Ginzberg Landau theory for superconductivity.pdf

Ginzberg Landau theory for superconductivity

1. Ginzburg-Landau Theory of Superconductivity
Luca Salasnich
Dipartimento di Fisica e Astronomia \Galileo Galilei", Universita di Padova
PhD School in Physics, UNIPD 2024

2. Summary
Basic superconductivity
Ginzburg-Landau phenomenological theory
Ginzburg-Landau vs Bardeen-Cooper-Schrieer
Ginzburg-Landau equation
Coupling with the magnetic eld
London penetration depth
Coherence length and quantized vortices

3. Basic superconductivity (I)
Superconductivity
and expulsion of magnetic ux elds occurring in certain materials when
cooled below a characteristic critical temperatureTc.
It was discovered in 1911 byHeike Kamerlingh Onnes.
In 1957John Bardeen,Leon CooperandRobert Schrieersuggested
that in superconductivity, due to the ionic lattice,
behave like , as somehow anticipated in 1950 byLev Landauand
Vitaly Ginzburg.

4. Basic superconductivity (II)
Critical temperatureTcof some superconductors at atmospheric pressure.
MaterialeSymbolTc(Kelvin)
Aluminium
27
13
Al 1:19
Tin
120
50
Sn 3:72
Mercury
202
80
Hg 4:16
Lead
208
82
Pb 7:20
Neodymium
142
60
Nb 9:30
In 1986Karl Alex MullerandJohannes Georg Bednorzdiscovered
high-Tcsuperconductors. These ceramic materials (cuprates) can reach
the critical temperature of 133 Kelvin.
For these high-Tcsuperconductors the mechanisms which give rise to
pairing of electronsare not fully understood.

5. Basic superconductivity (III)
Superconductors
of a magnetic material over a supercondutor (Meissner eect).
Some:
{ MAGLEV trains, based on magnetic levitation (mag-lev);
{ SQUIDS, devices which measure extremely week mgnetic elds;
{ very high magnetic elds for Magnetic Resonance in hospitals.

6. Ginzburg-Landau phenomenological theory (I)
In 1950, seven years before the Bardeen-Cooper-Schrieer (BCS) theory
1
,
Lev Landau and Vitaly Ginzburg proposed
2
a phenomenological approach
to describe the superconducting phase transition. The main idea is that,
close to the critical temperature, the Helmholtz free energy of a
superconducting material can be written as
F=Fn+Fs; (1)
whereFnis the contribution due to the normal component andFsis the
contibution due to the emergence of a superconducting complex order
parameter below a critical temperature. Ginzburg and Landau used the
words -theory to indicate their phenomenological theory.
1
J. Bardeen, L.N. Cooper, and J.R. Schrieer, Phys. Rev.106, 162(1957).
2
V.L. Ginzburg and L.D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950).

7. Ginzburg-Landau phenomenological theory (II)
Within the Ginzburg-Landau approach, for a D-dimensional system of
volumeL
D
, the super componentFsis given by
Fs=
Z
L
D
d
D
r

a(T)j (r)j
2
+
b
2
j (r)j
4
+jr (r)j
2

; (2)
where
a(T) =kB(TTc) (3)
is a parameter which depends on the temperatureT(kBis the
Boltzmann constant) and becomes zero at the mean-eld critical
temperatureTc, whileb>0 and >0 are temperature-independent
phenomenological parameters.

8. Ginzburg-Landau phenomenological theory (III)
Assuming a real and uniform order parameter, i.e.
(r) = 0; (4)
the energy functional (2) with Eqs. (3) and (4) becomes
Fs0
L
D
=a(T)
2
0+
b
2

4
0: (5)
MinimizingFs0with respect to 0one immediately nds
a(T) 0+b
3
0= 0; (6)
and consequently
0=
(
0 for TTc
q

a(T)
b
=
q
kB(TcT)
b
forT<Tc
: (7)
Thus, the uniform order parameter 0becomes dierent from zero only
below the mean-eld critical temperatureTc.

9. Ginzburg-Landau phenomenological theory (IV)
Potential energy the Ginzburg-Landau theory withT<Tc, which is the
typical energy landscape of second-order phase transitions: from Landau
to Higgs.

10. Ginzburg-Landau vs Bardeen-Cooper-Schrieer
In 1959 Lev Gor'kov showed that the phenomenological Ginzburg-Landau
theory can be deduced from the microscopic theory of
Bardeen-Cooper-Schrieer (BCS, 1957). In particular, the order
parameter 0can be identied with the BCS energy gap 0, while the
coecients,bandof the Ginzburg-Landau energy functional (2) are
directly related to the parameters of the BCS Hamiltonian.
We notice, however, that the Ginzburg-Landau theory is somehow better
than the the mean-eld BCS theory because, in general, the
Ginzburg-Landau order parameter (r) is not uniform, while the
mean-eld BEC energy gap 0is assumed to be uniform.

11. Ginzburg-Landau equation
Let us now consider the eects of a space-dependent Ginzburg-Landau
order parameter (r). Extremizing the functional (2) with respect to


(r) one gets the Euler-Lagrange equation
r
2
+a(T) +bj j
2
= 0: (8)
This equation is called Ginzburg-Landau equation, and it is formally
equivalent to the zero-temperature Gross-Pitaevskii equation with the
following identications:
!
~
2
2m

; (9)
a(T)! ; (10)
b!g: (11)
Herem

>0 is the eective mass of the particles described by the order
parameter, and~is the reduced Planck constant.
The connection with the BCS theory gives
m

= 2me; (12)
wheremeis the mass of the electron.

