The Cooper pair dispersion relation

Physica C 295 Ž1998. 93–100

The Cooper pair dispersion relation M. Casas a , S. Fujita b, M. de Llano a

c,)

, A. Puente a , A. Rigo a , M.A. Solıs ´

d

Departament de Fısica, UniÕersitat de les Illes Balears, 07071 Palma de Mallorca, Spain ´ b Department of Physics, SUNY, Buffalo, NY 14260-1500, USA c Instituto de InÕestigaciones en Materiales, UNAM, 04510 Mexico DF, Mexico ´ ´ d Instituto de Fısica, UNAM, 01000 Mexico DF, Mexico ´ ´ ´ Received 26 August 1997; accepted 22 September 1997

Abstract The binding energy of a Cooper pair formed with the BCS model interaction potential is obtained numerically for all coupling in two and three dimensions for all non-zero center-of-mass momentum ŽCMM. of the pair. The pair breaks up for very small CMM, at most about four orders of magnitude smaller than the maximum CMM allowed by the BCS model interaction, and its binding energy is remarkably linear over the entire range of the CMM up to breakup. q 1998 Elsevier Science B.V. PACS: 03.65.G; 71.10.Li; 05.30.Fk Keywords: Phonon mechanism; Electron–phonon coupling; Pair breaking

1. Introduction Pairing of fermions in superconductivity theories is commonly considered to be only among partners of equal and opposite linear momenta, i.e., into pairs with zero net center-of-mass momentum ŽCMM.. We investigate non-zero CMM pairing, and in particular the dependence on the CMM of the pair binding energy, mentioned in Schrieffer’s w1x monograph on superconductivity as being linear in the long wavelength Žsmall CMM. limit, in contrast to the expected w2x quadratic behaviour of a particle moving freely in vacuum. In this paper we first review the derivation of the Cooper pair eigenvalue equation for the pair binding energy as a function of its CMM. Next, this equation is reduced in two and three dimensions to double-integral equations that are solvable analytically for small CMM and numerically for arbitrary CMM. Finally, Cooper pairs are distinguished and compared with several familiar ‘‘collective modes’’.

)

Corresponding author. Tel.: q52 11 525 6224631; fax: q52 11 525 6161251.

0921-4534r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 9 2 1 - 4 5 3 4 Ž 9 7 . 0 1 7 4 0 - 1

M. Casas et al.r Physica C 295 (1998) 93–100

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2. Non-zero center-of-mass momentum pairing Besides the rest energy, the total energy EK of a ŽCooper. pair w3x of fermions of momentum "k 1 and "k 2 and with center-of-mass momentum ŽCMM. "K interacting pairwise and immersed in a background of N y 2 inert spectator fermions in a spherical Fermi surface Žin k-space. of radius k F is obtained as follows. Let the relative and center-of-mass coordinates be r ' r 1 y r 2 and R ' 12 Ž r 1 q r 2 ., respectively. The corresponding wave vectors are related through k s 12 Ž k 1 y k 2 . .

K s k1 q k 2 ,

Ž 1.

Define the kinetic energy operators Tˆi s y

"2

"2

"2

2m

m

4 m)

= 2 ; Tˆ1 q Tˆ2 s Tˆr q TˆR s y ) i

=2 y ) r

=R2 ,

Ž 2.

with m ) the effective fermion mass. The eigenvalue equation for the total energy EK is then

ž Tˆ q Tˆ q Vˆ / C r

R

K

K s EKCK

,

Ž 3.

where

CK s c Ž r . F K Ž R . ,

Ž 4.

F K Ž R . s ei K P R

Ž 5.

and VˆKCK s

X

Hdr V

K

Ž r ,rX . c Ž rX . F K Ž R . ,

Ž 6.

with VK Ž r,rX . a general non-local pair interaction. Expanding c Ž r . in a complete set of plane-wave states,

c Ž r . s Ý C k e i kP r

Ž 7.

k

the Cooper problem consists in setting C k s 0 for all k 1 , k 2 - k F or < k " 12 K < - k F .

Ž 8.

It is this restriction that distinguishes Cooper’s from Schrodinger’s equation, which is for two particles in ¨ vacuum while the Cooper problem refers to two particles in a medium of N y 2 other fermions satisfying the Pauli Exclusion Principle. Combining Eqs. Ž2. – Ž8. gives

Ý k

ž

"2

"2

m

4 m)

k2q )

/

K 2 e i kP r q

X

X

Hdr V

K

Ž r ,rX . e i kP r y EK e i kP r C k e i K P R s 0.

