V ol . 9 9 ( 1901)
A CT A P HY SIC A P O LO NIC A A
No . 6
E V OL U T I ON FR OM T HE B CS
TO TH E BOSE {E I N STE I N LI M I T
IN A
d -WAV E
SUP ER CON D U CTOR A T
T = 0
L .S. B o r ko w sk i a ; b an d C .A .R. SÇ
a d e Me l o b
a
Instit ut e of Physi cs, A . Mic kiew icz U niversity
U mu ltow ska 85, 61-614 Pozna ¥, Polan d
b
School of Physics, Geor gia Institute of T echnology , A tlanta, Georgia 30332, U SA
( Recei ved J anua r y 18, 2001 ; i n Ùnal for m May 7, 2001)
W e study the evolution from BC S to Bose limit in a two- dimensional
superconductor at zero temp erature and low density of charge carriers w ithin the mean- Ùeld theory . W e examine single quasipa rticl e prop erties
w hen particle density and attraction strength are v aried. F or su£ciently high
interaction strength there is a critical density b elow w hich the system has a
gap. T he spectral and thermo dynami c prop erties of the system do not evolve
smo othly from the BC S-like to the Bose- like regime.
PACS numb ers: 74. 20. {z, 74. 25. Gz, 67. 40.Db
d -w ave
1. I n t r o d u ct io n
T h e pro bl em of the evol ut i on fro m B CS to Bo se superconducti vi ty i s an ol d
one [1, 2] but recentl y i t ha s received considera bl e attenti on i n connecti o n wi th
hi gh tem p erature superconduct ors [3{ 12]. W hi l e the e˜ect of d - wa ve pa iri ng on the
op eni ng of a pseudogap abo ve T c was di scussed i n the l itera ture, there wa s a lack
of detai l ed studi es of the gro und sta te pro p erti es i n the i nterm ediate reg i me. It i s
wel l kno wn tha t the s - wave system exhi bi ts a sm ooth crossover b etween the weak
and stro ng coupl i ng reg i mes. Ho wever, pa irs wi th non- s -wa ve sym metry canno t
contra ct i n real space to poi nt b osons due to Ùnite ang ul ar m omentum of the pai rs.
Thus one m ay exp ect d -wa ve system s to b ehave i n a qua l ita ti vely di ˜erent way
fro m thei r s - wave counterpa rts as the bo soni c l i mi t i s appro ached. Here we di scuss
the singl e qua sipa rti cl e pro p erti es (exci ta ti on spectrum , m om entum di stri buti on,
and density of sta tes) as a functi on of attra cti on streng th o r parti cle density .
The weak coupl i ng (BCS) l i m it is chara cteri zed by a p ositi ve chem i cal potenti al ñ = ¯ F and a l arg e size of Co op er pai rs ( ¿ p a i r ƒ k F 1 ) , whi l e the stro ng
coupl ing (Bo se) regi m e is chara cteri zed by a l arg e and negati ve chemi cal p otenti al
À
‘
‘
, where E b i s the bi ndi ng energy of the two -b ody pro bl em i n the ‘ -th
ang ul ar m omentum channel , and by a sm al l size of pa i rs ( ¿pa ir § k F 1 ) .
ñ =
À
Eb
À
(6 91)
692
L .S. Bor kowski , C .A .R. SÇ
a de Mel o
2. T he m odel
W e start wi th the two - dim ensional Ham i l toni an
X
X
H =
¯kê
y
k¥
ê kk ¥ +
k¥
where
b
gul ar
= ê
y
b
(1 )
;
0
=
À
ê
#
=
"
. The i ntera cti on p otenti al
comp onents as
0
arccos( ^
=
‘
b
k k qq
mom entum
¢
V
^ )
1
i s the
V
1
=
‘
À1
angl e betwen the
V
i s expanded i n an-
V
‘
exp ( i ‘ ¢
vecto rs
),
where
and
0
and
d r r J ‘ ( k r ) J ‘ ( k r ) V ( r ) . The i ndex ‘ l abel s ang ul ar mom entum states
i n two spati al di mensions, wi th ‘ = 0 ; 1 ; 2 ; . . . corresp ondi ng to s; p; d ; . . .
channel s respecti vel y. A p ossibl e choi ce of the real space p otenti al i s V (r ) =
V È (R
r)
V È (r
R )È (R
r ) , whi ch i s repul sive at short di sta nces r < R ,
attra cti ve at i nterm edi ate di sta nces R < r < R , and vani shes f or r > R .
