Solid State Communications 144 (2007) 153–157
www.elsevier.com/locate/ssc
Metal–Insulator Transition and superconductivity in doped semiconductors
A. Therese Pushpam a,∗ , K. Navaneethakrishnan b
a Fatima College, Madurai, 625 018, India
b School of Physics, Madurai Kamaraj University, Madurai, 625 021, India
Received 9 April 2007; received in revised form 23 July 2007; accepted 26 July 2007 by F. Peeters
Available online 8 August 2007
Abstract
Using the effective mass theory and the variational method, the critical concentrations of shallow donors in Si at which the Mott Transition
occurs are obtained. Our theory uses the experimentally available donor ionization energies in the low-concentration regime and a mass variation
with interimpurity distance to account for the impurity bands. Since the experimentally available donor ionization energies are used, the centralcell effects and valley–orbit interactions are taken into account. Excellent agreement with the available experimental results is obtained for P, As,
Sb and Bi donors in Si. Since the critical concentration at which superconductivity occurs in B doped Si is established recently, we predict the
concentrations at which superconductivity should occur for these donors also.
c 2007 Elsevier Ltd. All rights reserved.
PACS: 71.55-i; 71.55.Cn; 71.55Eq; 71.55Gs; 71.30.+h; 71.70.Di
Keywords: A. Semiconductors; A. Doped semiconductors; D. Metal–Insulator Transition; D. Superconductivity
1. Introduction
Metal–Insulator Transition in semiconductors is drawing
considerable attention at present for several reasons. The
problem of impurity bands and their role in MIT and
superconductivity have not been completely understood [1–
5]. The predictions in the Hubbard model for the physically
interesting situation ( Ut ≈ 1) have eluded confirmations
by exact solution. Also contrary to the one electron scaling
theory [6], MIT has been observed in several low-dimensional
semiconducting systems throwing a challenge to the theory [7].
Several recent works have predicted superconductivity in
semiconductors and in alkali halide systems under hydrostatic
pressure [8]. Many of these results have been confirmed
experimentally. MIT has been shown to be the precursor for
superconductivity not only in high Tc superconductors, but also
in several other systems including doped semiconductors [3,
9]. The co-existence of ferromagnetism and superconductivity
and MIT in intense magnetic fields are other issues of recent
interest [10–13].
∗ Corresponding author.
E-mail address: tpushpu@yahoo.co.in (A. Therese Pushpam).
0038-1098/$ - see front matter c 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ssc.2007.07.036
Many-valley semiconductors pose considerable interest not
only in the formulation of effective mass theory but also in
the prediction of superconductivity for a long time [14–16]
Even today, no effective mass theory exists for these systems
that is free from criticism. In direct-gap semiconductors such
as GaAs, the simple effective mass theory which pictures the
donor system as a hydrogenic problem with an effective mass
for the electron pertinent to the conduction band minimum,
and the potential energy reduced by static dielectric constant
works well [17]. Several drawbacks of the effective mass theory
(EMT) as applied to a many-valley semiconductor such as Si
are attributed to central-cell corrections [18]. Because of these
deficiencies, the predictions of the critical concentrations at
which MIT occurs in these systems differ from experimental
values by one or two orders [19,20]. Other difficulties in
the theoretical formulations include randomness in impurity
distribution which triggers Anderson Transition and e–e
correlations leading to the formation of impurity bands [1,6].
In the present communication, we present a one electron
theory within the effective mass formulation using effective
masses obtained from the experimental, low-concentration
donor ionization energies, describing a mass variation with
the impurity concentrations and providing critical donor
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A. Therese Pushpam, K. Navaneethakrishnan / Solid State Communications 144 (2007) 153–157
Table 1
Ionization energies and effective masses for donors in Si
Donor
P
As
Sb
Bi
Table 2
Critical donor concentrations for MIT using Eq. (4)
E ion (meV)
experimenta
m ∗ (a.u.)
experiment
m ∗ (a.u.)
average
45.47
53.69
42.68
70.90
0.4577
0.5404
0.4296
0.7136
0.378
0.419
0.364
0.506
a Ref. [21].
concentrations at which MIT occurs for several donors in Si.
