Professor Dr Alex L Efros
A.F IOFFE Physico-Technical Institute,
Academy of Sciences of the USSR, Politekhnicheskaja, Leningrad 194021, USSR Translater Dr Serge Lunyi Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ 07974, USA Series Editors:
Professor Dr Manuel Cardona Professor Dr Peter Fulde
Professor Dr Hans-Joachim Queisser
Max-Planck-Institut fir Festkörperforschung, Heisenbergstrasse ] D-7000 Stuttgart 80, Fed Rep of Germany
Title of the original Russian edition:
Elektronniye svoistva legirovannykh poluprovodnikov © by *Nauka* Publishing House, Moscow 1979
ISBN 3-540-12995-2 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12995-2 Springer-Verlag New York Heidelberg Berlin Tokyo
Library of Congress Cataloging in Publication Data Shklovskii, B 1 (Boris Ionovich), 1944- Electronic
properties of doped semiconductors (Springer series in solid-state sciences ; 45) Translation of: Elekironnye svoistva legirovannykh poluprovodnikov Includes bibliographical references and index 1 Doped semi-
conductors, 2, Electron-electron interactions 3 Hopping conduction 4 Materials at low temperatures, I Efros, A L (Alex L.), 1938- II Title [II Series QC611.8.D66S5513 1984 537622 84-5420 This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, reuse of illustrations, broadcasting, reproduction
by photocopying machine or similar means, and storage in data banks Under § 54 of the German Copyright
Law, where copies are made for other than private use, a fee is payable to “Verwertungsgesellschalt Wort”, Munich
© by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany
The use of registered names, trademarks, etc in this publication does not imply, even in the absence
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and therefore free for general use
Offset printing: Beltz Offsetdruck, 6944 Hemsbach/Bergstr Bookbinding: J Schaffer OHG, 6718 Griinstadt
Trang 4Preface
First-generation semiconductors could not be properly termed “doped” — they were simply very impure Uncontrolled impurities hindered the discovery of physical laws, baffling researchers and evoking pessimism and derision in advocates of the burgeoning “pure” physical disciplines The eventual banish- ment of the “dirt” heralded a new era in semiconductor physics, an era that had “purity” as its motto It was this era that yielded the successes of the 1950s and brought about a new technology of “semiconductor electronics” Experiments with pure crystals provided a powerful stimulus to the develop- ment of semiconductor theory New methods and theories were developed and tested: the effective-mass method for complex bands, the theory of impurity states, and the theory of kinetic phenomena
These developments constitute what is now known as semiconductor phys- ics In the last fifteen years, however, there has been a noticeable shift towards impure semiconductors — a shift which came about because it is precisely the impurities that are essential to a number of major semiconductor devices Technology needs impure semiconductors, which unlike the first-generation items, are termed “doped” rather than “impure” to indicate that the impurity levels can now be controlled to a certain extent
New problems have arisen in the theory of the electronic states of doped semiconductors They concern electrons located not in an ordered field of crystal atoms, but in the chaotic field of impurities, and the potential energy of the latter is by no means small At low temperatures a doped semiconduc- tor crystal becomes a disordered system, which in its general characteristics resembles an amorphous system This is true for both lightly and heavily doped semiconductors: the lighter the doping, the lower the temperature at which these characteristics are exhibited
The aim of this book is to present in logical fashion the theory of electron- ic states and conduction in doped semiconductors at low temperatures, that is, in the region where the properties of the electronic states differ most from
those of Bloch waves
Trang 5for electron-electron interaction even at the lowest electron concentrations To this end, a nonlinear screening theory was developed, based on the self- consistent field method (Sect 3.4) This method does not, however, work in the vicinity of the Fermi level, where the density of states has interesting and peculiar features (Chap 10)