G eneral l y, i t i s not p ossibl e to Ùnd a separa bl e p otenti al i n mom entum space
V
=
Ñ w ( ) w ( ) , neverthel ess i n the spiri t of Ref. [2] we choose to study a
separa bl e potenti al tha t conta i ns most of the general features descri b ed abo ve. W e
consider o nl y sing l et superconducti vi ty , where the s -wa ve and the d - wave channel s
are studi ed separa tel y. We use p otenti al of the form V
=
Ñ ‘ w ‘ ( ) w ‘ ( ) . The
i ntera cti on term w ‘ ( ) can b e wri tten as a pro duct of two functi ons, w ‘ ( ) =
=
h ‘ ( k ) g ‘ ( ^ ) , where h ‘ ( k ) = (k = k ) ‘ = [ 1 + ( k = k ) ] ‘
contro l s the ra ng e of the
V
= 2¤
0
0
Ê
0
i ntera cti on and g ‘ ( ^ ) = cos( ‘¢ ) is the ang ul ar dependence of the i ntera cti on. Here
k
R
and k sets the scale at l ow mom enta . We assume tha t pai ri ng at T = 0
o ccurs wi th the same to tal m omentum = 0 onl y. Thi s sim pl i Ùcati on l eads to the
fol l owi ng saddl e p oi nt and numb er equati ons:
À
1
w‘ ( )
=
Ñ‘
2E ‘ (
n = 2
(2 )
;
)
(3 )
n‘ ( );
where
n‘ ( ) =
[1
[( ¯
ñ)
‘(
ñ ) = E ‘ ( )] = 2 i s the m omentum di stri buti on, E ‘ ( ) =
i s the sing le parti cle exci ta ti on energy, and  ‘ ( ) =
 ‘ w ‘ ( ) i s the o rder para m eter. For a gi ven i ntera cti on ra nge R
k
, the
tra nsiti on fro m the BCS li m it (l arg el y o verl appi ng pa i rs) to the Bo se l i m it of
(wea kl y overl appi ng pa irs) may o ccur either by chang i ng the attra cti on streng th
Ñ ‘ or the density n . In either case, thi s evol uti on can b e safely anal yzed wi th the
appro xi m atio ns used here pro vi ded tha t the system i s di l ute enough, i .e., n
k .
Thi s m eans tha t b elow a m axi mum density n
k
, the i ntera cti on ra nge R
i s much smal l er tha n the interpa rti cle spaci ng k
,R
k
, or equi val entl y
k =k
1. Thus we choose to scal e al l energ ies wi th respect to the maxi mum
Ferm i energy ¯
, whi ch Ùxes the m axi mum density n = n
= 2 £¯
, and
al l m omenta wi th respect to k
=
2 m¯
. The coupl i ng consta nt i s scal ed
wi th respect to the two- di mensi ona l density of sta tes £ . Fro m no w on we use thi s
scali ng .
+
Â
(¯
)
]
=
À
À
À
Evol uti on from t he BC S t o t he Bose{Ei nst ei n Li mi t
.. .
693
3. R esu l t s
Num eri cal sol uti ons for  0 ‘ and ñ , when k 1 = k 0 = 1 0 are shown i n Fi g . 1
for Ùxed density n = 1 , a nd changi ng Ñ‘ . Sim ila r pl ots can also b e m ade for Ùxed
i ntera cti on and varyi ng density n . In the weak coupl i ng l i mit the am pl itude of the
order pa ra m eter ( ¢ = 0 ) i s gi ven by
Â
‘
(k
ñ
)
¿
exp f
À
1
2[Ñ0‘ ( k ñ )
À
Ñ‘
À
1
2
]=h ‘ ( k
ñ
)g :
W i th our choi ce of h ‘ ( k ); Ñ 0 d ( k ñ ) ' 8 + ñ= 2 4 ¯ 1 + O [ ( ñ= ¯ ) ] , val id for ñ= ¯ § 1 ,
where ¯ = k . The ra ti os b etween  ‘ ( k ñ ) and the cri ti cal tem p erature T ‘ sati sfy
the usual rel ati ons  s ( k ñ ) = T s = 1 : 7 6 , and  d ( k ñ )= T d = 2 : 1 4 . The para m eters
 d and ñ ha ve conti nuo us Ùrst deri vati ves and di sconti nuo us second deri vati ves
as a functi on of Ñ d . Thi s b eha vi or al ways occurs when ñ = 0 in b oth  d a nd ñ ,
for varyi ng i ntera cti on Ñ d (see Fi g. 1) or varyi ng density n .