12. Coupling with the magnetic eld (I)
In general, to describe superconductors we must take into account also
the coupling with the electromagnetic eld. The minimal coupling reads
i~r! i~rqA(r) (13)
whereA(r) is the vector potential,qis the charge of each of the
composite bosonic-like particles described by the eld (r), andi=
p
1
is the imaginary unit. Thus, the Ginzburg-Landau functional becomes
Fs=
Z
L
D
d
D
r
n
a(T)j (r)j
2
+
b
2
j (r)j
4
+
~
2
2m

j

ri
q
~
A(r)

(r)j
2
+
1
20
jB(r)j
2
o
; (14)

13. Coupling with the magnetic eld (II)
where
B(r) =r^A(r) (15)
is the space-dependent magnetic eld.
The last term in Eq. (14) takes into account the free magnetic energy of
the system with0the magnetic permeability, assumed to be the vacuum
one.
The minimization of the functionalFs=Fs[ (r);A(r)] with respect to


(r) gives the so-called Ginzburg-Landau equation


~
2
2m


ri
q
~
A(r)

2
+a(T) +bj (r)j
2

(r) = 0; (16)
that is a nonlinear Schrodinger equation with cubic nonlinearity for the
order parameter (r), which contains the minimal coupling with the
vector potentialA(r).

14. Coupling with the magnetic eld (III)
The supercurrent charge densityjs(r) can be obtained by considering the
minimization of the functionalFs=Fs[ (r);A(r)] with respect toA(r).
In this way one nds
1
0
r^B(r) =i
q~
2
2m

(

(r)r (r) (r)r

(r))
q
2
m

j (r)j
2
A(r):
(17)
Remarkably, Eq. (17) can be rewritten as the familiar Ampere equation
r^B(r) =0js(r) (18)
setting
js(r) =i
q~
2m

(

(r)r (r) (r)r

(r))
q
2
m

j (r)j
2
A(r);(19)
which is identied as the supercurrent charge density.

15. Coupling with the magnetic eld (IV)
Taking into account that the order parameter (r) describes bosons, with
eective massm

and eective chargeq, which are made of Cooper pairs
(two electrons with opposite spin), we set
q=2e (20)
m

= 2me (21)
whereeis the negative electric charge of the electron (withe>the
electric charge of the proton) andmethe mass of the electron. We also
introduce, in full generality, the superuid local number density of
electrons as
ns(r) = 2j (r)j
2
: (22)

16. London penetration depth (I)
Assuming a real and uniform order parameter, see Eq. (4), the
supercurrent is strongly simplied and reads
js(r) =
q
2
m


2
0A(r): (23)
Clearly, from Eq. (22), in the case of a uniform and real order parameter
we have
ns= 2
2
0=

0 for TTc
2kB(TcT)
b
forT<Tc
: (24)
In addition, using Eqs. (20), (21), and (23), the supercurrent can be
rewritten as
js(r) =
e
2
ns
me
A(r): (25)
This is the, obtained for the rst time by the London
brothers in 1935.

17. London penetration depth (II)
The curl of the Ampere equation (18) gives
r
2
B(r) =0r^js(r); (26)
taking into account that
r^(r^B) =r
2
B+r(rB) =r
2
B (27)
due to the magnetic Gauss law
rB= 0: (28)
Inserting Eq. (25) into Eq. (26), and using Eq. (15), we get
r
2
B(r) =
e
2
ns0
me
B(r): (29)

18. London penetration depth (III)
Assuming thatB=B(x) the previous equation can be written as
@
2
@x
2
B(x) =
e
2
ns0
me
B(x) (30)
which has the following physically relevant solution forx0:
B(x) =B(0)e
x=L
; (31)
where
L=
r
me
e
2
ns0
(32)
is the so-called. The meaning of Eq. (31) is
that inside a superconductor the magnetic eld decays exponentially.
This is the Meissner-Ochsenfeld eect: the expulsion of a magnetic eld
from a superconductor, experimentally observed for the rst time in 1933.

19. London penetration depth (IV)

20. Coherence length and quantized vortices (I)
A very important characteristic length of the Ginzburg-Landau equation
is the so-called
=
s
~
2
2m

ja(T)j
: (33)
This is the distance at which there is a compensation between the
gradient term and the linear term of the Ginzburg-Landau equation:
j
~
2
2m

r
2
j '
~
2
2m


2
j j=ja(T)jj j: (34)
Alexei Abrikosov in 1957 showed that the is nothing
else than the
Ginzburg-Landau equation.
Invoking the presence of quantized vortices Abrikosov explaned type-II
superconductors, which were discovered by Rjabinin and Shubnikov in
1935.

21. Coherence length and quantized vortices (II)

22. Coherence length and quantized vortices (III)
Following an earlier intuition of Ginzburg and Landau, by solving the
Ginzburg{Landau equation Abrikosov found that a type-II superconductor
appears when
L

>
1
p
2
'0:7071; (35)
whereLis the is the.
Under this condition, there is the formation of Abrikosov quantized
vortices and the magnetic eld can penetrate deep inside the
superconductor.
Abrikosov found that the vortices arrange themselves into a regular array
known as a vortex lattice. A similar analysis was done for the problem of
vortex state in a rotating superuid by Lars Onsager and Richard
Feynman.

23. Coherence length and quantized vortices (IV)
Vortices in a YBCO lm imaged by scanning the intensity of the magnetic
eld with SQUID microscopy [F.S. Wellset al, Sci Rep.5, 8677 (2015)].

24. Coherence length and quantized vortices (V)
Taking into account that forT<Tcwe have
L=
r
me
e
2
0ns
=
s
meb
2e
2
0kB(TcT)
(36)
and
=
s
~
2
4meja(t)j
=
s
~
2
4mekB(TcT)
; (37)
it follows that
=
L

=
me
e
s
2b
0
: (38)

25. Coherence length and quantized vortices (VI)