IACS Institutional Repository

Advisory Board and Editors

Medhi, Amal

Basu, Saurabh

Kadolkar, C Y

Indian J Plm 82 (3). 251 -257 (2008) 
Variational Monte Carlo study of magnetic correlations 
in bilayer t-J model 
Amal Medhi,* Saurabh Basu and C Y Kadolkar 
Department of Physics, Indian Institute of Technology Guwahati 
Guwahati-781 039, Assam, India 
E-mail amedhi @ ntg ernet in 
Abstract We study the magnetic correlations in a t -J bilayer using Variational Monte Carlo (VMC) method, 
the variational wavefunction being the solution of the Hartree-Fock (HF) t-J Hamiltonian with antiferromagnetic 
lAF) order We calculate the spin-spin correlation functions and spin structure factors at half filling and slightly 
away from half filling The results show that, at half filling perfect antiferromagnetic long range order (AFLRO) 
f xists which starts to diminish as soon as holes are introduced It is observed that the holes drastically reduce 
the mterplanar spin correlations, whereas the effect on the planar correlations is rather small This is to be 
contrasted with the experimental results on the cuprates, which show that the AFLRO in the copper oxide 
planes gets completely destroyed even at a small hole concentration This is possibly due to the competition of 
AFLRO with the superconducting order, which has not been incorporated in our calculations We also examine 
the effect of the mterplanar parameters e g the hopping integral / and exchange constant, J} on the magnetic 
correlations We find that ft reduces planar correlations while the effect of J is small 
Keywords t-J model, VMC 
PACS Nos. 71 10 Fd, 74 25 Ha 
1. Introduction 
High temperature superconductivity discovered in the cuprate compounds first in 1986 [1] 
has been a subject of very intense research for a variety of reasons [2] Experiments 
have shown that the parent (undoped) compounds of high-7c superconductors are very 
good insulators with the spins of the Cu2+ ions ordered antiferromagnetically Upon doping 
with charge carriers (holes), the antiferromagnetic long range order (AFLRO) gets destroyed 
rapidly and the compounds show superconductivity with high transition temperatures. The 
basic feature in structure of the cuprate superconductors is the presence of copper oxide 
(Cu02) layers intervened by other atoms (e.g. La, Ba etc.). The properties of the high-7"c 
Corresponding Author 
© 2008 IACS 
252 Amal Medhi, Saurabh Basu and C Y Kadolkar 
superconductors are generally found to depend on the number of copper oxide layers 
present per unit cell. For example, the bilayer compounds (e.g. YBa2Cu306+x) have a spin 
gap [3-5] which is absent in single layer compounds, also its transition temperature, T is 
higher as compared to the single layer ones [6, 7], etc. 
Theoretically the t-J model have been shown to describe the properties of the high-T 
superconductors at least close to half filling reasonably satisfactorily. The model has 
been studied extensively using a variety of analytical and numerical techniques for two 
dimensional lattice. Application of numerical techniques, like the quantum Monte Carlo 
method is limited by the negative sign problem. On the other hand, analytical techniques 
can treat the constraint of no double occupancy in the t-J Hamiltonian only approaimately 
The Variational Monte Carlo (VMC) method is free from such problems, but it sensitively 
depends on the choice of the variational wavefunction. The VMC method has been used 
to study various phases of the t-J Hamiltonian, viz. antiferromagnetic, superconducting, 
metallic etc., for two dimensional lattice. In this work, we use VMC study magnetic 
properties of the bilayer t-J model by taking the Hartree-Fock antiferromagnetic state as 
the variational wavefunction. However, away from half filling this wavefunction is not expected 
to be the true ground state. We leave the incorporation other degrees of freedom 
(superconducting) in the variational wavefunction, for a future investigation. 
2. The model 
The t-J Hamiltonian for a bilayered lattice can be written as 
Ji Yl SrSk-±n,nk 
«u»v 4 ' 
<('.*»" 
(1) 
where t and J are the planar hopping and exchange integral respectively, while tL and 
J± are the corresponding interplanar parameters. cm \cjff J annihilates (creates) an electron 
of spin a at site /. S, is the spin operator at site /, given by $* = vj(%0a)vi. 
where Vi - and aa (a = x, y, z) are the Pauli spin matrices. The summation 
indices (/,;') and ((i,k)) indicate nearest neighbor pairs in the same plane and 
adjacent planes respectively. The Hamiltonian acts on a subspace of no doubly occupied 
sites. 
Variational Monte Carlo study of magnetic correlations in bilayer t-J model 253 
3. The variational wavefunction 
We take the following wavefunction with antiferromagnetic (AF) order as our variational 
ansatz for the ground state of the bilayer f-J Hamiltonian, 
m.r> = ^ | ^ F ( ^ ) > (2) 
where, 
k<kF 
(3) 
'j'G -11,(1- n/tA?4) is the Gutzwiller operator which projects out the states with no doubly 
