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Dielectric critical behaviour at a metal–insulator transition
under lattice compression
Shailesh Shuklaa, Deepak Kumara, Nitya Nath Shuklab, Rajendra Prasadb,*
aSchool of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, IndiabDepartment of Physics, Indian Institute of Technology, Kanpur 208016, India
Received 28 December 2004; revised 15 March 2005; accepted 15 March 2005
Abstract
We study dielectric critical behaviour around a continuous metal–insulator transition in crystalline Cesium Iodide induced by changing the
lattice parameter. The ab initio calculations of band structure and various quantities related to the dielectric response are performed in the
transition region, within the local density approximation of the density functional theory. These calculations allow us to establish the power-
law singularities of various quantities on two sides of the transition. The exponents obtained here are mean-field like due to the approximation
in which interactions and disorder are treated. The critical behaviour is discussed by applying the scaling principle to the wavevector and
frequency dependent dielectric function. We further investigate the effect of dielectric anomalies on optical properties by calculating the
reflectance around transition region taking the ionic contribution to the dielectric function also into account. We find that the reflectance as a
function of frequency shows very different kind of behaviour on both sides of the metal–insulator transition.
q 2005 Elsevier Ltd. All rights reserved.
PACS: 71.30.Ch; 71.20.Mq; 71.20.Nr
Keywords: D. Dielectric properties; D. Critical phenomena; A. Electronic materials; D. Optical properties
1. Introduction
Metal–insulator transition in solids under variation of
lattice parameter has been a subject of great theoretical
interest for a long time [1,2]. According to Mott, solids with
odd number of valence electrons like H, Na, etc. will
undergo transition from metal to insulator on lattice
expansion due to electron correlation effects [3]. Solids
with even number of electrons per unit cell can also show
metal insulator transition due to opening or closing of band
gaps at the Fermi level when their lattice parameter is varied
[3,4]. In this paper we study one such transition, which
occurs through the band overlap process and analyse
dielectric anomalies around the transition. Since one is
dealing with a quantum phase transition, one expects certain
universal features, which arise due to large scale spatial and
0022-3697/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jpcs.2005.03.005
* Corresponding author.
E-mail address: [email protected] (R. Prasad).
temporal quantum fluctuations that accompany a continuous
zero temperature transition [1,5,6].
There has been a tremendous amount of work toward
understanding the critical behaviour around metal–insulator
transition (MIT). The first set of ideas were introduced
through the work of Wegner [7] and Abrahams et al. [8] in
the context of disorder-induced metal insulator transition
(MIT) for noninteracting electrons. Since interactions are
inevitably present in any physical system, the work
following these pioneering papers has naturally focussed
on quantitative understanding of MIT in interacting systems
[1,2,9,10]. Much of this activity has focussed on Mott–
Hubbard transition induced by short ranged interaction as in
Hubbard Model. Our aim in this work is to complement such
studies by analysing a MIT associated with the closing of a
one-particle gap induced by pressure in a realistic system.
We should emphasize that this transition is being driven by
the one-electron properties of the periodic lattice system.
Though both interactions and disorder play a significant role
in determining the properties of the transition, neither
actually drives the transition. Nevertheless this MIT is
associated with many dielectric anomalies, which we
Journal of Physics and Chemistry of Solids 66 (2005) 1150–1157
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S. Shukla et al. / Journal of Physics and Chemistry of Solids 66 (2005) 1150–1157 1151
explore in the mean field spirit, in this paper. Such critical
properties have only been partially investigated in the
context of metal–insulator transitions induced by disorder or
interactions as in Mott–Hubbard models, due to inherent
limitations of the models studied. Thus our study provides a
broader base of physical properties that can be examined
theoretically and experimentally in the context of MIT.
In our formulation, the fundamental quantities are the
dielectric constant and the conductivity, which are exper-
imentally measurable quantities. This allows us to consider
long range Coulomb interaction, its screening and more
generally the critical behaviour associated with wavevector
and frequency dependent dielectric function. We apply the
principles of spatial and temporal scaling to it and obtain a
set of four exponents, which characterise the transition.
Scaling ideas have by now become a standard tool to
analyse continuous phase transitions [11]. Their application
to continuous quantum phase transitions has been recently
reviewed by Sondhi et al. [5]. Some key physical features of
the transition associated with opening (closing) of the one-
particle gap are different from the disorder-induced
Anderson transition, so there are natural differences in the
scaling relations formulated here from those in the work of
Wegner, Abrahams et al.