Using the recently observed superconductivity for B doped
Si, we predict the concentrations at which superconductivity
should be expected for P, As, Sb and Bi donors in Si.
2. MIT in n-type Si
For low-donor concentrations (≤1014 cm−3 ), the effective
mass theory in its simplest form leads to the solution of the
Schrödinger equation
"
#
− h̄ 2 2
e2
∇ −
ψ(r ) = Eψ(r ),
(1)
2m ∗
εo r
where m ∗ is the effective mass at the conduction band
minimum, which in the spherical band approximation is
0.2982 a.u. and εo is the static dielectric constant which is 11.7
for Si. ψ(r ) is called an envelope function [15]. To bring in the
impurity dependence into the theory, we use the experimentally
available donor ionization energies for the effective mass. These
values are given in Table 1. The usage of these effective masses
for different donors takes into account all central-cell effects
and the valley–orbit interactions. In the present work, we are
only interested in MIT and subsequent superconductivity and
not in the valley–orbit split excited states [21].
When the impurity concentration increases, overlap effects
become appreciable leading to the formation of impurity bands.
MIT occurs when the Hubbard gap closes as concentration
increases [1,4]. It has been recently shown in a series of works
that this simple one electron picture leading to MIT is consistent
with the Hubbard model calculations when concentrationdependent functions in the potential energy term such as
Thomas–Fermi and Hartree–Fock functions are used [22]. The
properties of the impurity bands are not completely understood
though considerable efforts are spent. Since the donor electron
enters into the impurity band when metallization occurs, we use
an effective mass which is the average of the low-concentration
impurity-dependent effective mass and 0.2982. A mass of
0.2982 for donors in Si is obtained by equating the donor
ionization energy in the spherical band approximation to the
numerical value obtained by using the longitudinal and the
transverse effective masses (m t and m l ) pertinent to the mass
ellipsoid near the conduction band minimum [23]. See the foot
note of Table 2. These masses for P, As, Sb and Bi donors in Si
are 0.378, 0.419, 0.364 and 0.506 respectively. Only when the
impurity band merges with the conduction band, the use of the
mass 0.2982 is justified. Using the Thomas–Fermi screening
n c in cm−3
Present
m ∗ = 0.2982
Semiconductor
Presenta
Experiment
1.74 × 1018
2.86 × 1018
1.43 × 1018
6.58 × 1018
3.85 × 1015
3.525 × 1018
3.74 × 1018
18
4.815 × 10
9 × 1018
3.147 × 1018
3.5 × 1018
8.458 × 1018
–
–
1.5 × 1016
hm
i
exp t +0.2982
a Corresponds to the concentration for m
.
average =
2
Si (P)
Si (As)
Si (Sb)
Si (Bi)
GaAs
function ε(r1 ) = 1ε e−r λ , we write the Hamiltonian as
e2
h̄ 2 k 2
− e−λr ,
∗
2m
rε
where
(2)
H=
λ=
"
12π e2 m ∗
h̄ 2 ε
# 12
1
−2
n 3
.
[3π 2 ] 3
γ
(3)
Here n is the electron concentration and γ is the number of
equivalent conduction band minima (6 in Si and 1 in GaAs) in
a semiconductor. Since the Schrödinger equation for H is not
amenable to an exact solution, we try an alternative approach
which has its origin in atomic physics. The stationary state is
characterized by an integral multiple of de Broglie waves filling
the circumference of the impurity orbital. Hence 2πan = nλ
and k = ann . For the ground state, n = 1. Replacing r by a and
minimizing H with respect to a leads to
ln a ∗ = ln a + ln(1 + aλ) − aλ,
(4)
εo h̄
where a ∗ = m
∗ e2 is the effective-Bohr radius.
As in Ref. [24] we have obtained the effective mass as a
function of interimpurity distance using values between m exp t
and 0.2982. This is shown in Fig. 1 for As in Si. We note that
when n c ≈ 1018 cm−3 , we obtain m ∗ ≈ 0.42 for As donor.