If the Fermi level is in the localized-state region, then conduction is due to electron hopping and is exponentially dependent on temperature and the im- purity concentration The hopping conduction phenomenon was identified long ago, but several major advances have taken place in the last decade A theory was developed which describes the temperature, concentration and magnetic field dependences quantitatively This theory is based on a new mathematical discipline known as “percolation theory” Today, the percola- tion method is as essential to the study of low-temperature conduction as the kinetic equation method is for band conduction; the term “percolation level” is as frequent in the relevant literature as “relaxation time” Good reviews of the percolation theory do exist — we cite many in Chap 5 — but these were written relatively long ago and are inappropriate to the study of hopping conduction For this reason we found it necessary to write a separate chapter detailing the main tenets of percolation theory (Chap 5), replete with bibliog- raphy on the topic
In all chapters devoted to hopping conduction there is a thorough compa- rison of theory and experiment, a comparison that we find, on the whole, favorable We have tried to point out discrepancies and theoretical problems that in our view remain unsolved
Although the book is devoted to crystallic semiconductors, many of the ideas and methods also apply to amorphous semiconductors, so much so that “amorphous digressions” are an integral part of the text Occasionally (see Chap 9) we use experimental data on amorphous semiconductors to support certain concepts
Our book is not intended solely as a specialists’ monograph, but also as an extension of an ordinary course in semiconductor theory that touches on a new range of problems Chapter 1 and Sects 4.1 and 11.1 serve to connect this book with standard courses in the theory of “pure” semiconductors The book is aimed at a wide readership: theoretical and experimental physicists, graduate students, and engineers acquainted with the basics of solid-state physics An easier version of our book can be obtained by omitting Sects 1.3, 2.3—2.5, 5.4, 5.5, 8.3, 10.1, 10.2, 11.3, 12.3, 13.4, It is useful to keep in mind that as a rule all questions are discussed twice, first qualitatively and then quantitatively For the reader not interested in mathematical detail the quali- tative explanation should suffice; sections which may be omitted are usually designated as such in the text
Trang 6Preface VII
the scaling theory of localization by Anderson with coworkers and Thouless We have incorporated this theory, as we understand it, into Chapter 2 of the present edition
New ideas have also emerged in the understanding of electron-electron in- teraction in disordered systems In 1975 we proposed the idea that a Coulomb gap may form in the vicinity of the Fermi level, which if correct would make it necessary to revise Mott’s law for variable-range hopping conduction In the Russian edition we timidly devoted only one section to this question (in Chapter 10), but since then, a number of authors have made both theoretical and experimental contributions to this subject, and we felt compelled to write a whole new chapter for the present edition (Chapter 14) It describes com- puter modelling of the Coulomb gap, the impurity-band structure, and hop- ping conduction
In 1979-1982 Al’tshuler and Aronov published a series of remarkable papers devoted to the role of electron-electron interaction in disordered sys- tems with delocalized states We were not able to consider these concepts in detail — otherwise we would have had to write a new book One reason not to do so was furnished by AI’tshuler and Aronov themselves, who discuss these topics in a chapter of the volume entitled Electron-Electron Interaction in Dis- ordered Systems ed by A L Efros and M Pollak (North-Holland, Amster- dam 1984), Other contributions to that volume also substantially complement our book,