occupied site. The wavefunction \*FAF(A)) is the ground state of the Hartree-Fock (HF) 
antiferromagnetic Hamiltonian, given by 
*AF = £ 
krr ck+Q 
-rtio)A 
-r,{o)A ck+Q (4) 
where Q= (n, n, n) is the perfect nesting vector, ek - -2r(cos kx + cos ky)- 2tl cos kz 
is the free electron dispersion relation and rj(a) = ±1 tor CT -T, 1 . A is proportional to 
the sublattice magnetization and is our variational parameter. The quasiparticle operators 
d\ are related to the electron operators by the following transformation, 
with 
cc„ ri(o)Pk 
-tl(cr) Pk cc 
1 / / i \1/2 
'k*Q 
(5) 
(6) 
Ac^iWVf^f2 (7) 
and k is such that ek < 0. 
The variational wavefunction describes the Neel anti-ferromagnetic phase for nonzero 
A, while for 4 = 0 , it reduces to the singlet liquid state [9]. 
254 Amal Medhi, Saurabh Basu and C Y Kadolkar 
4. Numerical details 
As with any variational calculation, we start by evaluating the energy expectation value 
V()= hu \u, \ (8) 
V v a r I ' var/ 
and optimizing the wavefunction with respect to the variational parameter A. The minimum 
energy thus found represents an upper bound to the true ground state energy of the 
system. For numerical evaluation of the expectation value, we write the expression as 
[10] 
C(R) (9) 
where \R) = cr1 ctj ...cr.x crii...|0) represents a configuration of the electrons in the real 
space representation of the wavefunction and C(R) is the weight of the configuration. The 
probability density for a configuration j R) is given by, 
mm- lcm? 
In the Monte Carlo procedure, the summation over R is replaced by the importance sampling 
with weight P(R), for which we use the Metropolis algorithm. 
Simulations are performed on a lattice of size 10 x 10x2. In one Monte Carlo sweep 
(MCS) through the lattice, we attempt Ns (number of lattice sites) random moves which 
consists of moving an electron to an empty site and exchanging two antiparallel spins. 
The Monte Carlo update of the configurations, after each successful move, is done by 
using the inverse update method due to Ceperley et al [8]. For calculation of the 
expectation values of observables, we make measurements on configurations chosen from 
about 10,000 - 50,000 Monte Carlo sweeps after warming up the system for about 5,000 
sweeps. 
5. Results 
The results shown here are for a lattice of size 10 x 10 x 2. As for the parameters, we 
choose values that are realized for various cuprate compounds, e.g. we take J/t = 0.35 
and the interplanar parameters are varied from, t± /t = 0.05 - 0.20 and JL jt = 0.05 - 0.35. 
First, we optimize the wavefunction with respect to the variational parameter, A for 
various values of interplanar parameters (fx, JL ) , at half filling and slightly away from half 
filling. For one set of parameter value, viz. (t1/t1 Jx/t) = (0.20,0.10), we have carried out 
Variational Monte Carlo study of magnetic correlations in bilayer t-J model 255 
the optimization to higher value of hole doping, in order to determine the stability of the 
antiferromagnetic phase against the singlet liquid state The results are shown in 
Figure 1 Our calculation shows that the antiferromagnetic phase yields lower energy upto 
20% hole concentration in a bilayer for the chosen parameter values For comparison, we 
have also calculated the optimal parameter, A as a function of hole concentration, s for 
a square lattice of size 10 x 10 and taking the value of J11, to be the same as that for 
the bilayer. We found that in this case, the stability of the AF phase against the singlet 
liquid state, extends upto about 14% hole concentration which is lower than that for the 
bilayer This shows that interplanar couplings sustain the AF phase till larger values of 
hole doping 
07 
04 
01 
[ (0 20 0 05) 
(0 20 0 15) 
(0 20 0^5) 
T M^ 
(0 05 0l0)-»"% 
(0 10 0 10)-*1-) 
(0 20 0 | 0 ) - * -
- —-—o \ 
0 02 
h 
(a) 
0 04 
00 
Figure 1. Optimal variational parameter A versus hole concentration & For (a) different interplanar 
parameter values in the underdoped region and (b) for one value of the parameter set at all hole 
concentration The paired numbers inside bracket in the figures indicate the values of the parameters 
CJJ 
Next, we show results for spin-spin correlation function, ( S ' S ^ calculated for the 
optimized wavefunction at various hole concentrations and for different parameter values 
Figure 2, shows the planar and interplanar spin correlations as a function of distance 
along the edge of a plane (x-direction), at and away from half filling for (f, tt J1 t) equal 
to (0 20, 0 10) The figure shows that perfect antiferromagnetic long range correlations 
(Neel order) exist at half filling Moreover, the magnitude of planar and interplanar correlations 
are almost equal albeit with J± < J Away from half filling, as soon as holes are introduced, 
both planar and interplanar correlations get diminished This reduction is more prominent 
1 2 3 4 
(r,-«p 
(a) Planar correlations 
5 6 0 2 3 4 
C-'j) 
(b) Interplanar correlations 