Ž 9.

X

Putting e l ' " 2 l 2r2 m ) , multiplying Eq. Ž9. by eyi k P r , integrating over r, using Hd r expŽiQ P r . s LddQ,0 where Ld is the system volume in d dimensions, and cancelling the e i kP R , leaves

Ž 2 e k q 12 e K . Ck q Ý VkK, k Ck s EK Ck X

X

X

Ž 10 .

X

k

where VkK, k X '

1 Ld

X

X

yi k P r

Hdr Hdr e

X

VK Ž r ,rX . e i kP r .

Ž 11 .

M. Casas et al.r Physica C 295 (1998) 93–100

95

The BCS model interaction to be used here assumes that VkK, k X s

½

ž(

yVu 2 k F2 q k D2 y K

if k F - < k " 12 K < , < kX " 12 K < - k F2 q k D2

(

/

Ž 12 . 0 otherwise with V ) 0, and " v D ' " 2 k D2 r2 m ) the maximum energy of a vibrating ionic lattice phonon. The step function reflects the fact that the interaction operates only if 0 F K F 2 k F2 q k D2 . This means that two fermions interact with a constant attraction yV when the tip of their relative momentum wavevector k points anywhere inside the overlap volume in k-space of the two spherical shells in Fig. 1. Using Eq. Ž12., Eq. Ž10. simplifies to

(

X

Ž 2 e k y EK q " 2 K 2r4m ) . Ck s Vu ž 2(k F2 q k D2 y K / Ý Ck ' yA K X

k

Ž 13 .

X

with the prime on the summation sign denoting the restrictions over k in Eq. Ž12.. Solving for C k leads to yA K Ck s . Ž 14 . 2 e k y EK q " 2 K 2r4m )

(

Multiplying this by Vu Ž2 k F2 q k D2 y K . and summing over k, restricted as in Eq. Ž12., gives

ž(

1 s Vu 2 k F2 q k D2 y K

X

/ Ý Ž2 e y E k

K q"

2

K 2r4 m ) .

y1

.

Ž 15 .

k

Setting the total energy eigenvalue EK ' 2 E F y DK , the pair is bound if DK ) 0 and Eq. Ž15. becomes an eigenvalue equation for the pair Žpositive. binding energy DK . For K s 0 it becomes g Ž e . de X EFq" v D y1 1 s V Ý Ž 2 e k y 2 E F q D0 . s V Ž 16 . 2 e y 2 E F q D0 EF k

H

from which one immediately obtains the familiar result 2"vD D0 s 2r l ™ 2 " v D ey2r l Ž 17 . e y 1 l™0 for the K s 0 pair binding energy, where l ' g Ž EF .V is a dimensionless coupling constant and g Ž E F . is the density of Žfermionic. states for each spin evaluated at the Fermi surface. The equality in Eq. Ž17. is exact in 2D for all coupling—as well as in 1D or 3D provided that " v D < E F so that g Ž e . f g Ž EF ., a constant that can be taken outside the integral in Eq. Ž16.. Note that since Eq. Ž17. yields only D0 F 11 K Žfor l F 1r2 and Q D f 300 K. compared with the total rest mass energy of two electrons which is ; 10 10 K, a Cooper pair is very weakly bound indeed when compared, say, with the deuteron for which these two energies are, respectively, about 2 and 2000 MeV, or with the pi-meson as made up of a ‘‘down’’ and ‘‘up’’ quark where the two energies are, respectively 560 and 700 MeV.

Fig. 1. Cross-section of overlap volume in k-space Žshading. where the tip of the relative wave vector k must point for the attractive BCS model interaction Eq. Ž12. to be non-zero, for a Cooper pair of CMM "K.

M. Casas et al.r Physica C 295 (1998) 93–100

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Fig. 2. Exact 2D Cooper pair dispersion relation calculated numerically from Eq. Ž18. for l s 0.5 and n s10y2 , compared with its linear approximation Eq. Ž19..

For a two-dimensional system Eq. Ž15. reduces Žsee Appendix A. to 1s

4l

pr2

H p 0

2

df

2

1r2

1r2

w1q n y k Ž1q n .sin f x y k Ž1q n . cos f d j j Ž D˜ q 2 Ž 1 q n . k Hw1y k Ž1q n .sin f x q k Ž1q n . cos f 2

2

1r2

k

1r2

2

y2q2j 2 .

y1

,

Ž 18 .

where g Ž EF . ' L2 m ) r2 p " 2 the 2D density of states; j ' krk F ; k ' Kr2Ž k F2 q k D2 .1r2 ; D˜k ' DK rEF ; and n ' " v D rEF s k D2 rk F2 . For small K, one obtains analytically from Eq. Ž18.

