disordered two-dimensional superconductors financial support: collaborators: felipe mondaini...
TRANSCRIPT
Disordered two-dimensional superconductors
Financial support:
Collaborators: Felipe Mondaini (IF/UFRJ) [MSc, 2008] Gustavo Farias (IF/UFMT) [MSc, 2009] Thereza Paiva (IF/UFRJ) Richard T Scalettar (UC-Davis)
Raimundo Rocha dos Santos
IF/UFRJ
Outline• Motivation• The model: disordered
attractive Hubbard model• Quantum Monte Carlo• Ground state properties• Finite-temperature properties• Conclusions
Motivation: disorder on atomic scales Sputtered amorphous films
Low coverage: isolated incoherent islands
High coverage: islands “percolate”
film thickness tracks disorder
How much dirt (disorder) can a superconductor take before it becomes normal (insulator or metal)?
Question even more interesting in 2-D (very thin films):
• superconductivity is marginal Kosterlitz-Thouless transition
• metallic behaviour also marginal Localization for any amount of
disorder in the absence of interactions (recent expts: MIT possible?)
A M Goldman and N Marković, Phys. Today, Page 39, Nov 1998
CR
ITIC
AL
TE
MP
ER
AT
UR
E T
c (ke
lvin
) SHEET RESISTANCE AT T = 300K (ohms)
Mo77Ge23 film
J Graybeal and M Beasley, PRB 29, 4167 (1984)
Sheet resistance:
R at a fixed temperature can be used as a measure of disorder
t
ℓℓ
ttAR
independent of the size of
square
Disorder is expected to inhibit superconductivity
Issues: Why does Tc behave like that with disorder? Is the transition at T = 0 percolative (i.e., purely geometrical)? If not, how does it depend on system details?
Metal evaporated on cold substrates, precoated with a-Ge: disorder on atomic scales.
D B Haviland et al., PRL 62, 2180 (1989)
Superconductor – Insulator transition at T = 0 when R� reaches one quantum of resistance for electron pairs, h/4e2 = 6.45 k
Quantum Critical Point
Bismuth
(evaporation without a-Ge underlayer: granular disorder on mesoscopic scales.1)
Our purpose here: to understand the interplay between occupation, strength of interactions, and disorder on the SIT, through a fermion model.First task: reproduce, at least qualitatively, Tc vs. R□
Generic phase diagram for Quantum Critical Phenomena:
T
disorder (mag field, pressure, etc.)
SUC
METAL
INS
QCP
Tc
T
Interlude: Phase transitions and critical phenomena
Long-ranged correlations at a phase transition singular behaviour of thermodynamic quantities e.g., order parameter of the transition:
magnetic transition: magnetization (1, 2 or 3-component) superconducting transition: gap (complex; macroscopic wave function)
ieTT |)(|)( kk 2-component
( T
)(0
)
T/Tc
s-wave isotropic
TTC
CTTC
specific heat
susceptibility CTT
and so forth
Universality (expt. and theory; 1970’s):main features of phase transitions (including critical exponents) do not depend on details (magnitude of interactions, etc.);
they depend on:symmetry of order parameter (# of components)dimensionality of lattice determine nature and number of excitations, can be so overwhelming to the point of
depressing Tc to zero (Mermin-Wagner thm)
magnets, superconductors, superfluids, fluids, etc., share common main features!
2-component order parameter in 2 space dimensions: M-W thm forbids long range order at T > 0 but phase with quasi--long-range order
possible below TKT: the Kosterlitz-Thouless transition
B. Berche et al. Eur. Phys. J. B 36, 91 (2003)
XY 2D
Stinchcombe JPC (1979)
Tc(
p)/T
c(1)
p
Heisenberg 3D
Yeomans & Stinchcombe JPC (1979)
Ising 2D
Dilute magnets: fraction p of sites occupied by magnetic atoms:Tc 0 at pc, the percolation concentration (geometry)
†( . ) ( )i j i i i i iìj i i
H t c c h c U n n n n
The homogeneous attractive Hubbard model
0.0 0.2 0.4 0.6 0.8 1.00.00
0.05
0.10
0.15
0.20
0.25
k BT
C/t
< n >[Paiva, dS, et al. (04)]
Homogeneous case
◊ particle-hole symmetry at half filling◊ strong-coupling in 2D:
• half filling: XY (SUP) + ZZ (CDW) Tc 0• away from half filling: XY (SUP) TKT 0
sites of 1fraction aon
sites of fraction aon 0
fU
fU i
Disordered case
particle-hole symmetry is broken
Heuristic arguments [Litak + Gyorffy, PRB (2000)] : fc as U
†( . ) ( )i j i i i i iìj i i
H t c c h c U n n n n
The disordered attractive Hubbard model
c 1- f
mean-field approx’n
Quantum Monte CarloCalculations carried out on a [square + imaginary time] lattice:
x
Ns
M
1M
T
Absence of the “minus-sign problem” in the attractive case
non-local updates: MNs2 Green’s functions
0 5 10 15 20 25
0
1
2
3
4
5
6
7
8
8X8 10X10 12X12 14X14 16X16
U=3 f=1/16
Ps
Typical behaviour for → :
correlations limited by finite system size
For given temperature 1/, concentration f, on-site attraction U, system size L L etc, we calculate the pairing structure factor,
iii
rrii ccPs with ,
averaged over 50 disorder configurations.