Figure 2. Spin spin correlation, \Sf$*) as a function of distance \r,-r,\ along x-direction In (a) sites 
/ and j belong to same plane, whereas in (b) / and y belong to different planes 
256 Amal Medht, Saurabh Basu and C Y Kadolkar 
is case of the interplanar correlations This can be understood as follows Since J{ < j 
the energy cost of disrupting a bond between two interplanar sites is less than that for a 
intraplanar bond So, the interplanar bonds of antiparallel spins are broken easily by the 
mobile holes. Another notable feature is that the magnitude of correlations at all distances 
get diminished uniformly by holes It does not decay appreciably with distance, except for 
the two larger hole concentrations shown in Figure, while planar correlations show a slow 
decay with distance. This result is to be contrasted with experimental results available for 
cuprates It is well known that the AFLRO in the copper oxide planes gets completely 
Figure 3. Spin structure factor S(q) as a function of q, at various hole concentration h The values of 
(^M*Qy) on the horizontal axes are the points in the triangular path in q- space, shown in (a) 
destroyed by as small as about 2% hole concentration, followed by the appearance of 
the superconducting phase. Our study here, where we consider a wavefunction only with 
antiferromagnetic order, shows that the holes alone are not effective in destroying long 
range correlations Thus we conclude that the AFLRO in the cuprate superconductors 
does not get destroyed, only due to the effects of mobile holes but possibly due to 
combined effects of holes and competition with other phases It may be mentioned that a 
number of phases are known to appear in the underdoped regime of high temperature 
superconductors [11]. These phases posses complex magnetic ordering and are found to 
be competing with each other. Further, the spin structure factors 
S(q) = (1//V) I ,
 y e
,qr
" (s? S*} as a function of q% calculated at various hole concentrations 
S«000itz«005-
tz-010-
12 « 0.20" 
v 
0.2 
0.1 
0 
-0.1 
-021 
««0.04 
I Ji-OIOJ 
tz-005-ntJ 
i te-0.1Q-«H 
ltz-020"-H 
V 
' J 
\ 3 5 7 
(b) 6 * 0.04 
Figure 4. Spin spin correlation function, (s?S*) for different values of interplanar hopping parameter 
t± , at and away from half filling 
Variational Monte Carlo study of magnetic correlations in bilayer f-J model 257 
are shown in Figure 3. The sharp peaks at q = (n, n, 7r)are indications of the existence 
cv antiferromagnetic long range order (AFLRO), which diminishes as one gos away from 
half falling There are also peaks in S(qx, qy1 0) at (n, /r, 0) in the S(q) versus q curves for 
1 - 0 However, S(qx, qy, 0) for all (qx, qy) increases with increasing hole concentration 
The increase in maximum in case of S(nf /r, 0) This is because of the reduction of 
interplanar antiferromagnetic correlations by the holes as remarked earlier 
Finally, we examine the effects of interplanar coupling on the magnetic correlations in 
the bilayer Figure 4 shows the planar spin correlations along x-direction for different values 
of interplanar hopping parameter, t± As seen in the figure, t[ plays no role at half filling 
as expected But away from half filling larger interplanar hopping reduces spin correlations 
•n a plane Interplanar correlations (not shown here) are not effected by tA On the other 
hanu the interplanar exchange constant, J{ is found to have negligible effects on both 
the planar and interplanar correlations 
6 Conclusion 
We have studied the magnetic correlations in the Hartree-Fock antiferromagnetic 
wavefunction for a bilayer t-J model using VMC method We find that, perfect 
antiferromagnetic long range order exists at half filling Away from half filling, the magnitude 
of correlations decreases with increasing hole concentration, but the long range order 
survives upto much higher hole doping in comparison to the range of AF phase in cuprate 
superconductors We also find that, interplanar hopping reduces planar correlations while 
the effect of interplanar exchange is small 
Acknowledgment 
One of us (AM) would like to thank CSIR, India for financial support under the Grant No 
F No 9/731 (022)/2003-EMR-
References 
[1] J G Bednorz and K A Muller Z Phys B64 189 (1986) 
[2) E Dagotto Rev Mod Phys 66 763 (1994) 
[3] J M Tranquada, P M Gehnng, G Shirane, S Shamoto and M Sato Phys Rev B46 5561 (1992) 
[4] B J Sternheb, G Shirane, J M Tranquada, M Sato and S Shamoto Phys Rev B47 5320 (1993) 
[5] S Shamoto, M Sato J M Tranquada, B J Sternheb and G Shirane Phys Rev B48 13817 (1993) 
[6] R lhara, R Sugise, M Tokumoto, Y Kumira and T Shinamura Nature (London) 334 510 (1988) 
[7] S S P Parkin, V Y Lee, A J Nazzal, R Savoy and R Beyers Phys Rev Lett 61 750 (1988) 
[8] D Ceperley, G V Chester and M H Kalos Phys Rev B16 3081 (1977) 
[9] H Yokoyama and H Shilba J Phys Soc Jpn 56 3570 (1987) 
[10] H Yokoyama and H Shiba J Phys Soc Jpn 56 1490 (1987) 
[11] H D Chen, S Capponi, F Alet and S C Zhang Phys Rev B70 024516 (2004) 


