6th
International Science, SocialSciences, Engineering and Energy Conference
17-19 December, 2014, Prajaktra Design Hotel, UdonThani, Thailand
I-SEEC 2014
http//iseec2014.udru.ac.th
Investigate the effect of pseudogap on the critical
temperature of superconductor
Thaipanya Chanpoom
Program of Physics and General Science Faculty of Science and Technology
NakhonRatchasima Rajabhat University
Nakonrachasima@yahoo.co.th
Abstract
Since the discovery of pseudogap in High-Tc cuprate superconductors, the
origin of the pseudogap has remained unresolved . In this research, we have presented
the two-gap behaviors in the high-Tc cuprates that superconducting gap and pseudogap.
The pseudogap is one of the superconductor’s gap that appear above the critical
temperature . The effect of pseudogap ( E0 ) on critical temperature ( Tc ) of high-Tc
superconductors is investigated. In our calculation, we assume that the pseudogap
occurs in the range T g and Tg   D where  D is the Debye cut-off energy. Within the
weak-coupling limit and E0  Tc , the exact formula of critical temperature equation
with pseudogap had derived. We find that the higher pseudogap, the lower critical
temperature is found.
Keywords: pseudogap, critical temperature
Introduction
Many experiments such as nuclear magnetic resonance (NMR) [1], angle
resolved photoemission spectroscopy (ARPES) [2], specific heat [3] show a narmal
state pseudogap on cuprate superconducytors. The pseudogap appearance below a
certain temperature T * where T *  Tc . Some measurement revealed T * depended on
2
carrier concentration p in the form T *  c / p n where c, n are constant [4]. [5] suggests
that the origin of pseudogap and superconducting gap are the same. Various scenarios
explained the behavios of pseudogap, in the strongly overdoped region pseudogap
merges with the superconducting gap and about optimal doping pseudogap line intersect
the superconducting dome [6] and show in Figure 1 [7].
Fig.1 The pseudogap depended on doping level
The existence of the pseudogap is the most important behavior of cuprates. [8]
show that the magnitude of a pseudogap at Tc is larger than superconducting gap at
T  0K . On cuprate superconductors the critical temperature depended on doping level
has the bell like curve. The BCS theory show that the critical temperature Tc depended
on coupling parameter  in the exponential form Tc  1.14D e1/  [9]. Many models
have been proposed to describe the effect of pseudogap on the superconducting
properties in cuprates superconductors. Chanpoom Chantrapakajee and
Udomsamuthirun [10] studied the effect of pseudogap on the critical temperature by
using two bands superconductors. Dahm [11] and Udomsamuthirun [12] studies the
isotope effect in the presence of a pseudogap one band model. The purpose of this paper
is to study the critical temperature of one band superconductors with pseudogap in
weak-coupling limit in the frame work BCS theory.
3
Model and calculation
To finded the energy gap of one band superconductors started with BCS Hamiltonian
[13]
H    k Ck Ck   Vkk CkCk C k Ck 
(1)
kk 
k
here  k is the kinetic energy of the electrons and Vkk  is the potential energy between
two electrons. Ck (Ck ) are the electron creation (annihilation) operators of momentum
k and spin   or  . The energy gap  defined is
   Vkk  C k Ck 
(2)
k
By using Green’ s function technic, the BCS energy gap is
1


D
0
  2  (T ) 2
tanh  k

2T

 k2  (T ) 2



d 
(3)
where  is the coupling constant parameter. With pseudogap taken into accout in the
model the energy gap (T )2 can be written in the form [14]
 (T ) 2   2s  E02
(4)
here  s and E0 are the superconducting gap and pseudogap respectively so the equation
(3) is
1


D
0
  2  2  E 2
s
0
tanh  k

2T

 k2   2s  E02



d 
(5)
4
The pseudogap is a normal state gap the temperature at which the pseudogap
vanish is usally labeled by Tg . All measurements in cuprates show that the magnitude of
pseudogap decreases with increasing doping level. Analyses of the Hall coefficient and
uniform susceptibility reveal a temperature scale Tg below which appears the pseudogap
at more underdoped Tg reaching as high value as 600K [Foltin]. At the critical
temperature equation  s  0 equation (5) expanded in the range of Tg and  D ,
 D  Tg  Tc
1


Tg
0
  2  E2
0
tanh  k
 2Tc

 k2   E02



d  

Tg

tanh  k
 2Tc
0
k


d  

D
0
 
tanh  k 
 2Tc d 
(6)
k
for condition E0  Tc and Tg  Tc the second term of equation (8) by using graph
3
2
 Tg 
 Tg 
 Tg 
fitting can be approximated 0.0108 
  0.163 
  1.02 
 so equation (6)
 2Tc 
 2Tc 
 2Tc 
is
3
2
 Tg 
 Tg 
 Tg

 0.0108 
  0.163 
  1.02 
 E0
 2Tc 
 2Tc 
 2Tc
1
Tg

 2D 
  ln 


  Tc 
(7)
the critical temperature can be written in the form
3
Tc  1.14D e
2
 Tg 
 Tg 
 Tg 
1 Tg
   0.0108
  0.163 2T  1.02 2T 
 E0
2
T
c
c




 c
(8)
5
Results and Discussions
The critical temperature of one band superconductors with taken into account
pseudogap varies on Tg , E0 as show in equation (11). To investigate the effect of
pseudogap on the critical temperature use parameter as D  750K The graph of
equation (11) is show in Figure 2. It was found that the critical temperature decreased as
pseudogap increased . At the same pseudogap and Tg the critical temperature increased
as the parameter  increased but the critical temperature increased as Tg decreased on
the same parameter  . This result corresponded to the reported of I. Tifrea and et al
[15].
80
=0.4,Tg=200K
=0.4,Tg=300K
=0.3,Tg=300K
=0.35,Tg=300K
Tc
60
40
20
0
0
50
100
E0
Fig.1 Plot graph of E0 and Tc
150
6
Conclusions
We have study the effect of pseudogap on the critical temperature one band swave superconductors in BCS frame work. The result found that all of parameters 
and Tg the critical temperature Tc decreases as the pseudogap E0 increases. The critical
temperature increased as Tg decreased on the same parameter  and E0 .
Acknowledgements
The author would like to thank Nakhonratchasima Ratjabhat University for
financial support.
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