We have chosen to study Cesium Iodide solid which is
experimentally well known to show insulator–metal
transition under pressure [12]. Several first-principle
electronic structure calculations for CsI exist, which
support the transition by the band overlap mechanism
under lattice compression. CsI has a direct band gap, and
the calculations show that it undergoes the transition by
closing of the direct gap. At this point it is relevant to
comment on the possibility of the excitonic phase, which
may exist in this situation in which the MIT occurs due to
the closing of a band gap. Mott first pointed out that such a
transition cannot be continuous due to the following
reason. As the MIT is approached from the metallic side
the screening gets weaker due to decreasing density of free
electrons and holes, and at some point this gas becomes
unstable to the formation of bound pairs [3]. For our
purpose an argument due to Knox, who considered the
situation from the insulating side, is of direct relevance.
The binding energy of the exciton is given by e4mr=320 [13],
where mr is the reduced mass and 30 denotes the static
dielectric constant. The exciton level lies within the energy
gap. The key point is how this level changes as the gap
closes. If the binding energy becomes larger than the gap
there is an instability toward exciton formation. This
possibility which has been extensively studied can be
realized when 3 variation is not significant as can happen
when an indirect band gap closes. This leads to interesting
intervening phases with charge and spin density waves. In
our case, the dielectric constant diverges due to the closing
of a direct gap and one needs to consider the relative
variation of the energy gap and the exciton binding energy.
We shall take up this point in a later section.
We have done ab initio calculations of various quantities
like dielectric constant, band gap, conductivity, density of
states at Fermi level, etc. for Cesium Iodide in CsCl
structure around the transition region. The calculations are
done in the local density approximation, so that the
exchange and correlation effects have been treated in a
mean field approximation. We have also included
disorder in our calculations of dc conductivity as well as
the frequency dependent dielectric constant. However, the
disorder has also been included in the Drude like
approximation. So any effects like localization due to
disorder or weak-scattering corrections have been ignored.
From the analysis of the data, we have extracted the critical
behaviour of the quantities associated with the dielectric
response. The approximate values of various exponents
characterising the transition have been determined. Since in
our calculations the interactions are treated at a mean field
level, we expect the exponents to be simple fractions. Thus
we believe that the simplest fractions closest to our
numerical results are the correct exponents. We also explore
the effect of dielectric anomalies on infra-red (IR) optical
properties by calculating the reflectance in the transition
region. As the incident photons also interact with the
vibrational modes of the ionic crystal we take ionic
contribution to dielectric function also into account. This
is found to yield interesting features in the spectrum which
should be observed in optical measurements.
The organisation of this paper is as follows. In Section 2,
we deal with anomalies in the dielectric function and its
scaling analysis near the transition. This is followed by
Section 3 containing details of our calculation. In Section 4,
we present our results and its discussion. Finally, we
conclude this work in Section 5.
2. Scaling analysis for the dielectric function
The wavevector-frequency dependent dielectric function
of a solid, 3ððq;u; aÞ provides a unified description of charge
response of both metals and insulators. Here a denotes the
lattice constant, and is the drive parameter for the transition.
We first consider static dielectric function 3ððq; 0; aÞ. In
metals, due to screened coulomb interactions, 3ððq; 0; aÞdiverges as (1/L2q2) in limit ðq/ ð0. Here ‘L’ is the
screening length, which in Thomas Fermi Approximation
is given by LK2 Z4pe2NðEfÞ, where N(Ef) is the density of
states at Fermi level, Ef. In insulators as ðq/ ð0, 3ððq; 0; aÞgoes to a finite limit 30(a), the static dielectric constant. In
approaching the critical point denoted as ac, L and 30 are
expected to diverge according to power laws in
tZ ðaKac=acÞ. To relate power laws on the two sides of
the transition, we follow the scaling arguments due to Imry
et al. [14], which were developed in the context of the
disorder-induced MIT (Anderson transition). Here the basic
assumptions are that (i) there exists a characteristic
correlation length, x in each phase which diverges as
S. Shukla et al. / Journal of Physics and Chemistry of Solids 66 (2005) 1150–11571152
the transition is approached,
x Z x0jtjKn (1)
(ii) the static dielectric function obeys the homogeneity
relation,
3M=Iðq; 0; aÞ Z x2KhfM=IðqxÞ (2)
where the subscripts M and I refer to metallic and insulating
phases, respectively. Behaviour of f(x) for small x in the
metallic and insulating phases can be obtained from the
above considerations; fMðxÞf ð1=x2Þ for a metal and fI(x)fconstant for an insulator. To obtain large x behaviour, we
adopt the physical argument proposed by Imry et al. [14]
that at length scales much smaller than x, the dielectric
behaviour in the two phases should be independent of x.