English

Variational Monte Carlo Study of Magnetic Correlations in Bilayer t- J Model

Forthcoming Publications

Choudhury, Dilip Kumar

Bhattacharjee, Buddhadeb

Indian J. Phys. 82 (5), 495-496 (2008) 
DGOQMI] 
D 
Q% ^OREWORD 
Proceedings (Parti) of the 5th Biennial Conference of the Physics Academy 
of the North East (PANE) held at Department of Physics, Gauhati University 
Guwahati-781 014, Assam, India during March 1-2, 2007. 
The conference was inaugurated by Prof. A. Choudhury, Vice Chancellor, Gauhati 
University followed by a key note address on "Use of Low Energy Accelerator" by 
Prof. Amit Roy, Director of the Inter University Accelerator Centre, New Delhi. The 
inaugural ceremony was presided over by Prof. A. Rahman, Head, Department of 
Physics, Gauhati University. The conference received a large number of contributions 
on various topical areas of physics. Altogether, 113 research contributions had been 
accepted for presentation of which 76 papers were finally presented in the form of 
poster and oral presentation. Total 175 delegates from the North East India as well 
from other parts of the country participated in it. It was a wonderful experience for 
the teachers and students community of the university to be with such a distinguished 
gathering of physicists and enthusiastic young research scholars. 
Keeping the tradition of earlier conferences of the series, Professors J. Hatibarua, 
B.K. Barua and T.D. Goswami, the three distinguished and senior physicists of North 
East India were felicitated in the conference for their life long contribution to research 
and teaching in physics. 
The history of the PANE spans little less than a decade. It was inaugurated formally 
by renowned Physicist Prof. G C Deka on April 6,1998 in the premises of IIT, Guwahati 
with the aim of promoting basic research and physics education in North East India, 
besides Providing a platform of interaction among different groups of physicists 
working in different corners of North East India. The genesis of the academy can be 
traced to a seminar celebrating the hundred years of the discovery of electron held 
at NT, Guwahati on November 11,1997 when a small group of enthusiastic physicists 
of North East India led by Prof. K M Pathak and Y V G Murti gave a call to start such 
an organization. 
© 2008 IACS 
496 Foreword 
With these objectives, PANE organized the first conference of physics research 
at IIT, Guwahati during 1998, to be followed by the similar ones in Cotton College 
(2000), Dibrugarh University (2002) and G.C. College, Silchar (2004). The keynote 
speakers of these earlier conferences include persons like Prof. J.V. Narliker, Prof 
Asoke Sen (FRS), Prof. S. Duttagupta and Prof. K. D. Krori. 
On physics education sector, the academy arranged a workshop on new models 
of Post Graduate Programmes of physics at Tezpur University during 1999 followed 
by another on a theme "Physics Education in Undergraduate and Postgraduate levels 
now requires some Need Based Orientation" in the year 2005 jointly held at Gauhati 
and Dibrugarh University. 
The Academy has also been awarding talented young research scholars for their 
best presentations in the conferences since 2004. They include Ms. R. Gogoi, Mr 
S. Sengupta, Mr. P. Nath of Gauhati University and Mr. J. Choudhury of Assam 
Universiy, Silchar. 
We are thankful to the members of the advisory and organizing committees for 
their valuable suggestions for planning various aspects of the conference. We are 
also thankful to all the students and non-teaching members of this department for 
their significant contributions in organizing the conference. We acknowledge with 
gratitude the financial assistance of Gauhati University; DST, New Delhi; CSIR, New 
Delhi; BRNS, Mumbai; ASTEC, Guwahati; UCIL, Shillong; Bipin Pal Das Trust and 
Amiya Bala Bhattacharjee Trust. We express our heartfelt thanks to all the concerned 
persons at the respective funding agencies. We express our deep gratitude to the 
Vice Chancellor of Gauhati University for providing all possible help and constant 
guidance in holding the conference here at Gauhati University. 
Dilip Kumar Choudhury Buddhadeb Bhattacharjee 
Executive President, PANE Convener, LOC, 
Vth PANE Conference, 
Department of Physics, 
Gauhati University, 
Guwahati-781 014, 
Assam, India 
and 
Guest Editor 


