146
Supplement of the Progress of Theoretical Physics, No. 57, 1975
Metal-Nonmetal Transition in Doped Semiconductors
Kazuo MORIGAKI and Fumiko YONEZAWA *
Institute for Solz"d State Physics, University of Tokyo
Roppongi, Tokyo 106
*Department of Physics, The City College of CUNY
New York, N. Y. 1003Jtl
(Received January 21, 1975)
The metal-nonmetal (M·NM) transition observed in some doped semiconductors is
discussed on the basis of the Matt-Hubbard-Anderson scheme. A classification of
characteristic donor (acceptor) concentration regions is given. The M-NM transition
concentration is defined as a concentration at which the states corresponding to the Fermi
level become delocalized. Several experimental data near the transition concentration
are analyzed and interpreted as lending support to the above-mentioned scheme. A
possible picture concerning the position of the Fermi level relative to the mobility gap is
proposed and some experimental results are described, which are explained in consistent
with this picture.
§ 1.
Introduction
In some non-crystalline systems in which atomic configurations are
disordered in some way or other, the metal-nonmetal (M-NM) transitions are
observed when appropriate physical parameters such as densities, impurity
concentrations, electric fields, temperatures, pressure are changed. Doped
semiconductors which we shall discuss here are one of the most typical and best
studied examples of those disordered systems that show the M-NM transitions.
Other examples are metal-ammonia solutions, 1> metal-rare gas mixed solids, 2>
supercritical metallic fluids, 3) high density excitons, 4 > etc. To a certain
extent, electrons in these materials are treated on more or less the same ground
in view of the essential feature, common to most of these disordered systems,
that electrons are interacting with one another as well as with fluctuating
potential due to randomly distributed atoms. So far, two major theoretical
schemes have been put forward to explain the M-NM transitions and the
related properties of electrons in these systems; that is, the Mott- HubbardAnderson scheme 5)-7) and the percolation theory for inhomogeneous conduction
regions. 8)' 9)
t> On leave from Department of Applied Physics, Tokyo Institute of Technology,. Meguro-ku, Tokyo
152.
Metal-Nonmetal Transition in Doped Semiconductors
147
The purpose of this short article is to study electronic properties of doped
semiconductors near the M-NM transition concentration Nc along the line
of the Matt-Hubbard-Anderson scheme. Because of the limited space allowed
for us, we specifically confine ourselves to the donor concentration region
N D;SNc of n-type doped semiconductors without compensation although most
of the arguments given here are, with appropriate modifications, equally
applied top-type doped semiconductors and to the case where samples are
compensated. It must be noted, however, that in actual samples a certain
amount of compensation exists inevitably even when the purity and the doping
of the impurity material are highly controlled, and this small amount of compensation enables the hopping conduction to take place in the low concentration
region. In §2, we classify the donor concentration regions according to the
behaviour of the states which provide theoretical basis for the Matt-Hubbard
picture in disordered systems. In §3, experimental results in the concentration
region below Nc, i.e., N D;SNc are studied in order to see how they are explained within the framework of the Mott-Hubbard-Ander.son scheme. In
§ 4, we suggest a possible position of the Fermi level with respect to the mobility
gap near the transition concentration and some concluding remarks will be
given in §5.
§2.
Characteristic concentration regions and
M-NM transition
As is clearly shown in the famous and almost classical paper by Fritzsche,lO)
the donor concentration regions of doped semiconductors are classified into
three major characteristic categories according to the types of impurity conduction; i.e., the low concentration region where the conduction is due to the
hopping between donor centers; the intermediate concentration region where
the conduction is observed mainly of an activation type with an activation
energy which is termed e2; and the high concentration region where the
conduction is nearly metallic. Among several possible mechanisms which may
be responsible for the activation energy e2, the most probable candidate is the
activation across the energy gap between the donor band and the D- band, the
latter of which is the band grown from the energy level of an electron added
to an isolated neutral donor atom. This idea of the D- band1°) is essentially
the same as that of electron correlations proposed for an explanation of the
so-called Mott transition, 11 > formulated later by Hubbard 6> for a simplified
model system. These two bands are now generally referred to as the lower
and upper Hubbard bands. Besides the effect of electron correlations, there
is another mechanism, possible only for disordered systems, which is expected
to hinder electrons from moving around freely. This is generally known as
Anderson's localization-delocalization mechanism 7> due to potential fluctuation.