It is a pleasure to us that Springer-Verlag has undertaken to publish the English translation of our book We thank Professor Mike Pollak, our long- time friend, and Dr Serge Luryi for their hard and selfless labor in preparing this edition We are grateful to our foreign colleagues B N Butcher, J Chroboczek, E Guyon, J J Hauser, S Kirkpatrick, Y Imry, P A Lee, R Mansfield, N F Mott, G A Thomas, H E Stanley, J P Straley, and many others who regularly send us reprints of their works, thus keeping us abreast with the latest developments
Trang 7
Part I Lightly Doped Semiconductors
1 The Structure of Isolafed Impurity States
1.1 Shallow Impuritie§ cu cu 1.2 Impurity Levels Near a Nondegenerate Band_
1.2.1 Extremum at the Center of the Brillouin Zone
1.2.2 Several Equivalent Extrema
1.3 Impurity Levels Near a Point of Band Degeneracy
1.4 Asymptotic Behavior of the Impurity-State Wave Functions
2.2 Anderson Transition c
2.3 Examples of Transition from Localized to Delocalized States Conductivity Near the Transition Point
2.4 The Scaling Theory of Localization
2.5 Localization in the LifshitzModel
3 The Structure of the Impurity Band for Lightly Doped Semiconductors
3.1 General Remarks c co
3.2 The Impurity Band at Low Degrees of Compensation
3.3 Long-Range Potential at Low Degrees of Compensation
3.4 The Impurity Band at High Degrees of Compensation
4 A General Description of Hopping Conduction in Lightly Doped Semiconduefors
4.1 Basic Experimental Facts c
Trang 8x Contents
5.4 Theory of Critcal Exponent$S : 118
5.5 Electric Conductivity of Random Networks of Conducting Bonds Infinite Cluster Topology - 123
5,5,1 Dead EHỞS vn kg nh nh nh ki bó ng 125 5.5.2 The Nodes and Links Model 126 5.5.3 The Scaling Hypothesis and Calculation of the Conductivity 00%) eee eee 129 5.6 Percolation Theory and the Electric Conductivity of Strongly Inhomogeneous Media 130
6 Dependence of Hopping Conduction on the Impurity Concentration and Strain in the Crysfal 137
6.1 Resistivity p; for Semiconductors with Isotropic Impurity Wave Functions 20 cece ce cece eee rete ees 137 6.2 Resistivity p; for Semiconductors with Anisotropic Impurity Wave FUNCHONS 0.0 ccc ốŒARAÁAAÁAÁAÁAÁAÁAÁAAR 144 T Honping Conduction in a Magnefic Field 155
7.1 The Elementary Resistance Ry in a Magnetic Field 156
7,2 Evaluation of the Magnetoresistance and Discussion of Experimental Data 0.6.0 cece cence een cuc 163 § Acfivation Energy for Hopping Conduction 180
8.1 Activation Energy ¢;at Low Degree of Compensation 180
8.2 Activation Energies e, and ¢,at High Compensation 186
8.3 The Perturbation Method in Percolation Theory General Theory of the Activation Energÿy£i 191
9, Variable-Range Hopping Conduction 202 9.1 Mott’sLaw << ents 202 9.2 Magnetoresistance in the Region of Variable-Range Hopping Conduction cuc 210 9.3 The Dependence of Hopping Conduction in Amorphous Films on Film Thickness 216
9.4 The Preexponential Factor in Hopping Conductivity 222
10 Correlation Effects on the Density of States and Hopping Conducfion ch hoc 228 10.1 Coulomb Gap in the Density of States., 228
10.1.1 The Theory of the Coulomb ap 228
40.1.2 Possible Manifestations of the Coulomb Gap 238
10.2 Many-Body Correlations in Hopping Conduction 244
10.2.1 Hubbard Current Correlations - 246
Trang 9
Part II Heavily Doped Semiconductors
11 Electronic States in Heavily Doped Semiconductors 255
11.1 Linear Screening Theory 255
11.2 Density of States Near the Bottom of the Conduction Band 259 11.3 Derivation of the Quasiclassical Formula for the Density
of States .ố eee 264
12 The Density-of-States Tail and Interband Light Absorption 268 12.1 The Optimum Fluctuation Method 268 12.2 The Uniformly Charged Sphere Approximation
The Spectrum of the Majority Carrlers 270 42.3 Exact Distribution of Impurities in Optimum Fluctuations 277 12.3.1 Derivation of Basic Equations 277 12.3.2 Derivation of (12/2/12) cu vua 279 12.3.3 Density-of-States Asymptotics in the Absence of
Impurity Correlations 2.0.00 00 ccc ec ees 281 12.4 The Spectrum of the Minority Carriers 283 12.4.1 Classical Case (y » Eÿ) Qua 284 12.4.2 Quantum Case (y « Eg) 285 12.5 The Theory of Interband Light Absorption 287
12.5.1 On the Relation Between the Density of States and
the Interband Light Absorption Coefficient (ILAC) 287 12.5.2 Light Absorption Induced by Gaussian Fluctuations
in the Absence of Carrier Degeneracy 289 12.5.3 Discussion of Experimental Results 29 13 The Theory of Heavily Doped and Highly Compensated
Semiconducfors (HDCS) 295 13.1 Uncorrelated Impurity Distribution 295 13.1.1 Qualitative Discussion 295 13.1.2 Basic Equations and the Asymptotic Properties `