Foreword

Baruah, K K

Indum J. Phvs. 82 (5), 513-517 (2008) 
riflloS 
Bianchi type-1 string cosmology with a scalar field 
K K Baruah 
Department of Mathematics, North Gauhati College, 
Guwahati-781 031, Assam. India 
E-mail kkbng-lc sifycom 
Abstract . The Bianchi type-1 anisotropic cosmological model containing a self - interacting scalar field with 
an exponential potential of the form V{&) - eM> m the context of cosmic strings have been studied Physical 
features of this model are briefly discussed 
Keywords Strings, Scalar field, Bianchi-1 cosmology 
PACS Nos. 98.90, 40 20 
1. Introduction 
The study of 'cosmic strings' has attracted a lot of attention, as they are believed to give 
a compelling and plausible description of the early stage of the universe. Gauge theories 
with spontaneous symmetry breaking in elementary particle physics have given rise to 
an intensive study of cosmic strings. It appears that after the big-bang the universe may 
have experienced a number of phase transitions. These phase transitions can produce 
vacuum domain structures such as domain walls, strings and monopoles [1]. Of all these 
structures, cosmic strings are of considerable interest. Letelier [2] studied a model of 
massive strings in the context of general relativity. He observed that the early stage of the 
universe was dominated by massive strings and during the evolution of the universe; the 
strings disappear leaving only particles. Self-interacting scalar fields play a central role in 
studies of inflationary cosmology [3-6]. It is observed that the scalar fields which drives 
inflation for F R W models will not make the anisotropic cosmological models inflate 
[3]. 
The objective of the paper is to study cosmic strings along with a self-interacting 
scalar -field with an exponential potential of the form V(&) ~ eM> , where k is a non-
negative constant to Bianchi-type-1 cosmology . This model has been chosen because 
© 2008IACS 
514 K K Baruah 
Bianchi type -1 cosmology is supposed to be a reasonable representation of the 
early universe . The scale factors are assumed to have power- law behaviour. This would 
give power- law inflation in an F R W model .We here show that unlike F R W model, the 
power- law solutions for the anisotropic model (here, we consider) shows neither inflation 
nor isotropization and further we propose to show that during the evolution of the universe 
the strings disappear leaving only particles and ultimately the scalar field helps the formation 
of galaxy. 
2. Einstein equations coupled to a system of strings and a scalar field 
The Einstein equations for a system of strings with scalar field are 
fl/iv " g 9nv R = PU/iuv - ^ xv + 9„0 dv0 - g^ \ - sl^K 0./* + W0) [ • (2 1) 
Here p is the rest energy density of the system of strings with massive particles 
attached to them . P = pp + K pp being the rest energy density of particles and / 
is tension density of the system of strings. The vector u^ describes the system 
four velocity and x^ represents a direction of anisotropy; 0 is a scalar field ana 
1/(0) is a potential function. We have here [2] 
u^u^ =-x^x^ =1 and i/x^ = 0 . (2 2) 
The contracted Bianchi identity for (2.1) is equivalent to 
V^j/^j-AxX = 0 (2 3) 
V^Ax^-pi/x^O 
H£(p(>v-A jr) = 0 
(2 4) 
(2 5) 
where xn' = x^VM(xv) and uv = i /%(u v ' ) . 
And H(f is the 'Projection operator1 that projects in the direction that are 
perpendicular to both x^ and ( / • The scalar field 0 satisfies the equation 
d^v<p-g^AlgaPQ,a0,p+V((t>) . = 0 (2.6) 
Bianchi type-1 string cosmology with a scalar field 515 
Equation (2 3), (2 4) and (2 5) are the evolutions for the system of strings These 
equations are also the integrability conditions for (2 1) The deceleration parameter q is 
given by 
q = -30* 9 + U2 
3 (2 7) 
where o - ua
 a , volume expansion of the fluid and ,' means the covanant derivative 
3 Field equations and their integrals 
The line element for Bianchi type -1 universe is 
cfe2 = dt2 - a2(t) (dx'f - 4(t) (dx2f-a2(t) (dx3)2 (3 1) 
where ay a2, a3 are functions of t only 
From equations (2 1), (2 2), and (3 1), we may write 
a" =u^ =(1,0,0,0) (3 2) 
d c) () 
And x^ must be taken along any of the three directions —y» —j» —j 
Without losing generality, we choose x'' parallel to
 1 , so that 
<)x 
x" = ( 0 , 3 / , 0,0.) (3 3) 
Applying equations (3 1), (3 2), and (3 3) in the Bianchi identities (2 3), (2 4) (2 5) and 
(2 6) one gets 
p + (p-X) Up 
a i 
a2
 +
 a3 
^
 a2 a3 
= 0 (3 4) 
and 0 0 + 
31
 t ^2 t a3 
( a i % ^ J 
0 2 -v(0) = O (3 5) 
Here the dot ( ) denotes the derivative with respect to / only Also it is to be noted 
that as a consequence of the Einstein equations p and A are functions of t only 
The Einstein field equations (2 1) for the metric (3 1) are 
A3&
 + ^ £ L = p +(V-.y ( 0 ) 
a i a2 a2 a3 a3 a i 
(3 6) 
516 K K Baruah 
a2 ^ ^3
 +
 a2 % = A -
a 2 a 3 a2 a3 
202 + V(0) 
a1 a3 a1 a3 
a i
 +
 a 2 ^ a i a 2 
= - | ^ 2 + v W 
r
^
2 +
 ^ ) 
a-j «2 a^  a2 
The solutions of the equations (3.5) to (3.9) are 
a, = f m , m > 0 
1^ 
&2 - 33 - t ,2m-1 m> 
A = (m-1)(5m-3)r 2 
m(m-1)(5m-3) ._2 
P = 5m-4 r 
2(7m2-8m + 2) 
5m-4 
log? 
1/(0) = - (5m-3x7m
2
-8m+2) 
(5m-4) 
= _4(m-1)2(5m-3)r2 
(5m-4) 
q = -
5(5m-2)3(m-1) 
r 
(37. 
(3.8. 
(3 9, 
(310, 
(311, 
(312. 
(313, 
(314, 
(315i 
(316) 
(3.17) 
The solutions (3.10) to (3.16) identically satisfy the equation (3.4). 
4. Discussion 
From solutions (3.10) to (3.17), it is observed that for m in the interval 3/5 < m< 4/5, 
X < 0, p > 0, pp > 0, q > 0 and $ is real. 
Bianchi type-1 string cosmology with a scalar field 517 
Also we have p^ - j ^ Z f > 1 for
 g < m< g (4 1) 
Thus from (4.1), we conclude that throughout the process of evolution, the universe is 
dominated by massive strings for 3/5 < m < 4/5 Since q is positive for 3 5 < m < 4 5 , 
there is no inflation at any stage Further, a, * a2 = a^ , so the universe is anisotropic 
throughout the process of evolution For m = (4+ V2) 7, 0 and 1/ vanish and the 
expansion of the universe is guided by a particular value of m 
We have considered Bianchi type-1 anisotropic model universe filled with self- interacting 
scalar fields In the solution of the Einstein equations, we have assumed that the scale 
factors exhibit powers of time enabling us to find exact solutions with the scalar field 
having an exponential potential It is observed that in the era of cosmic strings, the scalar 
field which led to inflation and isotropy in F R W models, do not lead to one in which 
anisotropy is present from the beginning We have seen in the solution of Bianchi type -
1 model that the scalar fields which, are used for F RW model, fail to inflate and isotropize 
in the early stage of the universe It is also interesting to note that due to the presence 
of scalar field, the power index lnf of the metric co-efficient has a range of values and 
when </> vanish 'rrf has a fixed value Thus we may conclude that the scalar field (</>) 
plays a significant role in the process of evolution of the universe 
References 
111 T W B Kibble J Phys A(9) 1387 (1976) 
[2] P S Leteher Phys Rev D(28) (10) 2414 (1983) 
[3] A Feinstein and J Ibanez Class Quant Grav 10 93 (1993) 
(4| G F R Ellis and M S Madsen Class Quant Grav 8 667 (1991) 
[5] M Heusler Phys Lett B253 33 (1991) 
[6| M S Madsen Class Quant Grav 5 627 (1988) 