Eq. (4) is an exact result. More sophisticated Hamiltonians
including overlap and correlations lead either to approximate
results or numerical procedures [5]. Eq. (4) is solved for
different impurity concentrations. When these values of a are
used in Eq. (2), we obtain donor binding energies. The results
are presented in Fig. 2 for As donor in Si. For other donors, the
critical concentrations are presented in Table 2.
If the relativistic correction is used, then
"
#1/2
h̄ 2 n 2
e2
2
H = mc 1 + 2 2 2
(5)
− e−λr
rε
m c r
2
and we obtain
"
#
1
h̄ 2
ln a = ln a + ln(1 + aλ) − aλ − ln 1 + 2 2 2
2
m c a
∗
2
(6)
in place of Eq. (3). However, since m 2h̄c2 a 2 ≺≺ 1, Eq. (5)
leads to the same result for ‘a’ obtained earlier from Eq. (4).
A. Therese Pushpam, K. Navaneethakrishnan / Solid State Communications 144 (2007) 153–157
155
Fig. 1. Variation of effective mass with interimpurity distance for As in Si.
Fig. 3. Polarizability versus concentration for As in Si.
Fig. 4. Variation of diamagnetic susceptibility with concentration for As in Si.
Fig. 2. Donor ionization energy as a function of As donor concentration in Si.
Hence we find no change either in the ionization energies or
in the critical concentration due to relativistic effects. This is
physically due to the fact that in semiconductors, the effective
Bohr radii are very large and the binding energies are weaker.
Hence, relativistic corrections appear smaller.
The variation of donor polarizability α D with impurity
concentration is also of great interest for a long time [25].
At the critical concentration, the donor polarizability has
been observed to diverge indicating nonlocalization of the
carrier [26]. In our model, α D = 4.5a 3 [27]. Hence, we have
plotted in Fig. 3, the variation of α D with concentration. The
rapid variation at n c is evident.
The diamagnetic susceptibility is given by χdia =
2
− 6mce 2 ε hr 2 i for a single donor [28]. The variation of χdia with
o
concentration is shown in Fig. 4. The divergence at n c is again
clear from this figure. Similar works have appeared recently on
low-dimensional semiconductor systems and our results are in
agreement with these results in the proper limit [29].
Whether the MIT takes place in the conduction band or
in the impurity band is a highly debatable question. Whether
an impurity band can support metallic conduction is also
not clearly established. However, by observing the 1s → 2p
transition of a shallow donor, Romero et al. [30] have concluded
that the impurity band merges with the conduction band of
GaAs for a donor concentration of 8 × 1016 cm−3 . Similar
conclusion in a GaAs/Ga1−x Alx As superlattice has been arrived
at by Helm et al. [31]. In contrast, it has been shown that the
Hubbard model results are in agreement with the mean-field
calculations (Thomas–Fermi or Hartree) as far as the critical
concentration values are concerned if proper corrections are
made for the effective masses due to overlap effects [22,26].
3. Superconductivity
Superconductivity in semiconductors is not new. Traditionally there have been two approaches to superconductivity in
semiconductors: (i) In undoped samples, by applying hydrostatic pressure, closure of the band gap leading to metallization
and subsequent increase in pressure enhancing the density of
states at the Fermi level leading to BCS like superconductivity [32]. The Tc values observed are low (≈3 K). (ii) By increasing the impurity concentration, the formation of impurity
bands and the merging of the impurity band with the conduction band leading to superconductivity [33,34]. Since the conduction in the impurity band is by hopping, and in view of the
recent experimental evidence for superconductivity in B doped
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A. Therese Pushpam, K. Navaneethakrishnan / Solid State Communications 144 (2007) 153–157
Si at n of the order of 1021 cm−3 , it is clear that superconductivity occurs in the valence band due to holes with the conventional BCS-type mechanism [9]. Further experimental support
for BCS mechanism in the conventional band structure of Si
follows from similar observations in B doped diamond [35].
From the available theoretical and experimental works on
pressure induced superconductivity on several metals, alkali
halides and semiconducting systems, it appears to us that
the high frequency, long wave vector phonons are eliminated
encouraging conduction electrons to undergo Cooper pairing
in the presence of low energy and long wavelength phonons.