Of the Potential co 299 13.2 Correlated Impurity Distributon 303 13.3 Kinetic Properties of HDCS_ 305 13.4 Completely Compensated Semiconductor 309 Part II1 Computer Modelling
14 Modelling the Impurity Band of a Lightly Doped Semiconductor
and Calculating the Electrical Conductivity 316
14.1 Minimization of the Total Energy and Calculation
Trang 10XII Contents
14,2.1 Intermediate Degrees of Compensation 324
14.2.2 Strong Compensation (1— K 4f1) 326
14.2.3 Low COmpP€NSALOD uc nhọ nhở 328 14.3 Distribution of Electric Fields on Neutral Donors 329
14.4 Activation Energy of Hopping Conduction 333
14.5 Variable-Range Hopping Conduction 344
14.6 Activation Energy ø¡ of Band Conduction 344
44.7 Some Other Applications of the Minimization Program 350 ẢDDPGHÏX cee nee ener e ener n ene es 353
References 0 ce cee eee teens 359
Trang 11Part I
Trang 121 The Structure of Isolated Impurity States
This chapter is a brief introduction to the theory of impurity centers in semiconductors, The reader interested in more detail is recommended a
review by Kohn [1.1], the more recent one by Bassani et al [1.2], and the
book by Bir and Pikus [1.3]
1.1 Shallow Impurities
At low temperatures most of the electronic properties of semiconductors are
determined by impurities An impurity can be of either donor or acceptor type A donor impurity can be comparatively easily ionized in the crystal medium by donating an electron to the conduction band These electrons can then participate in transport processes while the impurity centers become positively charged Donors give rise to the electronic type of conductivity in semiconductors When the impurity concentration is not too large, electrons are captured by donors at a sufficiently low temperature and become neutral This phenomenon is sometimes called the "freezing-out" of conduction electrons
The most important characteristic of an impurity is its ionization energy, ie, the energy necessary to move one electron from the donor level to the
bottom of the conduction band The freeze-out temperature is mainly determined by this energy On the energy-level diagram of a semiconductor
the donor tevels are located in the forbidden gap (Fig 1.1)
Ec
Eo Fig, 1.1 Band diagram of a semiconductor Zc and Ey are,
respectively, the edges of the conduction and the valence
TT re rẽ Ca bands; Ep and E, are the energies of the donor and accep- Ey tor levels
A donor impurity is called shallow if its level is close to the bottom of the conduction band, i.e., when the ionization energy is small compared to the
energy gap Shallow impurities play a special role in semiconductor physics
Trang 13An acceptor impurity has the property of capturing one electron from the crystal The impurity center becomes negatively charged while a hole appears
in the valence band Acceptors in semiconductors are responsible for
conductivity by holes, or p-type conductivity At low temperatures holes are frozen out, each hole being localized near an acceptor Acceptor levels are also located in the forbidden gap (cf Fig.1.1) Shallow acceptors are those whose levels are close to the top of the valence band
Whether a given impurity is a donor or an acceptor is determined in many cases merely by its position in the periodic table Thus, for example, in
semiconductors of Group IV (Ge, Si), impurities which belong to Group V (P, Sb, As) are generally donors This rule is related to the tetrahedral lattice structure of the Group IV semiconductors Each atom is bound to its
four nearest neighbors with covalent chemical bonds formed by four sp?
orbitals The elements of Group V have five valence electrons, and when
placed in a tetrahedral host structure, these atoms easily lose the excess electron, i.e., become donors In contrast, Group II] elements lack one electron, which they can easily capture from the host, giving rise to a mobile
hole in the valence band of the latter Therefore in germanium and silicon
the Group III elements (B, Al, Ga, and In) become acceptors We note that the type of impurity is not always determined by properties of the impurity atom itself For example, in gallium arsenide an atom of Ge or Si can become either a donor or an acceptor, depending on whether it substitutes for gallium or arsenic.!