Bianchi Type-1 String Cosmology with a Scalar Field

Meitei, Irom Ablu

Singh, K Yugindro

Indian J Phvs 82 (5), 519-522 (2008) 
The space distribution of QSOs 
Irom Ablu Meitei* and K Yugmdro Singh 
Department of Physics Manipur University Canchipur 
Imphal 795 003 Manipur India 
E mail iromablufr/ rediffmail com 
Abstract We present a study of the space density of the quasi stellar objects (QSOs) expressed in the 
form of the optical luminosity function (OLF) and its cosmological evolution with redshift using over 9600 QSOs 
n the 10k catalogue of the 2dF QSO Redshift Survey (2QZ) of the Anglo Australian Telescope (AAT) Two 
Degree Field (2dF) 
Keywords Luminosity function QSO lookback time comoving volume 
PACS Nos 95 75 Pq 95 80 +p 95 85 Kr 98 54 h 98 54 Cm 
1 Introduction 
The QSOs being a sub-class of the active galactic nuclei (AGN), the QSO OLF and its 
evolution with redshift provide valuable information on the study of the AGNs, in general 
In this paper we present the OLF of over 9600 QSOs in the 2QZ 10k catalogue in the 
magnitude range -13 < MB< -28 The evolutionary scenario of the QSOs with redshift is 
also discussed We have analysed the 2QZ 10k catalogue using the Data Interface Tool 
developed under the Virtual Observatory India (vo-i) initiative Details of the catalogue can 
be found in [1] 
2. Results 
2 1 Cumulative number-magnitude distribution, N(b): 
The cumulative number-magnitude distribution of the QSOs is shown in Figure 1 The 
steeper slope at the brighter end (dlogNfb^ = 1 72) indicates that the population of QSOs 
is evolving with time 
Corresponding Author 
© 2008 IACS 
520 Irom Ablu Meitei and K Yugindro Singh 
1000 r 
100 k + + 
«-x 10 
z 
g1 
Figure 1. Cumulative number-magnitude distribution, N(b) of 2QZ 10k QSOs with 0 01 < z <- 4 0 
2.2. Number-redshift distribution 
The number-redshift distribution shown in Figure 2 also shows that the QSO population is 
evolving with time with maximum number of QSOs at z - 1 6 It may be inferred that 
either we have seen an epoch of QSO formation (density evolution), or the OLF of the 
QSOs evolves with time (luminosity evolution), or both The number of QSOs in the 
600 
O 
CO 
O 
Figure 2. Number-redshift distribution for 2QZ 10k catalogue QSOs with 0.01 < z < 4 0. 
The space distribution of QSOs 521 
catalogue plotted as a function of the lookback time is shown in Figure 3 It shows that 
the QSOs might have appeared when the age of the Universe was 10 per cent of the 
2 
present age which is taken as ——, H0 being the Hubble constant 
1000 r 
03 04 05 06 07 08 
Lookback time (in units of the age of Universe) 
1 0 
Figure 3 Number of 2QZ 10k QSOs as a function of lookback time (in units of the age of the Universe) 
2 
for qf = 0 5 , H = 50 km s 1 Mpc 1 The age of the Universe = —— 
3rVrt 
le-08
 r 
1e-09 
1e-10 
le-11 
1e-12 
Figure 4. The optical luminosity function for 2QZ 10k QSOs with -28 < MB < -18 and z upto 4 0 
522 Irom Ablu Meitei and K Yugindro Singh 
2.3 QSO optical luminosity function: 
The optical luminosity function of the QSOs with -28 < MB < -18 and z upto 4o i$ 
shown in Figure 4 
3. Conclusion 
The QSO population is evolving with time. A proper estimation of the OLF of the QSOs 
will help in understanding the nature of its evolution and of the AGN population as h 
whole 
Acknowledgment 
The 2QZ was based on observations made with the Anglo-Australian Telescope and UK 
Schmidt Telescope We would like to thank vo-i. We are also indebted to Dr Ranjeev 
Misra and Prof. Ajit Kembhavi of IUCAA, Pune. Further, I. Ablu Meitei acknowledges the 
CSIR for providing financial assistance as a Senior Research Fellow (SRF). 
References 
[1] S M Croom, R J Smith, B J Boyle, T Shanks, N S Loanng, L Miller and I J Lewis MNRAS 322 U 
(2001) 
[2] B J Boyle, T Shanks, S M Croom, R J Smith, L Miller, N S Loanng and C Heymans MNRAS 317 
1014 (2000) 


The Space Distribution of QSOs

Konwar, K

Gogoi, P

Baishya, B

Naphthacene and Pentacene Based Organic Thin Film Transistors with La<SUB>2</SUB>O<SUB>3</SUB> as the Gate Insulator