This requires that at large x, fM=IðxÞf ð1=x2KhÞ. From the two
asymptotic behaviours one obtains the divergence beha-
viours for screening length and static dielectric function in
the critical region as
L2ðxÞfxh f jtjKnh (3)
and
30ðaÞfx2Khf jtjKnð2KhÞ (4)
Next we consider the frequency dependent properties.
The scaling with respect to frequency should be important,
as the transition is associated with an energy scale related to
the band gap. The imaginary part of dielectric function, 3 00 is
related to the real part of conductivity s 0 according to
300ððq;u; aÞ Z4ps0ððq;u; aÞ
u(5)
Let us consider various quantities at zero wavevector and
in small frequency limit. While limu/0s 0(0,u,a)Zsdc, the
dc conductivity in metals, in insulators one finds that
3 00(0,u,a) vanishes as Au in the small frequency limit when
a finite lifetime to the electronic states is allowed. We
assume, following the principles of dynamical scaling, [11]
that (i) there exists a frequency scale, U, in each phase
which vanishes as the critical point is approached according
to a power law in t, and may be written as,
U Z U0xKz (6)
where z is termed as the dynamical exponent, (ii) the
dielectric function satisfies the homogeneity relation with
respect to the frequency scale U,
300M=Ið0;u; aÞ Z UygM=I
u
U
� �(7)
In order to conform to the low frequency behaviour of the
conductivity and the dielectric constant as discussed above,
as x/0, gI(x)fx for insulators, and gMðxÞf ð1=xÞ for
metals. To determine the large x behaviour of g(x), we again
invoke the argument that for frequencies, u[U the
dielectric behaviour for both the phases is independent of
U. This specifies the form gM/I(x)fxy in large x limit.
It follows from the two asymptotic behaviours that the
divergence of the dielectric function while approaching the
critical region from insulating side is according to,
300ð0;u; aÞfuUyK1 fujtjKnzð1KyÞ (8)
Similarly the conductivity behaviour on approaching
transition from metallic side is obtained as
sdcðaÞfUyC1f jtjnzðyC1Þ (9)
Clearly these conclusions require K1!y!1 to be
satisfied. The four exponents n, h, y, and z fully characterise
the metal–insulator transition driven by lattice parameter.
At this point it is pertinent to mention that the above
discussion is specific to transition driven by the change of
lattice constant and for systems with some disorder so that
the conductivity is finite in the metallic phase. We envisage
a situation of weak disorder so that the transition is not
driven by disorder in three-dimension. Thus our analysis is
different from the scaling analysis for pure systems in which
the transition is driven by interaction parameter or by the
filling fraction [1,2] or the disorder driven-Anderson
transition [7,8].
3. Method of calculation
The macroscopic dielectric function of the crystal is
calculated ab initio using the expression (for uO0) [15]
300ðuÞZ4pe2
m2u2
Xðk ;i;j
jhi;ðkjpxjðk; jij2f
iðk ð1Kfjðk ÞdðEjðk KE
iðk KZuÞ
(10)
It gives the diagonal term of the imaginary part of the
dielectric function for the crystal in RPA, ignoring the
local field effects. Here i and j indicate the band indices, ðkthe wavevector, jikO is the eigenstate with wavevector ðkand band index i, E
iðk , the corresponding eigenvalue, and
fiðk is the occupation number, ‘px’ the x component of
momentum transfer. Through Kramer–Kronig inversion
of 300(u), the calculation of dielectric constant, 30Z30(u)ju/0, where 30(u) is the real part of dielectric
function, becomes possible with an accuracy limited by
the frequency range of our calculation. We allow a finite
life-time ‘GK1’ to the transition between states, by
replacing the delta function in the above expression by
a Lorentzian of width G in the numerical calculation. In
realistic systems, one expects the single-particle states to
have finite lifetimes. The contribution to the life-times
come from disorder as well as electron–electron inter-
actions [16]. The latter are ignored in the RPA
approximation. This procedure was introduced earlier by
Philipp and Ehrenreich [17], and was reasonably success-
ful in achieving agreement with the optical data in some
semiconductors.