In the Matt-Hubbard-Anderson scheme, these two mechanisms of electron
148
K. Morigaki and F. Yonezawa
correlations and Anderson's localization are regarded to be the most essential
cause for the M-NM transitions in disordered systems in general, although
detailed interplays of these mechanisms are of course different for different
disordered materials. In doped semiconductors, the dominant factor is
considered to be the electron correlations while the actual transition is explained to be the Anderson transition.5),1 2) Both for theoretical and experimental analysis' sake, it is convenient to subclassify the intermediate and
high concentration regions as follows:
(i) Intermediate concentratz"on region I;
in which there exists a real gap between the upper and lower Hubbard bands
as shown in Fig. 1 (a). The shadowed regions in the figures denote that the
states in these energy regions are localized in Anderson's sense.
(ii) Intermediate concentration region II;
in which there is no longer a real energy gap, but a mobility gap still remains
(see Fig. 1 (b)). In both cases (i) and (ii), s2 is defined as IEF- Ec I, where
Ec is the upper edge of the mobility gap.
(iii) High concentratz"on region I;
in which the states at the Fermi level is delocalized but still in such an energy
region as is characteristic of the impurity bands (see Fig. 1 (c)).
(iv) High concentration region II;
where the tail from the conduction band covers well over the energy region to
which the Fermi level belongs (see Fig. 1 (d)).
(a)
(b)
(c)
(d)
E,
Fig. 1. Schematic diagram of the density of states for the impurity band
and the conduction band versus energy. Figures (a)-(d) approximately correspond to the situations for the concentration regions (i)
-(iv) respectively.
Metal-Nonmetal Transition z'n .Doped Semiconductors
149
The predicted behaviour of the density of states as shown in Fig. l is
theoretically confirmed by calculating the spectra for these systems in the
"so-called first principle" manner. Our numerical results are shown in Figs.
2 to 4_13)-15) We have used the Hubbard model for electron correlations. The
spectra in Fig. 2 are the conduction band and the two Hubbard bands relevant
to the impurity conduction of a system in which host atoms (A) on regular
lattice sites are substituted for by donor atoms (B). They have originally
been calculated for a disordered binary system Al_zBx (x=0.4), 13> in which
the effect of disorder is treated by the coherent potential approximation.16)
In actual doped semiconductors, the effective Bohr radius of donor electrons
is large compared with the atomic spacing of the host lattice, so that one should
regard the distribution of donor atoms to be structurally disordered. However, in an approximate sense, the variation of the spectra with the donor
concentration (or the average distance between donors) can be seen in Fig. 2
by changing the value of Ll/ U, where Ll and U denote the bandwidth and the
repulsive Coulomb interaction of two electrons with opposite spins on the same
site respectively. The large value of Ll/ U corresponds to the situation of the
higher donor concentration. Figure 3 has been obtained by regarding the
distribution of donor atoms to be structurally disordered, but not taking into
account the conduction band. A single-site theory1 7) proposed for this kind of
disorder has been used. In view of the fact that the effective Bohr radius of
a donor atom in doped semiconductors is comparable to the mean distance
between donor atoms, this assumption of totally disordered atomic distribution
E/U
Fig. 2. Calculated density of states of the two Hubbard bands and tbe conduction band for X= 0.4 and various values of LJ / U. E A and E B designate the
energies of tbe neutral donor level and tbe center of the gravity of the conduction band respectively. Small triangles in the figures indicate the position
of the Fermi level. (After Ref. 13).)