Wary, G

Rahman, A

Indian J. Phys. 82 (6), 723-728 (2008) 
wm 
^^m^ 
Comparative studies on optical parameters of CdTe 
and ZnO thin films 
G Wary1* and A Rahman2 
1
 Department of Physics, Cotton College. Guwahati-781 001, Assam, India 
2
 Department of Physics, Gauhati University, Guwahati-781 014, Assam, India 
E-mail • ganesh_wary(f/yahoo co in 
Abstract : The optical parameters of vacuum evaporated (p)CdTe and (n)ZnO thin films were studied and 
compared using uv-visible spectrophotometer equipped with an integrating sphere The main optical parameters 
studied are. transmittance, reflectance, absorbance, refractive index, extinction co-efficient etc Comparison at 
wavelength 500 nm shows that refractive index 'ri as well as imaginary part of dielectric constant, V 'of 
(n)ZnO films are smaller than that of (p) CdTe films which indicates the more transparency of (n)ZnO films and 
more efficient semiconductor for using as transparent conducting oxide (TCO) in most of the junction devices 
Static (dc) electrical conductivity is observed to increase slowly towards the IR spectrum as imaginary part of 
refractive index 'k' increases towards higher wavelength 
Keywords : Activation energy, complex dielectnc constant, extinction coefficient. TCO 
PACS Nos. : 71.55.Gs, 78.20.Ci, 78.40.Fy 
1. Introduction 
ZnO is found as a good candidate of transparent conducting oxide (TCO) and therefore 
can substitute indium tin oxide (ln203: Sn) and tin oxide (Sn02) in conductive electrodes 
of other semiconductor solar cells. Due to its high stability [1] in hydrogen plasma, strain 
free, nontoxicity, controllable resistivity and low cost of the constituent elements make 
ZnO very attractive for photovoltaic applications [2]. On the other hand CdTe (energy gap 
[3] =1.5 eV) an ll-VI intermetallic compound is a direct band gap semiconductor. It has a 
high absorption co-efficient, which is required for good theoretical conversion efficiency. 
This has led to the investigation of CdTe based heterojunctions and Schottky Barrier 
devices for using as photovoltaic (PV) devices. The l-V characteristics under illumination 
of the ZnO/CdTe/Ag junction were reported [4] without PV efficiency calculation. Sputter-
Corresponding Author 
©2008IACS 
724 G Wary and A Rahman 
deposited (n)ZnO on (p)Zn3P2 devices have been reported [5] with PV efficiencies of only 
2.0%. A thin film ZnO/p-CdTe photovoltaic device, which showed an efficiency of 3.7% 
was reported [6]. Recently PV efficiency of vacuum deposited (p) CdTe/(n)ZnO 
heteorojunction has been reported [7,8] to be 4.01% using hydrogenation for Na = 1.03 x 
1015 cm"3 A/d = 9.95 x 1014 cm'3. 
2. Experiment 
Films of equal area (1.5 x 1.5) cm2 of pure CdTe powder (99.999%, Aldrich Chem. Co., 
USA) mixed with 2.53% of Sb and ZnO powder (99.995%, Aldrich Chem. Co., USA) 
mixed with 4.08% of Al were deposited by thermal evaporation from a molybdenum boat 
on chemically cleaned glass substrates under pressure 10~5 Torr. The substrate temperature 
was maintained at 423 K at the time of film deposition. The films were then annealed for 
5 hours allowing maximum temperature of 528 K and maintaining the pressure at 10~4 
Torr. For transmittance, absorbance and reflectance measurement, Cary-300 Scan uv-
visible spectrophotometer (Varian, Victoria-3170, Australia) was used. The measurement 
of the type and carrier concentrations were carried out by the traditional technique involving 
Hall effect and C-V measurements on the structures of AI/(p)CdTe and Sn/(n)ZnO Schottky 
junctions were also made. 
3. Result and discussion 
The optical transmittance (T) and absorbance (A) spectra of a typical (p)CdTe and (n)ZnO 
annealed films have been shown in Figure 1 and Figure 2 respectively. In both the cases 
absorbance has its maximum value at the higher energy and decreases with lower energy 
towards the visible region. 
Absorption coefficients in the interference-free region were calculated from relation [9], 
z r = exp[a(f2 - f1 ) ] (1) 
where, T^ and T2 the transmittances for the samples of thickness ^ and f2 respectively 
and a, the absorption coefficient. Since CdTe is a direct band gap semiconductor, its 
band gap was determined from (hva)2 versus hv plots using relation [10], 
ahv = B(hv-Eg)V2 (2) 
Figure 1(c) gives a typical plot of (ahvf versus hv for (p)CdTe film from which 
extrapolation of data to the {ahvf ->0 axis gives the band gap energy Eg = 1.564 eV 
and for annealed (n)ZnO film, this value is 3.35 eV [Figure 2(c)]. 
Values of refractive index (n) of (p)CdTe thin film in the interference zone at all 
transmittance minima [Figurel(a)] were determined from the relation [11], 
Tmjn = 6n2/(n4 + 3.25n2 + 2.25) (3) 
Comparative studies on optical parameters of CdTe and ZnO thin films 725 
and in the interference-free region by extrapolation of the n values obtained in the interference 
region using Cauchy's dispersion relation, 
n=A + B/k2 (4) 
where, A and 8 are constants and can be obtained from the n versus 1 A2 plots fFioure 
1(d) and 2(d)]. 
From the knowledge of a , the extinction coefficient k(k = oX An) and then real and 
400 500 600 700 
Wavelength (nm) 
800 400 800 500 600 700 
Wavelength (nm) 
Figure 1(a). Transmission spectra of annealed Figure 1(b) Absorption spectra of annealed 
(p)CdTe film (p)CdTe film 
1 4 1 6 1 2 0 2 2 2 4 2 6 2 8 3 0 3 2 
X(nm) 
4 0 
\2(\xm) 
Figure 1(c). (tva)2 versus tv plots of (p)CdTe Figure 1(d). n versus x* plots of (p)CdTe film 
film 
•160 
22-T 
450 500 550 600 650 700 750 800 
X(nm) 
Figure 1(e). n and k versus A plots of (p)CdTe film 
726 G Wary and A Rahman 
imaginary parts of the dielectric constants e' and e* were calculated using relations 
[12], 
E' = rf-k 2 u2 
and 
respectively 
e' = 2nk Ana 
ID 
(5) 
(6) 
(n)lnO 
001 -J 
0 00 
(n)ZnO 
600 000 
Wav«l«ngth (ran) W»veH»noth (nm) 
Figure 2(a). 
