

In the ergodic regime, l # L, these results enable us to evaluateexplicitly the energy dependence of the interaction matrixelements in chaotic cavities.

014510-13

ANTONIO M. GARCIA-GARCIA et al. PHYSICAL REVIEW B 83, 014510 (2011)

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