n =1
Ground State Properties
0.00 0.05 0.10 0.150.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
f= 0 f= 1/16 f= 2/16 f= 3/16 f= 4/16 f= 5/16
U=4P S
/L2
1/L
Spin-wave–like theory (two-component order parameter) Huse PRB (88):
zero-temperature gap
2
2sP C
L L
n = 1
F Mondaini, T Paiva, RRdS, and RT Scalettar, PRB 78, 174519 (2008 )
0.0 0.1 0.2 0.3 0.40.0
0.5
1.0
1.5
2.0
U=2 U=2.5 U=3 U=4 U=6
f
We estimate fc as the concentration for which 0;
can plot fc (U )...
normalized by the corresponding pure case
For 2.5 < U < 6, a small amount of disorder seems to enhance SUP
~~n = 1
F Mondaini, T Paiva, RRdS, and RT Scalettar, PRB 78, 174519 (2008 )
0 1 2 3 4 5 60.0
0.1
0.2
0.3
0.4
f c
U
fc increases with U, up to U ~ 4;
mean-field behaviour sets in above U ~ 4?transition definitely not driven solely by geometry (percolative):
fc = fc (U )
(c.f., percolation: fc = 0.41)
n = 1
F Mondaini, T Paiva, RRdS, and RT Scalettar, PRB 78, 174519 (2008 )
GJ Farias, MSc thesis, UFMT, (2009)
U = 6 For n = 1: f =0 CDW+SUP f >0 SUP dirt initially enhances SUP
For n < 1:f =0 only SUP f >0 still only SUP dirt always tend to suppress SUP
GJ Farias, MSc thesis, UFMT, (2009)
?
fc with n for fixed U(at least for
U 4):less
electrons to
propagate attraction
Finite-temperature propertiesFinite-size scaling for Kosterlitz-Thouless transitions
KTusual
line of critical points ( = ∞)
Barber, D&L (83)
Lg
LL )(
c
L1/1 L2/2L1/1 L2/2
KT
2
0
2 1~
with ,
Lr
rd
ccP
L
s iiir
rii
Finite-size scaling at T > 0: KT transition
For infinite-sized systems one expects
KT
KT
TTTT
A,exp~ 21
LfLLPs 2),(
0 2 4 6 8 10 12 14 16 18 200.00
0.01
0.020.03
0.04
0.05
0.06
0.07
0.080.09
0.10
0.11
8 10 12 14
U = 4 f = 2/16
Ps/L
2-
Kosterlitz-Thouless transition: Curves should cross/merge at βc for η(Tc)=1/4:
Repeat this for other values of f...
0.0 0.1 0.2 0.3 0.40.00
0.05
0.10
0.15
0.20
0.25
Tc
f
n = 1
F Mondaini, T Paiva, RRdS, and RT Scalettar, PRB 78, 174519 (2008 )
Somewhat arbitrary Check against independent method
Superfluid density, s, ( helicity modulus) measures the system response to a twist in the order parameter [M.E. Fisher et al. PRA 8, 1111 (1973)]
need current-current correlation function [DJ Scalapino et al. PRB 47, 7995 (1993)]
very costly: imaginary-time integrals
At TKT-, s has universal jump-discontinuity [D.R. Nelson and J.M.
Kosterlitz, PRL 39, 1201 (1977)]: KTs T
2
determined is ,2
when
)(plot
KTs
s
TT
T
0.00
0.05
0.10
0.15
0.20
0.00
0.05
0.10
0.15
0.20
0.00
0.05
0.10
0.15
0.20
0.0 0.1 0.20.00
0.05
0.10
0.15
0.20
L=12 U = 4(a) f =1/16
(b) f =2/16
s (c) f =3/16
(d) f =4/16
T
n = 1 2T/s
F Mondaini, et al. PRB 78, 174519 (2008 )
0.0 0.1 0.2 0.3 0.40.00
0.05
0.10
0.15
0.20
0.25
Tc
f
n = 1
F Mondaini, T Paiva, RRdS, and RT Scalettar, PRB 78, 174519 (2008 )
0.0 0.1 0.2 0.3 0.40.00
0.05
0.10
0.15
0.20
0.25T
c
f
U = 3 Ps
U = 4 Ps
U = 4 s
U = 6 Ps
n = 1
F Mondaini, T Paiva, RRdS, and RT Scalettar, PRB 78, 174519 (2008 )
GJ Farias, MSc thesis, UFMT, (2009)F Mondaini, T Paiva, RRdS, and RT Scalettar, PRB 78, 174519 (2008 )
Different concavities?
Need to refine: s
Possibly concavity
changes with n ?
Possibly non-linear relation
between R□ and f ?
ConclusionsAt half filling, small amount of disorder seems to initially favour SUP in the ground state; not for other fillings.
fc depends on U transition at T = 0 not solely geometrically driven; quantum effects; correlated percolation?
for given U, fc with n (need more calcn’s for U < 4)
Two possible mechanisms at play:
• MFA: as U increases, pairs bind more tightly smaller overlap of their wave functions, hence smaller fc.
• QMC: this effect is not so drastic up to U ~ 4 presence of free sites allows electrons to stay nearer attractive sites, increasing overlap, hence larger fc.
• QMC: for U > 4, pairs are tightly bound and SUP more sensitive to dirt.
n=1: A small amount of disorder allows the system to become SUP at finite temperatures; as disorder increases, Tc eventually goes to zero at fc.
n <1: Tc decreases steadily with f. Concavity???