S. Shukla et al. / Journal of Physics and Chemistry of Solids 66 (2005) 1150–1157 1153
To calculate the dc conductivity, we use Eq. (5) and take
the limit u/0 in Eq. (10) to obtain [18]
sdc Ze2
3G
Xn;ðk
v2
n;ðkdðE
n;ðk KEfÞ (11)
where ðvn;ðk Z ðVE
n;ðk , is the group velocity for the eigenstate
with quantum numbers n; ðk. Note that in this approximation,
the single-particle relaxation rate instead of the transport
relaxation rate occurs in the expression for the conductivity.
Here this only changes the conductivity by a constant factor.
In addition to these quantities, we calculate the direct
band gap, Eg, and the density of states at the Fermi level,
N(Ef), in insulating and metallic sides, respectively. For all
the numerical calculations the Full Potential Linearised
Augmented Plane Wave method [19] incorporated in
WIEN2k package by Blaha and co-workers [20] is used.
The method is a highly accurate scheme for electronic
structure calculations in solids with the following principal
features (i) The unit cell is divided into two parts: atomic
spheres and interstitial regions, which are described by
atomic like functions and plane waves, respectively. A
suitable linear combination of them is used as the basis set
for expanding Kohn–Sham orbitals. (ii) A modified
tetrahedron method is used for the Brillouin zone inte-
grations (iii) Exchange-correlation effects are treated within
the density functional theory using the Local Spin Density
Approximation (LSDA) [21].
4. Results and discussion
4.1. Electronic properties
Cesium Iodide has a high compressibility and a band gap
(w6 eV), which is the smallest among the alkali halides. CsI
Fig. 1. Band structure of CsI along principal symmetry directions at lattice consta
begins (aZ6.59 a.u.).
has also been subjected to experimental study of metalliza-
tion under pressure. During compression intermediate
phases with different structural symmetries occur at
different pressures. But it is shown experimentally as well
as supported theoretically, that even allowing for structural
transformations, CsI becomes metallic due to overlap of
filled 5p like Iodine band with the empty 5d like Cesium
band at the G—point in the Brillouin zone at a pressure
z100 GPa [12,22,23]. This is equivalent to a volume ratio
of z.50. Such a symmetric and controlled change in
electronic structure of CsI with pressure, in spite of the fact
that ionic character decreases on compression, makes it
ideally suited to our purpose.
Our results for the band structure calculation are shown
in Fig. 1. The calculations have been done in a relativis-
tically self-consistent manner by varying the lattice
constant. The two panels on left and right show the energy
band dispersion along the principal symmetry axes in
the Brillouin zone at equilibrium (8.63 a.u.) and at the
point of metal–insulator transition (MIT) (6.59 a.u.).
The equilibrium band gap is underestimated (3.6 eV)
compared to its actual value of 6.3 eV. This is not surprising
since it is well known that the DFT-LDA always under-
estimates the band gap. The MIT occurs at a lattice constant
equal to 6.59 a.u. This is equivalent to a metallization
volume of 0.44 of the uncompressed volume which is in
good agreement with previous calculations [12,22–24]. It is
interesting to note that the transition volume obtained here is
rather close to the Herzfeld’s polarization catastrophe
criterion, which gives a fractional volume of 0.42.
The results of the calculations of 3 00(u) at small
frequencies in lattice parameter range 6.63–6.605 a.u. are
shown in Fig. 2. The linear behaviour with the frequency (in
the limit u/ 0) and the increasing slope with the approach
to the critical lattice parameter is clearly seen in the figure.
The variation of the slope A in the linear region with t, the
nts corresponding to (a) equilibrium (aZ8.63 a.u.) (b) when metallisation
6.63 a.u.
6.61 a.u.
6.62 a.u.
6.615 a.u.
ω (eV)
ε"(ω
)
6.605 a.u.
0
2
4
6
8
10
0 0.2 0.4 0.6 0.8 1
Fig. 2. Imaginary part of dielectric function of CsI versus frequency for
lattice parameters between 6.605 a.u. (top curve) and 6.63 a.u. (last curve at
the bottom) for GZ0.02 eV.
t (10–3 units)
Eg
(10–3
Ryd
)
2
6
10
14
18
22
3 5 7 9 11 1
Fig. 4. Band gap variation as a function of t. The dashed line shows the
power law fit to t.92.