D K ™ D0 y K™0

Ž1qn .

2

1r2

q e 2r l

e 2r l y 1

p

"Õ F K q O Ž K 2 .

Ž 19 .

which for weak coupling l ™ 0 reduces to

D K ™ D0 y K™0

2 p

"Õ F K q O Ž K 2 . .

Ž 20 .

Fig. 2 compares DK in the linear approximation Eq. Ž19. to the exact dispersion relation obtained numerically from Eq. Ž18., for l s 0.5 and n s 10y2 . Indeed, the linear approximation is very good for moderately small l and n over the entire range of K values for which DK G 0. Pair breakup, specifically DK - 0, occurs at a relative small value of K, about four orders of magnitude smaller than the maximum value 2 k F2 q k D2 allowed by the interaction Eq. Ž12. for the values of l and n exhibited in the figures. In three dimensions, assuming n < 1 and the 3D density of states g Ž ´ . s Ž L3rp 2 " 3 . m ) 3´r2 f g Ž EF ., Eq. Ž15. becomes Žsee Appendix A.

(

(

pr2

1s2l

H0

2

2

1r2

1r2

w1q n y k Ž1q n .sin f x y k Ž1q n . cos f dj j Hw1y k Ž1q n .sin f x q k Ž1q n . cos f

d f sin f

2

2

1r2

1r2

2

D˜k q 2 Ž 1 q n . k 2 y 2 q 2 j 2

y1

Ž 21 .

which for small K, assuming from Eq. Ž17. the weak coupling expression D˜0 f 2 n ey2 r l which is very accurate for the typical range of l and n values Žsee Fig. 4 below., gives

DK ™ D0 y e 4r l K™0

'1 y n ey2 r l Ž 2ey2r l q n ey2r l q 1. e 2r ln  ln A q ln B 4 q n ln Ž AB . q 2e 2r l '1 y n ey2r l 1

=

Že

2r l

q 1 y '1 q n .

"Õ F K q O Ž K 2 . ,

Ž 22 .

M. Casas et al.r Physica C 295 (1998) 93–100

97

Fig. 3. Exact 3D Cooper pair dispersion relation calculated numerically from Eq. Ž21. for l s 0.5 and n s10y2 , compared with its linear approximation Eq. Ž22..

where A ' Ž'1 q n y '1 y n e y 2r l .r Ž'1 q n q '1 y n e y 2r l ., and B ' Ž1 q '1 y n e y 2r l .r Ž1 y '1 y n ey2 r l .. For n < 1, l ™ 0 and K ™ 0, Eq. Ž22. reduces to the result cited in Ref. w1x, namely

DK ™ D0 y 12 "Õ F K q O Ž K 2 . . K™0

Ž 23 .

Results in 3D are qualitatively similar to those in 2D and are illustrated in Fig. 3. In Fig. 4 are compared the weak coupling asymptotic Cooper pair binding energy in Eq. Ž17. to the exact value D0 for a range of n values and for several l values. In 2D, D0 is given exactly by the equality in Eq. Ž17., while for 3D it was obtained numerically by solving Eq. Ž21., with k s 0. The two are seen to agree very well for small l for the relevant empirical values of n f 10y3 in 3D, at least for conventional Želemental. superconductors, and for all n in 2D.

3. Collective modes in a superconductor Collective modes in a superconductor have indeed been discussed since the late 1950s by Nambu, Anderson, Rickayzen, and Bardasis and Schrieffer. A review of the early work by Martin is available w6x, as is a more

Fig. 4. Fractional deviation of the weak coupling asymptotic value 2 " v D ey2 r l in Eq. Ž17. of the zero-CMM Cooper pair binding energy, to its exact value D0 , as function of n f Q D r TF for three values of l, in both 2D Žthin straight lines. and 3D Žthick curves.. In 2D the exact D0 is given by the equality in Eq. Ž17. for any n . In 3D D0 must be obtained numerically from Eq. Ž21. with k s 0. For 2D-like cuprate superconductors n is empirically w4x 0.03 to 0.07; for 3D-like conventional Želemental. superconductors n f10y3 .