(n)ZnO film 
Transmission spectra of annealed Figure 2(b). Absorption spectra of annealed (n)ZnO 
film 
4 6 -
40-
E 35-
> 30-
* 25-O 
O 20-
5 -J 
0 -
e Unannealed film 
-Annealed film 
(n)ZnO 
E -3 32 eV 
/ 
E8«3 35«V -fd 
—r—, , , . r-^ . , . . . , , ,—J 
1 73 
1 72 
1 71 
1 70 
1 69 
1 68 
1 6* 
1 66 -j 
1 65 H 
1 64 
(n)ZnO I 6 6 + ( 0 1 K / X
1 ) 
2 6 3 0 3 5 
hv (cV) 
4 0 4 5 6 7 
X 2 (^ im) 2 
Figure 2(c). 
film 
(Aim)2 versus rv plots of (n)ZnO Figure 2(d). n versus K2 plots of (n)ZnO film 
1 71 
1 70 
1 69 H 
1 68 A 
1 67 
0 07 
0 06 
0 05 
0 04 
350 400 450 500 550 600 650 700 
A.(nm) 
Figure 2(e). n and k versus X Plots of (n)ZnO film 
o 03 
Comparative studies on optical parameters of CdTe andZnO thin films 727 
Where, co is the angular frequency of the incident light and o is the static (dc) 
electrical conductivity 
Variations of n and k with wavelength have been shown in Figure 1(e) and Figure 2(e) 
Normal dispersion is pronounced in both the cases The change of imaginary part of 
refractive index (K) with wavelength is found to be interesting phenomena in (p)CdTe films 
Imaginary part of refractive index, k (and so, imaginary part of dielectric constant,
 t" ) is 
observed to increase slowly up to the wavelength 550 nm, slows down to 625 nm and 
then increases rapidly towards the IR region [Figure 1(e)] However, it increases slowly tn 
case of ZnO films These show that static (dc) electrical conductivity of both (p)CdTe and 
(n)ZnO films increase towards the IR spectrum[12] Values of some optical parameters of 
typical (p)CdTe and (n)ZnO films are shown in Table 1 and 2 respectively The comparison 
chart of some optical parameters of these films for same order of doping concentration 
and at same wavelength (500 nm) has been shown in Table 3 Companson at wavelength 
Table 1 Values of some optical parameters of a typical (p)CdTe film (doping concentration Nt - 2 87 x 
1016 cm-3) for film thickness 5304 A 
k 
(nm) 
400 
450 
500 
550 
600 
650 
700 
800 
T% 
27 324 
54 336 
75 436 
78 134 
70 232 
80 054 
75 587 
84 044 
n 
2 838 
2 504 
2 320 
2 120 
1 997 
1 948 
1 725 
1 725 
(/ 
(104 cm 1) 
3 10 
3 06 
2 83 
2 60 
2 33 
2 18 
2 24 
2 44 
Table 2. Values of some optical parameters of a 
1017 cm3) for film thickness 3050A 
X 
(nm) 
350 
400 
450 
500 
550 
600 
650 
T% 
89 286 
94 805 
95 129 
95 779 
96 428 
96 429 
97 362 
n 
1 710 
1 698 
1 690 
1 684 
1 680 
1 677 
1 674 
a 
(104 cm"1) 
1 364 
1 319 
1 285 
1 257 
1 211 
1 152 
1 121 
k 
(10"* ) 
98 676 
109 578 
112 602 
113 796 
111 249 
112 761 
124 777 
155 335 
typical (n)ZnO 
k 
0 038 
0 042 
0 046 
0 050 
0 053 
0 055 
0 058 
/ 
8 045 
6 744 
5 369 
4 481 
3 976 
5 095 
3 972 
2 952 
film (doping 
i 
2 922 
2 881 
2 853 
2 833 
2 819 
2 809 
2 799 
i" 
0 560 
0 544 
0 522 
0 482 
0 444 
0 509 
0 498 
0 536 
E<,l*V) 
1 56 
concentration Nd = 2 28 x 
t" 
0 129 
0 143 
0 155 
0 169 
0 178 
0 184 
0 194 
£ff(eV) 
3 35 
728 G Wary and A Rahman 
500 nm shows that refractive index n as well as imaginary part of dielectric constant,
 t* 
of (n)ZnO films is smaller than that of (p)CdTe film which indicates more transparency of 
(n)ZnO film. This indicates that (n)ZnO is more efficient semiconductor for using as 
transparent conducting oxide (TCO) as well as window layers in most of the junction 
devices. 
Table 3. Comparison of some optical parameters of typical (p)CdTe, and (n)ZnO films (for same order of 
doping concentration and in same wavelength =500 nm). 
Sample 
(p)CdTe 
(n)ZnO 
N 
(1016cnr3) 
2.87 
2.88 
a 
( *T 1 cm'1) 
8.47 x 10-8 
3.37 x 1(T7 
a 
(cm"1) 
2.83 x 104 
1.26 x 104 
n 
2.32 
1.68 
e' 
5.37 
2.83 
f ' 
0.52 
0.17 
4. Conclusion 
Optical parameters of vacuum deposited CdTe and ZnO thin films have been determined 
and compared. Higher absorption coefficient has been observed in (p)CdTe film, which is 
required for a good conversion efficiency and has led to the investigation of CdTe-base 
heterojunctions and Schottky barrier devices for its use as a photovoltaic (PV) device. On 
the other hand lower refractive index of (n)ZnO film in comparison to (p)CdTe indicates 
that (n)ZnO is more efficient semiconductor for using as transparent conducting oxide 
(TCO) in most of the junction devices. 
References 
[1] B Joseph, K G Gopchandran, P K Manoj, P Koshy and V K Vaidyan Bull. Mater. Sci. 22 921 
(1999) 
[2] K L Chopra, S Manoj and D K Pandya Thin Solid Film 102 1 (1983) 
[3) J J Lofesrki J. Appl. Phys. 27 777 (1956) 
[4] B R Mehta, S Kumar, K Singh and K L Chopra Thin Solid Films 164 265 (1988) 
[5) K W Mitchell Physics of Solar Cells, (eds.) S C Join, S Radhakrishna and T R S Reddy (1st 
edition) (International Council of Scientific Union, Madras, India) 159 (1984) 
[6) M S Tomar Thin Solid Films 164 295 (1988) 
[7] G Wary, T Kachary and A Rahman Intl. J. Thermophysics 27 332 (1) (2006) 
[8) G Wary, T Kachary and A Rahman Indian J. Pure & Appl. Phys. Vol. 44 754 (2006) 
[9] F Abeles Optical Properties of Solids 23 (Amsterdam : North Holand Publishing Co.) (1972) 
[10] J I Pankove Optical Processes In Semi-conductors pgs. 36, 94 (New York : Dover Publications, 
Inc) (1971) 
[11] I Martil and G Gonzalez Diaz Am. J. Phys. 60(1) 83 (1992) 
[12] K L Chopra Thin Film Phenomena pgs. 729, 734 (New Work : MacGraw Hill Book Co.) (1969) 


Comparative Studies on Optical Parameters of CdTe and ZnO Thin Films

Indian J. Phys. 82 (6), 809-812 (2008) 
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Foreword

Bianchi Type-1 String Cosmology with a Scalar Field

The Space Distribution of QSOs

Naphthacene and Pentacene Based Organic Thin Film Transistors with La<SUB>2</SUB>O<SUB>3</SUB> as the Gate Insulator

Comparative Studies on Optical Parameters of CdTe and ZnO Thin Films

Notes for Contributors

Advisory Board and Editors

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