σ dc
(a)
0.2
0.4
0.6
S. Shukla et al. / Journal of Physics and Chemistry of Solids 66 (2005) 1150–11571154
distance from the critical point is shown in Fig. 3a. The
analysis of this data (Fig. 3a) shows a power-law variation,
A(a)fjtjKx. In order to confirm this behaviour and
determine the power law exponent, x, we have done this
calculation by varying the dissipation parameter, G, from
0.1 to 0.002 eV. We find that the exponent increases with
decreasing G, but reaches an asymptotic constant value of
2.40G.03 at GZ0.02 eV. We take this to be the correct
value in consistency with the RPA approximation. The error
limit for the exponents quoted here are the ranges over
which we obtain the best fits of the available data. By using
the dispersion relation, we have also estimated the static
dielectric constant. Its variation with t is shown in Fig. 3b.
The exponent associated with its divergence is .54G.02.
However, this value has a bigger error margin than other
exponents obtained in our computation due to a finite
frequency range of integration in the dispersion relation. In
Fig. 4, we plot the band gap on insulating side around
transition point with respect to critical point distance. It is
seen to vanish as jtj.92 with error limits of the exponents G.02. Note that the value of the exponent for the static
A(a
)
ε 0
t (10–3 units)
(a)
(b)
0
4
8
12
16
4 6 8 10 2 0
4
12
16
8
Fig. 3. (a) Slope of the curves shown in Fig. 2 in region u/0 as a function
of tZ ðjaKacj=acÞ. The dashed line shows the power law fit by tK2.4. Note
the Y-scale for this curve is on left as shown by arrow. (b) Dielectric
constant of CsI as a function of tZ jaKacj=ac. The dashed line shows the
power law fit by tK.54. Y-scale for this curve is on the right.
dielectric constant is inconsistent with the expectation based
on the Penn’s relation 30fEK2g [26].
In Fig. 5a, we present the results for the calculation of the
dc conductivity as a function of lattice constant. One finds
that it also has a power law behaviour sdcfjtj2.1 with G.12
as the error on exponent. The variation of density of states
(DOS) at Fermi level on metallic side is shown in Fig. 5b,
and it varies around the critical point as jtj1.1 within G.14
limits of exponent. It is interesting to note that the exponents
for the conductivity and DOS are different. In this transition,
which is driven by band overlap and hence the DOS at
Fermi level, one expects the conductivity to follow
the exponent for DOS. However, the difference in
t (10-3 units)
(b)
N(E
f) (
10–2
Sta
tes/
eV)
0
1
2
3
4
5
8 12
0
4 0
Fig. 5. (a) Static conductivity as a function of t. The dashed line shows the
power law fit by t2.1. (b) Density of states as a function of t. The dashed line
shows the power law fit by t1.1.
S. Shukla et al. / Journal of Physics and Chemistry of Solids 66 (2005) 1150–1157 1155
the behaviours is considerable. According to Eq. (11)
sdc f hv2n;kiNðEfÞ, N(Ef) being density of states at Fermi
level. If group velocity hv2n;ki did not vary as Fermi surface
vanishes, conductivity is indeed proportional to DOS. But
when the bands just overlap and give rise to small Fermi
surfaces, hv2n;ki does seem to depend on the Fermi surface
size as our computation in the transition region indicates.
The results therefore indicate that the mobility variation in
the transition region is also crucial even for the band overlap
transition.
Let us recall that the calculations have been performed in
the LDA which may not precisely predict the position of
metal–insulator transition and the exact behaviour of
various quantities around it. The many body correlation
effects [16] especially in the low energy range may strongly
affect the electronic band structure in the critical region.
Thus keeping in mind the mean-field character of the LDA
with regard to electron–electron interactions, one expects
integer or simple fractional values for these exponents. Our
numerical accuracy in the determination of exponents is
obviously limited by the following factors. On the metallic
side, in the transition region the Fermi surface becomes very
small and we expect errors in the number of k-points used in
sampling it. Similarly on the insulating side the energies in
the denominator become very small, and errors due to finite
grid of k-points are quite likely. Furthermore, the critical
behaviour is limited to a rather small range of lattice
constants and thus the number of points available to
determine power law exponents is small. In view of these
possible errors, it would be well within the accuracy of our
calculations to take the nearest rational exponents to be the
correct values. The values for the various exponents
determined in the calculation along with their closest
rational approximations are summarized in Table 1. From
this data we have also obtained the exponents for the
divergence of the length and frequency scales introduced
above. These are also listed in Table 1. Note that the
analysis of the Drude weight and the compressibility in case
of transitions controlled by filling yields the correlation
length exponent n!(1/2) and the dynamical exponent zO2
in correlated systems [1].