150
K. Morigaki and F. Yonezawa
~
0~-A~llW~L-~~~~J--llllW~--~
,::
0
0
EF
u=0.3
0
EF
Fig. 3. Calculated density of states of two Hubbard bands for various
values of u which denotes U in units of the binding energy of donor
electrons. The shaded r.egion designates the region with a localized
character. (After Ref. 14).)
ll./U=0.4
Un(E)
0.6
.6
E/U
E/U
Fig. 4. Calculated density of states of two Hubbard bands with each
different bandwidths. The model corresponds to the case where the
width of the .D- band is three times larger than that of the donor
band. (After Ref. 15).)
is considered to be almost adequate. Figure 4 shows the result of the calculation in which the original Hubbard model is extended so as to take into account
the difference of the band widths of the donor band (the lower Hubbard band)
and the D- band (the upper Hubbard band). Since the effective Bohr radius
Metal-Nonmetal Transi#on in Doped Semiconductors
151
for the D- band is nearly 2 to 3 times larger than that for the donor band, a
proper care of this difference must be taken so that the theoretical discussion
can be more reliable. A characteristic feature is an asymmetry of the two
Hubbard bands, the explanation of which we intend to give in a forthcoming
paper.
§ 3.
Experimental evidence for the Mott-HubbardJlnderson scheme near ]Vc
If the behaviour of the density of states accompanying the change of
the donor densities is as described by Fig. 1, experimental data are expected
to give some indications for the existence of the mobility gap at N D~Nc.
Let us see, in the following, whether this is the case.
3.1.
Existence of the gap
According to our prediction that the Hubbard bands (upper and lower)
are separated from each other either by a real gap in the intermediate concentration region I or by a mobility gap in the intermediate concentration
region II, the existence of the gap must be observed. The experimental
fact for the activation type conduction through e2 in this concentration region
is nothing but the most direct and certain evidence for the gap. Besides
this almost complete evidence, we can give a few other experimental results
indicating the existence of two kinds of electron systems, which are compatible with the existence of the gap. That is, for a finite temperature,
electrons are partially excited from the lower Hubbard band to the upper
Hubbard band across the gap. Since the states in the lower Hubbard band
have a more localized nature than the states in the upper Hubbard band, we
have consequently two kinds of electrons, one being conductive and the other
being rather localized.
In the intermediate concentration regions I and II, ESR enhanced
conductivity 1 s>~ 21 > is measured as shown in Fig. 5, which is explained as
follows :1 9),*) If we assume to have a donor spin system which is responsible
for ESR and a system of mobile electrons which contributes to the conduction,
energy from the microwave applied to the system is first absorbed by localized
spins at resonance and thereafter transferred to mobile electrons via interaction between these two kinds of electrons, which brings about the increase
of the kinetic energy of mobile electrons and accordingly the increase of the
mobility. Another experimental support for the two kinds of electrons is
given by the Raman spectra 2 3) for Ge: As on both the sides of Nc. The
*> Recently, Mott and Kamimura5),12),22l have suggested that the result of ESR enhanced conductivity
provides an evidence for the existence of the mobility gap (the pseudogap) in which the variable
range hopping conduction may be enhanced by absorption of microwave energy from excited spins
at resonance.
152
K. Morigaki and F. Y onezawa
J
f
0
I
'\
:~'~!
1
-12dB
\
lQ-4
~
\
lQ-5
1:-
I
lQ-6
1Xl0 18
No (cm- 3 )
Fig. 5. Resistivity change due to ESR plotted as a function of ND at
1.5 K and at the microwave power level of -12 dB for Si: P.
(After Ref. 21).)
spectra are analyzed to be the superposition of the Raman scattering line
related to the valley-orbit splitting and the background spectra of singleparticle excitation. The result can be understood by regarding the former
due to localized electrons and the latter due to conductive electrons.
3.2.
The mobility gap
If the transition is of Anderson's type, the gap observed on the nonmetallic side of Nc must be the mobility gap rather than the real density of
states gap. This is guaranteed if we have a nonzero density of states at the
Fermi level before the transition; i.e., n(EF)~O for Nn-:5Nc. One evidence
for this is the fact that a variable range hopping 24> is observed in this region
for uncompensated Si 25 > and Ge,26) and also for compensated Ge 27),28) with
the T-114 dependence of the logarithmic resistivity at low temperatures, which
possibly occurs only when n(EF) ~0. Another evidence is that the Pauli spin
susceptibility29 >· 30> has no discontinuous change across the transition concentration Nc. This is consistent with the argument that, when a transition
is the Anderson transition, the density of states at EF on the nonmetallic side
is nonzero and undergoes no discontinuity as N n is increased onto the metallic
concentration. 5), 12>
Metal-Nonmetal Trans£tion in Doped Semz'conductors
§4.