Â
Â
694
L. S. Bor kowski , C .A .R. SÇ
a de Mel o
W e Ùrst l ook at the singl e qua sipa rti cle exci ta ti on spectrum E d ( k ) . For ñ > 0 ,
i ncl udi ng the BCS l im i t, the excita ti on spectrum i s gapl ess at k ñ al ong the special
di recti ons ¢ = Ï ¤ = 4 ; Ï 3 ¤ = 4 , near whi ch the excita ti on spectrum di sperses l i nearl y
wi th mom entum . The energy gap at k = k ñ and ¢ = 0 ; E g (k ñ ) = j  d ( k ñ ) j i s a
no nm ono to ni c functi on of k ñ for Ùxed density , and thus a no nm ono toni c functi on
of Ñ d . The ma xi mum E g (k ñ ) i s reached at interm edi ate values o f ñ > 0 . At ñ = 0 ,
the m i nimum gap i s E g (0 ) = j  d (0 ) j = 0 , and occurs at the sing l e poi nt k = 0 .
In thi s case the exci ta ti on spectrum i s E d ( k ) = ( ¯ 2k + j  d ( k ) j 2 ) 1 = , whi ch b ehaves
qua dra ti cal ly for sm al l mom enta at any gi ven ang le ¢ , since  d ( ) ¿ k cos(2 ¢ )
and ¯ = k = 2 m . The shri nki ng of the energy gap to zero at = 0 i s a consequence
of the di m i nishi ng pai ri ng i ntera cti on h d ( k ñ ) for ñ ! 0 . As soon as ñ < 0 , i ncl udi ng
the Bo se l i m it, a f ul l gap i n the exci ta ti o n spectrum app ears, but the mi ni mal gap
rem ai ns at = 0 , E g ( 0) = j ñ j , since  d (0 ) = 0 , see Fi g. 2 [3].
Fi gure 3 shows the l i nes where ñ = 0 o n the gra ph of n vs. Ñ ‘ . The l ow
density l i mit of the s -wa ve system i s al ways Bo se-li ke, i .e., a tw o-b ody b ound
sta te app ears at arbi tra ri l y smal l Ñ s . The d -wa ve system i s qual i ta ti vely di ˜erent:
i t i s BCS- li ke for Ñ d < Ñ cd and Bo se-l i ke for Ñ d > Ñc d , where the cri ti cal coupl i ng
Ñ c separa ti ng the two regi mes i s Ùnite, i .e., the app eara nce of a two -b ody b ound
sta te i n the d -wa ve case requi res Ùnite Ñ d .
Let us bri eÛy di scuss the b ehavi or of the mom entum di stri buti on at l ow k
for three di ˜erent regi m es: ñ > 0 ; ñ = 0 , and ñ < 0 . For p ositi ve ñ the m om entum di stri buti on i s n s ( k ñ + £k ) ' [ 1 À 2 k ñ £k = Â s ( k ñ )] = 2 near k ñ . At l ow k ,
ho wever,
˜
= Â
n s(k )
s =( ñ
+ Â
ati ve ñ; n s ( k ) =
Ob vi ously, n s ( k )
'
˜ k = 2 k )] = 2 , where Ûp =
and sma ll k ; n s ( k ) ' (1 À
[ 1 À Ûn (1 + ˜ k = 2 k )] = 2 for smal l k , wi th
i s a conti nuous functi on of ñ for al l k .
[1 +
s).
For
Ûp (1 +
ñ =
0
ñ=
ñ
k =Â
Ûn =
s
j
s , and
For negñ + Â s.