At high pressures the following effects are expected. (i) The
band gap closes leading to metallization. (ii) Increase in Fermi
energy and the consequent increase in the density of electron
states. (iii) Enlargement of the Brillouin zone associated with
increase in phonon energies including Debye energy and the
consequent increase in phonon density of states. However, at
low temperatures high-energy phonons are not excited and
superconductivity occurs by BCS mechanism.
In doped systems however, high impurity concentrations are
required to attain the superconducting state. From studies such
1
as B doped Si [9], approximately 10
th of the host atoms are
replaced by impurity atoms. In such a situation, the phonon
dispersion relations of the host material require enormous
modification in the discussion of superconductivity in the BCS
theory. Such a problem has not been addressed so far. Roughly
speaking, in an impurity atom with a heavier mass such as
V group donors in Si, one expects a reduction in the phonon
frequencies in addition to the defect modes. Since the impurity
band merges with the conduction band when superconductivity
occurs [9,35], the consequent increase in Fermi energy and the
elimination of large momentum and large energy phonons from
the scenario at low temperatures leads to superconductivity as
in the pressure applied case.
Using the experimentally available donor concentrations
in Si, for the onset of super conductivity, we predict the
following concentrations for P, As, Sb and Bi donors in Si.
Since the ionization energies vanish when MIT occurs, we
use the experimentally observed critical concentration of 2.8 ×
1021 cm−3 for B in Si, and the critical donor concentration
ion (B)
for P, As, Sb and Bi should be EEion
(D) n c (B), where D refers
to any donor and E ion (B) is 45 meV. Since Bi is the heaviest
atom, we predict donor concentrations of 2.77 × 1021 cm−3 ,
2.35 × 1021 cm−3 , 2.95 × 1021 cm−3 and 1.78 × 1021 cm−3 for
P, As, Sb and Bi donors respectively.
In an earlier work, Rasolt predicted superconductivity
in many-valley semiconductors in the presence of intense
magnetic fields [16]. Typical values obtained by Rasolt for n
are of the order of 1018 cm−3 and Tc of the order of 0.5 K
for a magnetic field of 1 MG. Recently, it has been shown
that before entering into the superconductivity state the doped
semiconductor passes through a metallic state in every magnetic
field [36]. In view of recent discovery of superconductivity in
ferromagnetic systems, we believe that it is worthwhile to study
experimentally the magnetic field induced superconductivity.
In an earlier work Cohen has shown that Tc values enhance
when the number of conduction band valleys increase [15].
In the last few years, superconductivity in B doped Diamond
has drawn the attention of several physicists [9]. The results
of Ref. [36] may further be improved when the appropriate
screening function relevant to the host semiconductor is used.
4. Results and discussion
The critical donor concentrations at which metallization
occurs is presented in Table 2 for different donors in Si.
We find an excellent agreement with experimental values.
Such an agreement obtained for the first time confirms that
metallization occurs in the impurity band since the mass
at n c is different from the mass at the conduction band
minimum, namely 0.2982. This observation is in agreement
with several experiments performed recently indicating the role
of impurity bands [37]. It also demonstrates that the correlation
effects among electrons, the central-cell effects and valley–orbit
interactions on the donor states are correctly taken in to account.
Sharp increase in the effective mass near the critical density
has also been suggested recently by Shaskin et al. [38]. This
suggestion is in agreement with the masses used in the present
work, displayed in Table 1. We believe that this increase in the
mass from the conduction band minimum value of 0.2982 is
due to impurity bands.
The prediction of donor concentration above which superconductivity should occur in doped Si requires experimental
support. The concentrations predicted are 2 to 3 orders higher
than what Rasolt conjectured in a strong magnetic field [16].
Cohen earlier predicted the donor concentration of the order of
1020 cm−3 in many-valley semiconductors [15]. However, the
recently observed impurity concentrations in Si are of the order
of 1021 cm−3 .
We conclude that though the Tc values observed so far
for any semiconductor (except High Tc materials) is less than
10 K, these systems offer a scope for understanding impurity
bands, MIT and the mechanism for superconductivity. Since
nanotechnology involving semiconductors is expected to give
enormous impetus to technology, understanding the phenomena
in these systems will also be of great interest. Hence MIT and
superconductivity in lower dimensions should also be pursued
with great interest.
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