153
The Fermi level and the mobility gap near Nc
The Anderson transition from nonmetal to metal is defined by the
statement that the states at the Fermi level become delocalized. We notice
that there are three possible ways in which this delocalization of the Fermi
level occurs. These are schematically described in Fig. 6. The first possibility as shown in (a) is the case where the Fermi level remains in the mobility
gap until the mobility gap itself disappears, which is quite unlikely since the
spectra are far from symmetric especially where the impurity levels are shallow
and the influence of the conduction band is important. The second possibility
as given in (b) is the case where EF escapes upwards from the mobility gap
through the top mobility edge while, in the third possibility (c), EF gets out
of the mobility gap downwards through the bottom mobility edge. In what
follows, we give a couple of experimental results which are consistently explained on the assumption of the third case.
(1)
Anomalous microwave hot electron eifect21)
When the microwave is applied to a sample of an intermediate donor
concentration under the non-resonant condition, the resistivity decreases due
to an increase of the mobility of mobile electrons with the absorption of the
microwave. On the other hand, when the donor concentration in Si : P is
increased into the high concentration region I across Nc, a sharp increase of
resistivity as shown in Fig. 7 is observed just above Nc accompanying the
application of the microwave to the sample. When N D is increased further,
the enhanced portion of resistivity .r::Jp/p becomes smaller monotonically and
disappears around N D::::::::: 2 X 1019 cm-3, which is nearly the critical value at
which the system transforms from the high concentration region I to II. 31 >
This result is explained by assuming the third possibility of Fig. 6 (c). The
sharp increase of the resistivity just above Nc is related to the microwaveassisted excitation of electrons from the region around the Fermi level into
that of the mobility gap where the states are localized. This apparently
EF
(a)
~t
EF
(b)
EF
(c)
Fig. 6. Schematic diagram showing the relative relation between the
positions of E F and the mobility gap. Ec and Ec• designate the top
and bottom mobility edges respectively.
154
K. Morigaki and F. Yonezawa
1.5K
0
OdB
H=O
w-3
-
Q:
~
w-4
""
J0-5
5
6 7 8 9 10 19
No (cm- 3 )
1. 5
2
Fig. 7. Resistivity increase due to microwave irradiation plotted as a
function of N D in zero magnetic field at 1.5 K and at the microwave
power level of 0 dB (the microwave power at 0 dB is about 30mW)
for Si: P. (After Ref. 21).)
works to increase the net resistivity.
(2)
Dependence of
e2
on donor concentration
Mott 12> argues that e2=! Ec-EF I approaches zero linearly as N D
approaches Nc from below. If the M-NM transition occurs in the way as
shown in Fig. 6 (b), this conjecture of Mott sounds reasonable. But, if the
situation is as expressed by Fig. 6 (c), s 2 changes from a finite value to zero
discontinuously when the Fermi level escapes from the mobility gap downwards and the transition from nonmetal to metal takes place. It is difficult
to see whether e2 goes continuously zero or undergoes a discontinuous change
at N 0 , partly because of the experimental difficulties and partly because the
variable range hopping becomes important near N 0 • However, our careful
analysis of experimental data1 0),18),19),32)-35) shows that they seem not to
exclude the possibility of a discontinuous drop of e2 at Nc.
§ 5.
Discussion
We have studied experimental data of doped semiconductors near the
M-NM transition concentration Nc in order to see how these experimental
results fit in the Mott-Hubbard-Anderson scheme. By analyzing some
observed properties of electrons carefully, we have also proposed a possible
picture for a detailed behaviour of the Fermi level near the transition point.
So long as existing experimental results, known to the authors, in this concentration region are concerned, the Mott-Anderson theories successfully work
Metal-Nonmetal Transition in Doped Semiconductors
155
to give theoretical explanations of the transition mechanisms. A full discussion of all concentration regions will be given elsewhere, which also seems to
indicate that the M-NM transition in doped semiconductors is satisfactorily
understood by the Mott-Hubbard-Anderson scheme.
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