+ Â
) =2.
ñj=
Evol uti on from t he BC S t o t he Bose{Ei nst ei n Li mi t
Fig. 3.
T he line
ñ =
0
for b oth
s
- and
d - wave
order parameters for
k
1
695
.. .
=
k
0
= 10
.
The
m omentum di stri buti on in the d -wa ve case ha s the f orm
sgn ( k 2 À ñ )] al ong the di recti on of the no des ( ¢ = Ï ¤ = 4 ; Ï 3 ¤ = 4 ).
Near k ñ we ha ve n d ( k ñ + £k ) ' [ 1 À 2 k ñ £ k = Â d (k ñ )] = 2 , for k cl ose to k ñ , and
n d (k) '
1 À (Â 20 d = ñ )( k = 4 k ) for smal l k . W hen ñ vani shes, at k = 0 i s
=
n d (0 ) ' (1 À ç ) = 2 , where ç = (1 + Â d = k )
. Fi nal l y, when ñ b ecom es negati ve,
n d ( k ) ' ( Â d = ñ )( k = 4 k ) for smal l k . The di sconti n ui t y of n d ( k ) at ñ = 0 and
l ow k , see Fi g. 4, coi nci des wi th the col l apse o f the four D i ra c p oi nts to a sing le
p oi nt at k ñ = 0 , and wi th the app eara nce of a f ul l gap as soon as ñ < 0 . Sim i lar
b ehavi or of n d ( k ) was al so fo und recentl y i n R ef. [13, 14] for a l atti ce mo del wi th
attra cti ve i ntera cti on of nearest- nei ghb o r parti cles.
The qua l i ta ti ve chang es i n E ‘ ( ) and n ‘ ( ) , as a functi on o f ñ , a˜ect substa nti al l y the qua siparti cle density of sta tes N ‘ ( ! ) = N ‘ ( ! ) + N ‘ ( ! ) , where
n
d
(k) =
N
[1
‘
À
( ! ) = (2 ¤ )
d
[1
À
n ‘ ( )] £ [ !
À
E ‘ ( )]
(4 )
corresp onds to addi ng a quasipa rti cle, and
N
‘
( ! ) = (2 ¤ )
d
n ‘ ( ) £ [ ! + E ‘ ( )]
(5 )
corresp onds to rem ovi ng a qua sipa rti cl e. At l ow frequenci es N d ( ! ) chang es di sconti nuo usly fro m l inear i n ! for ñ > 0 , where E d ( ) i s l i near in m omentum close
to the nodes, to a consta nt at ñ = 0 (where E d ( ) / k at l ow k ), to zero for
ñ < 0 (where E d ( ) ' j ñ j + O ( k ) for sm al l k ), as can b e seen in Fi g. 5. In the
cal culati on of the densi ty of sta tes we ha ve negl ected the e˜ects of quasipa rti cle
l i feti m es . The lack of pa rti cle{ hol e symm etry seen i n Fi g. 5 i s a general pro p erty
696
L. S. Bor kowski , C .A .R. SÇ
a de Mel o
Fig. 4. T he momentum distributi on of quasipartic les for ¢ = 0 ; n = 1 ; k 1 = k 0 =
and several v alues of ñ for a d -w ave order parameter. T he inset show s results for ñ ç
Fig. 5.
k
Density of states for a
, and v arying Ñ .
d -w ave
order parameter
near
ñ =
0,
for
n =
1;
10
,
0.
k
of superconduct i ng system s wi th smal l chem i cal p otenti al (see R ef. [15] fo r resul ts
i n the no rm al sta te of a b oson{ ferm i on mo del i n the regi m e of p ositi ve ).
The contri buti ons fro m qua siparti cles to speciÙc heat
and spin suscepti bi l ity
chang e f rom
, and
for
, to
, and
const for
, and to
exp
, and
exp
for
. The slopes
of
and
wi th respect to tem pera ture are di sconti nuous at
when
.
Evol uti on from t he BC S t o t he Bose{Ei nst ei n Li mi t
697
.. .