We can return now to the question of the existence of
intervening excitonic phase in the present situation.
Spontaneous formation of excitons is favoured if the
binding energy of the electron–hole pair exceeds the band
gap as the band gap closes. The exciton binding energy
goes as the inverse square of 30 [13]. Given that the
exponent of divergence of 30 is .54 while band gap
vanishes as t.92, it is seen that the binding energy decreases
Table 1
Power law dependence of A, 30, s, N(Ef) and Eg on t and the values of critical ex
A 30 s N(Ef) Eg
K2(K2.4) K(1/2)(K.54) 2(2.1) 1(1.1) 1(.92)
Values obtained from the actual calculations are given in the brackets while thos
a little faster than the gap. Thus the excitonic phase and the
associated discontinuous transition is marginally excluded
in this case. However, since the exponents have been
obtained from a mean field type calculation and the
excitonic binding energy scales approximately same as the
band gap under lattice compression, resolving the issue of
excitonic instability requires a more subtle and rigorous
study.
4.2. Reflectance studies
We explore optical properties around metal–insulator
transition by calculating reflectivity and examine its relation
with divergence of dielectric function. We restrict ourselves
to IR region of spectrum in which total dielectric function
has contribution from both electrons and ions (eZelectronic, iZionic):
3ðuÞ Z 3eðuÞC3iðuÞ (12)
The reflectance for the normal incidence is given by the
well-known Fresnel’s formula [27],
RðuÞ ZðnðuÞK1Þ2 Ck2ðuÞ
ðnðuÞC1Þ2 Ck2ðuÞ(13)
where nðuÞC ikðuÞZffiffi3
pðuÞ is the complex refractive index
function.
Let us take insulating and metallic cases separately.
4.2.1. Insulating phase
The total dielectric function in insulating phase at small
frequencies can be given as
3ðuÞ Z 30 C iAu CU2
p
ðU2T Ku2Þ
(14)
The first two terms constitute the electronic part in limit
u/0 while the third term is the ionic contribution ignoring
its absorptive part. Here Up is the ionic plasma frequency,
Up Z4pn
Me*2
T (15)
in which n is the number of dipoles per unit volume, M the
reduced mass, and e*T the effective charge including the
deformation from the deformable shells of the electrons. UT
is the transverse harmonic frequency of the phonon, which
couples to the light. To find Up at lattice parameters away
from the equilibrium value we use the relation [28]
e*T Z
1
3ð30 C2Þe*
s (16)
ponents
n h y z
3/4(.82) 4/3(1.34) 0(K.06) 8/3(2.74)
e outside it are the rationalised ones.
6.54 a.u.
6.56 a.u.
6.58 a.u.
ω (eV)
R (ω
)
0.4
0.6
0.8
1
0 0.02 0.04 0.06 0.08 0.1
Fig. 7. Reflectivity spectra for CsI at different lattice constants on the
metallic side of the transition.
6.66 a.u.
6.62 a.u.
6.605 a.u.
ω (eV)
R (ω
)
0
0.2
0.4
0.6
0.8
0 0.02 0.04 0.06
Fig. 6. Reflectivity spectra for CsI at different lattice constants on the
insulating side of the transition.
S. Shukla et al. / Journal of Physics and Chemistry of Solids 66 (2005) 1150–11571156
where e*s is the Szigetti charge whose variation with volume
V is given as
d log e*s
d log Vx:6 (17)
Similarly the variation of phonon frequency UT with
volume can be obtained from the relation[28]
d log UT
d log VZKg (18)
where the Gruneisen parameter gZ2 for cesium halides at
TZ0. Knowing the variations of 30, A, Up, UT as a function
of volume, we calculate reflectance using Eq. (14) for the
dielectric function. Fig. 6 shows the infra-red reflectivity
spectra at lattice parameters aZ6.66, 6.62, and 6.605 a.u.,
respectively. It is seen to have a strong peak at uZUT,
where 3(u) possesses a real pole. This is the well known
Reststrahalen region in which there occurs strong inter-
action of the photon with the transverse optic (TO) phonon
as they are in resonance. In this case, the major contribution
to dielectric function comes from the lattice part and it
follows from Eq. (13) that reflectance R/1.