M. Casas et al.r Physica C 295 (1998) 93–100

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Fig. 5. Dispersion curves for: the plasmon Eq. Ž24. Ždashed.; the weak coupling Žrepulsive interaction. zero-sound phonon Eq. Ž25. Ždot-dashed.; the weak coupling Žattractive interaction. Anderson mode Eq. Ž28. Ždotted.; and the weak coupling 3D Cooper pair dispersion D0 y DK from Eq. Ž23. Žfull curve..

recent treatment by Belkhir and Randeria w7x. However, we do not deal here with ‘‘collective modes’’ but rather with Žnon-zero center-of-mass. ‘‘Cooper pairs’’ which can Bose–Einstein condense while collective modes cannot. Cooper pairs are entities distinct from collective modes such as zero-sound phonons or plasmons since they: Ža. are bounded in number Žbefore the thermodynamic limit is taken., and Žb. are fixed in number as they carry a fixed constituent fermion number Žnamely two., while phonons or plasmons, say, do not share either property. Fig. 5 compares and contrasts them in the long wavelength limit Ž K ™ 0.. The dashed quadratic curve is the plasmon dispersion relation Žsee Ref. w5x, p. 180., for an electron gas 2

v K s v P 1 q 109 Ž KrK TF . q . . .

Ž 24. .xy1 r3

2

2.

in the ‘‘ring ŽRPA. approximation’’ valid for rs ' r 0ra0 ' wŽ4 pr3. n rŽ " rme < 1, where r 0 is an average electron spacing, n ' k F3 r3p 2 being the electron number density, a 0 the first Bohr radius " 2rme 2 with m the electron mass, while the plasmon frequency is v P ' 4 p ne 2rm and the ‘‘Thomas–Fermi inverse screening length’’ is K TF ' 6 p ne 2rEF with EF ' " 2 k F2 r2 m as before. The dot-dashed curve is the weak coupling zero-sound phonon dispersion ŽRef. w5x, p. 183., curve for repulsive interactions between fermions at T s 0, and is given by

(

(

v K , w 1 q 2 exp  y 2 p 2 " 2rmk F n Ž 0 . q 2 4 Õ F K 2

3

Ž 25 .

yi qP r

for n Ž0. < " rmk F , where n Ž q . ' Hd r e V Ž r . and V Ž r . the Žrepulsive. interparticle interaction potential. The slope of this straight line rises as coupling is increased, and assumes the form

v K , n Ž 0 . r3p 2 Ž " 2rmk F .

1r2

ÕF K

Ž 26 .

2

for n Ž0. 4 " rmk F . Note that Eq. Ž26. can be rewritten as

v K2 ,

n Ž 0. 3p

2

Ž " 2rmk F .

Õ F2 K 2

Ž 27 .

and becomes the plasmon frequency squared v P2 if n Ž K . is taken as the Fourier integral of the Coulomb interaction, 4 p e 2rK 2 . On the other hand, for attractive interactions V - 0 between the fermions one has the so-called ‘‘Anderson mode’’ w6x, 1 v K , 1 y 4 g Ž EF . < V < Ž 28 . '3 Õ F K in the weak coupling limit, which is shown as the dotted curve in Fig. 5. Finally, the weak coupling Cooper pair dispersion relation D0 y DK from Eq. Ž23. is represented by the full curve. In 2D, the Anderson mode Eq. Ž28.

M. Casas et al.r Physica C 295 (1998) 93–100

99

carries w7x a factor 1r '2 instead of the 1r '3 of 3D; it thus also lies higher than the 2D Cooper pair dispersion relation Eq. Ž20. since 1r '2 ) 2rp. Of possible interest is that the quadratic correction term in Eq. Ž19. just after the linear term diverges exponentially like e 2r l in the limit of zero coupling, as was verified by computer-algebra expansion of DK in powers of K in 2D. This suggests that Eq. Ž19. does not possess a power expansion in K beyond and including the K 2 term for all coupling. An analogous divergence, though much weaker, also occurs with the plasmon dispersion relation Eq. Ž24., rewritten explicitly as

ž

vK s v P 1 q

"2

9p

40 m ) e 2 k F

K 2q...

/

Ž 29.

in the limit of weak coupling e 2 ™ 0.

4. Conclusions The Cooper pair binding energy for the BCS model interaction acting pairwise between fermions decreases almost linearly to zero with the pair center-of-mass momentum ŽCMM. as was determined numerically for all coupling from the Cooper eigenvalue equation in both two and three dimensions. This linear behaviour contrasts with the quadratic dispersion relation of a bound composite particle moving in vacuum, e.g., a deuteron.