4 . Co n cl u si o n s
In sum m ary, we studi ed the evol uti on fro m BCS to Bo se l i mi t for varyi ng i ntera cti on streng th i n a d -wa ve superconducto r. The gro und sta te pro p erti es of thi s
system change signi Ùcantl y when the chemical p otenti al ñ chang es sign. The enti re
m omentum di stri buti on n d ( k ) i s redi stri buted, wi th l arg est chang es o ccurri ng at
l ow k . Thi s reorg ani zati on in m omentum space i s rel ated to the tra nsiti on f rom
an extended to a l ocal chara cter of the pa i r wave functi on. The symm etry of the
wa ve functi on i s preserved but i ts to p ol ogy is al tered . The chara cter o f spectro scopi c and therm o dyna m ic pro perti es changes fro m a p ower-l aw to an exponenti al
b ehavi or, as ñ b ecomes negati ve.
For consta nt pai ri ng streng th Ñ and varyi ng pa rti cl e density , qua nti ti es such
as pai r size, correl ati on l ength a nd com pressibi l i ty di verg e at ñ = 0 i n the saddl e- poi nt appro xi m atio n. Thi s mi ght i ndi cate the existence of a qua ntum pha se tra nsiti on. We wi l l publ i sh these and other results separatel y [16]. In order to answer
the questi on whether these di sconti nui ties i ndi cate the qua ntum pha se tra nsi ti on
or are simpl y an arti fact of the m ean- Ùeld theo ry one needs to i ncl ude the Ùnite
l i feti m es of the qua sipa rti cles.
y
Ac kn owl ed gm en t s
W e woul d l i ke to tha nk the G eorg i a Insti tute of T echnology for Ùnanci al
supp ort. Som e o f the num eri cal cal culati ons were p erf orm ed at the Po zna¥ Sup ercomputer and Netwo rki ng Center (PCSS). An earl i er versi on of thi s pap er i s
a vai l able i n the xxx. la nl .gov archi ve as prepri nt cond- m at / 9810370.
R ef er en ces
[1] A .J . Leggett,
in: M oder n T ren ds i n the Th eor y of Con densed M atter,
Eds. A . Pek alski, J . Przystaw a, Springer- V erlag, Berlin 1980; J . Ph y s. ( Pa ri s)
Col loq. 41, C 7- 19 (1980 ).
[2 ] P. N ozie̊res, S. Schmitt- Rink, J. L ow-T em p. Ph y s. 59, 195 (1985).
[3 ] M. Randeria, J .- M. Duan, L. -Y . Shieh, Ph y s. R ev . B 41, 327 (1990).
[4 ] S. Schmitt- Rink, C .M. Varma, A .E. Ruckenstein,
[5] R. Micnas, J . Ranning er, S. Robaszkiew i cz,
445 (1989).
113 (1990).
[6] M. Randeria, in:
, Eds. A . Gri£n,
S. Stringari , C ambridge U niv. Press, C ambridge 1995, p. 355.
[7] M. Drechsler, W. Zw erger,
9004 (1997).
[8 ] F. Pistolesi, G. C . Strinati,
15 (1992); S. Stintzing,
15168 (1996).
D.
Snoke,
W. Zw erger,
698
L. S. Bor kowski , C .A .R. SÇ
a de Mel o
[9] B. J anko, J . Maly , K . Levin, Ph ys. Rev. B 56, R11407 (1997).
[10] S.K . A dhik ari, A . Ghosh, Ph ys. Rev. B 55, 1110 (1997).
[11] C.A .R. SÇ
a de Melo, M. Randeria, J.R. Engelbrecht, Ph ys. R ev. L ett. 71, 3202
(1993); J .R. Engelbrecht, M. Randeria, C .A .R. SÇ
a de Melo, Ph ys. R ev. B 55,
15153 (1997).
[12] J.R. Engelbrecht, A . N azarenko, M. Randeria, E. Dagotto, Ph y s. Rev . B 57, 13406
(1998).
[13] B. C. den Hertog, Ph ys. Rev. B 60, 559 (1999).
[14] N . A drenacci, A . Perali, P. Pieri, G. C . Strinati , Ph y s. R ev. B
[ 15] J. Ranninger, J .M. Robin, M. Eschrig,
[16] L. S. Borkow ski, C .A .R. SÇ
a de Melo, to be publish ed.
12410 (1999).
4027 (1995).