The behaviour in the region u!UT can be inferred from
the following analysis. We can write
3ðuÞx30 C iAu CU2
p
U2T
(19)
In the small u regime, where Au/ 30CU2
p
U2T
� �, the
refractive index n[k, and the reflectivity R may be
approximated as
Rx1 K4n
ðn C1Þ2(20)
For example at aZ6.605 a.u., 30Z16.07, U2p=U
2T x:5
yields value of R(u)Z.37, which agrees well with the
calculation. R(u) is nearly independent of u over a broad
range below UT as seen in the calculations. The reflectance
increases towards unity as the MIT boundary is approached.
4.2.2. Metallic phase
In metallic phase, small frequency behaviour of elec-
tronic dielectric function is dominated by intraband
contribution, which is basically conductance. Including
the lattice part, it is given as
3ðuÞ Z 30 C i4psdc
uC
U2p
ðU2T Ku2Þ
(21)
and sdc is the static conductivity. Here we also assume that
even in the metallic phase of this ionic solid, the optical
phonon exists with the volume variation of the parameters
as described above. Taking the values of sdc from the ab
initio calculations, we obtain reflectivity using above
expression (21) for dielectric function. The results for the
metallic phase at lattice parameters aZ6.58, 6.56, and
6.54 a.u. of CsI are displayed in Fig. 7. Note that the
behaviour of R(u) for metallic phase is very different from
Fig. 6 for insulating phase particularly near uZUT.
The reflectance is seen to decrease as the MIT is
approached from the metallic side. This is in accord with the
fact that higher conductivity implies good reflectivity. In the
region u!UT!sdc, the behaviour of R is of Hagen–Ruben
kind in respect to frequency as well as the static
conductivity:
R Z 1 K2u
psdc
� �1=2
(22)
This is well obeyed by the detailed computation. Below
the electronic plasma frequency, the TO phonon frequency
again gives rise to a feature in the reflectance spectrum. At
uZUT, this spectrum also shows a pronounced peak.
Interestingly the peak gets widened with the decrease in
lattice parameter and finally appears to coalesce with the
Hagen–Ruben behaviour. This is understandable, since at
higher compression free electron like character supercedes
the role of lattice vibration. In this region, with loss of polar
character of CsI, the free electron Drude absorption
becomes as important as the absorption due to resonant
optical mode.
S. Shukla et al. / Journal of Physics and Chemistry of Solids 66 (2005) 1150–1157 1157
R(u) for uOUT but !up (upZelectronic plasma
frequency) saturates to a constant value. This can be
understood by noting that the dielectric function in this
region can be written in the form,
3ðuÞ Z 1 Ku2
p
uðu C iGÞC
U2p
ðU2T Ku2Þ
(23)
where we have used the Drude form for the conductivity.
Then for large frequencies, the reflectivity R is approxi-
mately given as
Rx1 K2G
up
(24)
which matches the calculation. It becomes larger with
increase in up i.e. enhanced metallicity.
5. Summary and concluding remarks
We have examined the critical behaviour of the
wavevector and frequency dependent dielectric function
around the lattice-driven metal–insulator transition in
Cesium Iodide. We have obtained exponents for the
critical behaviour of conductivity, imaginary part of
dielectric function in the small frequency limit, screening
length and band gap. We have discussed these results in
terms of the ideas of dynamical scaling appropriate to this
quantum phase transition. The exponents obtained here
are mean field like as (a) the exchange-correlation effects
are treated in the local density approximation; (b) the
effect of disorder has been included in the conductivity in
the Drude approximation, and in the frequency dependent
calculation by allowing for a level-independent broad-
ening. We have also examined the possibility of the
existence of an excitonic phase. Our present estimate of
exponents barely exclude the exciton phase, however,
given the inaccuracy of these exponents, this question
needs further analysis.
We have also studied IR reflectivity in the transition
region as an observable property, which is strongly affected
by the dielectric anomalies. In this analysis we include the
effects due to ionic part coming from the optical phonon and
find an interesting feature at the resonance whose obser-
vation should provide interesting knowledge about the
critical dielectric behaviour.
Acknowledgements
We are indebted to Prof. Deepak Dhar for some useful
remarks on this work. We also gratefully acknowledge
financial support from the Department of Science and
Technology, New Delhi under project Nos. SP/S2/M-46/97
and SP/S2/M-51/96.
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