Acknowledgements M.C. and M. de Ll. acknowledge partial support from grant PB95-0492 and SAB95-0312 ŽSpain.. M. de Ll. and M.A. S. acknowledge partial support from UNAM-DGAPA-PAPIIT-IN103894 ŽMexico ´ .. M. de Ll. thanks R.M. Carter, N.J. Davidson, D.M. Eagles and O. Rojo for extensive discussions andror correspondence, and NATO ŽBelgium. for a research grant.

Appendix A We sketch the derivation of working Eq. Ž18. in 2D and Eq. Ž21. in 3D, from the Cooper pair eigenvalue Eq. Ž15.. Since EK ' 2 E F y DK , the latter equation can be written in d-dimensions as 1sV

L

ž / 2p

d

dk

X

H " Žk 2

2

y k F2

. rm

)

q DK q " 2 K 2r4m )

Ž A.1 .

where the prime on the integral sign again denotes the restrictions in Eq. Ž12.. These restrictions are identical in two and three dimensions, and can be written as k F2 - < k y 12 K < 2 s k 2 y kK cos f q 14 K 2 , 1 2

2

Ž A.2 . 1 4

k F2 q k D2 ) < k q K < s k 2 q kK cos f q K 2

Ž A.3 .

with f the angle between k and K, since it can be limited to values between 0 and pr2 because of symmetry. Manipulating Eqs. ŽA.2. and ŽA.3. we obtain k 2 y kK cos f q 14 K 2 y k F2 ) 0,

Ž A.4 .

k 2 q kK cos f q 14 K 2 y k F2 y k D2 - 0.

Ž A.5 .

M. Casas et al.r Physica C 295 (1998) 93–100

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These conditions can be studied separately but must be satisfied simultaneously. The left side of Eq. ŽA.5. is an increasing parabola for k G 0, and so is equivalent to k - k max '

Ž k F2 q k D2 . y 14 K 2 sin2f

1r2

y 12 K cos f ,

Ž A.6 .

where k max is found by requiring the left side of Eq. ŽA.5. to be zero and choosing the positive solution. The left side of Eq. ŽA.4. corresponds to a parabola with a minimum at k s 12 K cos f . For this value of k the left hand side of Eq. ŽA.4. becomes 14 K 2 sin2f y k F2 . If k F2 - 14 K 2 sin2f Eq. ŽA.4. is satisfied for all k G 0; otherwise we must consider the range of k values making the parabola positive. This parabola intersects with the x-axis at the points k 0 ' y Ž k F2 y 14 K 2 sin2f . k min ' Ž k F2 y 14 K 2 sin2f .

1r2

1r2

q 12 K cos f ,

Ž A.7 .

q 12 K cos f . 2

Ž A.8 . r4 y k F2 .

k F2 ) K 2r4

The intersection point with the y-axis is K So if the restriction Eq. ŽA.4. becomes k ) k min ; otherwise k ) k min or k - k 0 . Thus, Eq. ŽA.4. can be rewritten as 0-k-`

if

k F2 - 14 K 2 sin2f ,

k ) k min

if

k F2 ) 14 K 2 sin2f ,

if

k F2 ) 14 K 2 sin2f ,

k ) k min or k - k 0

and

k F2 ) 14 K 2 ,

and

k F2 - 14 K 2 .

Ž A.9 .

Taking into account that pairs break up for very small CMM ŽFigs. 2 and 3., we consider only the restriction k ) k min , which together with Eq. ŽA.6. gives k min - k - k max ,

0 - f - pr2.

Ž A.10 .

Note that for larger values of K there exists a minimum value of f below which k min ) k max . Using the dimensionless variables defined below Eq. Ž18., the d s 2 and d s 3 expressions for g Ž EF . given in the text, and the restrictions Eq. ŽA.10., one finally obtains Eq. Ž18. for d s 2 and Eq. Ž21. for d s 3.

References w1x w2x w3x w4x w5x w6x w7x

J.R. Schrieffer, Theory of Superconductivity, Benjamin, New York, 1964, p. 33. J.M. Blatt, Theory of Superconductivity, Academic Press, New York, 1964, and references therein. L.N. Cooper, Phys. Rev. 104 Ž1956. 1189. D.R. Harshman, A.P. Mills, Phys. Rev. B 45 Ž1992. 10684. A.L. Fetter, J.D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, New York, 1971. P.C. Martin, in: R.D. Parks ŽEd.., Superconductivity, vol. 1, Marcel Dekker, New York, 1969. L. Belkhir, M. Randeria, Phys. Rev. B 49 Ž1994. 6829.