There is a situation in which the structure and energy of an impurity state are almost independent of its particular chemical nature This occurs in the very important and common case of shallow impurities To be specific we shall discuss this situation in the instance of donors, while bearing in mind that for acceptors the argument is similar The proximity of a donor level to the bottom of the conduction band implies that an excess clectron is weakly bound to the donor center, and located far from it on the average This means that the atomic structure of the impurity center has little influence on
the state of the extra electron, which is bound to the center only because of
the positive charge on the center We can therefore regard the impurity center as a point charge and assume a central potential for the electron motion, viz
U(r) = er/kr , (1.1.1)
where e is the electron charge, r the distance to the center, and « the dielectric permittivity of the lattice It is permissible to use « when the radius
of an impurity state greatly exceeds the lattice constant In low-symmetry
crystals « is a tensor We note that since an impurity center does not move,
1 It is possibte for a single atom to have many impurity levels of cither kind, e.g., gold in
Trang 144 1 The Strueture of Isolated Impurity States
only the static permittivity enters (1.1.1), even though the dispersion (œ)
may be appreciable at frequencies corresponding to the ionization energy of the center (In this regard the impurity-center theory differs from the theory of excitons, in which both negative and positive charges must be considered mobile, which may lead to dynamical effects in lattice screening.)
In assuming the potential (1.1.1), we have excluded from the picture all the individual features of an impurity center associated with its chemical
nature Clearly, in this approximation the levels of all donors must be identical Of course, they do depend strongly on properties of the host semiconductor, not only through its dielectric constant « but also through the
dispersion law for electrons
The characteristic distance at which an electron can be localized near a
shallow impurity center (localization radius) is much larger than the lattice
constant Therefore in a Fourier expansion of an electronic wave function, wave vectors small compared to the reciprocal lattice vector will be dominant
Consequently, the electron dynamics will receive a contribution only from a
portion of its energy spectrum corresponding to wave vectors near the bottom
of the conduction band These ideas are mathematically expressed in the
effective-mass method, which forms the foundation of the electron theory of semiconductors, This method is reviewed in the present chapter
In Sect 1.2 we consider the case of a nondegenerate band and describe in detail the structure of donor levels in germanium and silicon The degenerate band case and the structure of acceptor levels in typical semiconductors are discussed in Sect 1.3 The main purpose of this chapter is to derive the asymptotic properties of the impurity-state wave functions at large distances
from the center These properties will be important in developing the theory
of hopping conduction Asymptotic properties of the wave functions wil be considered in Sect 1.4
1.2 Impurity Levels Near a Nondegenerate Band
A band is called nondegenerate if it possesses no more than a twofold spin degeneracy near its extremal point Such is the case for the conduction band of typical semiconductors (Ge, Si, A3 B; compounds)
In an ideal lattice the single-electron Hamiltonian is of the form
RP
2mg
where mo is the free-electron mass and V(r) is the periodic potential of the
lattice; A is the Laplace operator The solution to the Schrédinger equation is
provided by the Bloch functions
Hạ =— A +V(), (1.2.1)
đa = — Une , (1.2.2)
Trang 15wave vector, » labels the band and the spin, and Vo represents a normalization volume We shall restrict the wave vector to the first Brillouin
zone and assume that within this zone we know the eigenvalues £,,(k) of the Hamiltonian (1.2.1)
Our problem is to solve the Schrédinger equation for the Hamiltonian
H=H)+ U(r), (1.2.3)
with U(r) given by (1.1.1) Consider first the simplest case
1.2.1 Extremum at the Center of the Brillouin Zone
In this case the minimum of &, (k) occurs at k= 0 We assume that in the vicinity of this point the function is isotropic, i.c., for small k we have
fk?
2m
The quantity 7 is usually referred to as the cflective mass We shall seek a solution to the equation E,(k) = (1.2.4) (Hạ + UY = EY (1.2.5) in the form ý= DY Bek bye @ « (1.2.6) alk’
Substitute (1.2.6) into (1.2.5), multiply (1.2.5) by 4,0, and integrate over
r Taking into account the normalization of the Bloch functions, we find
LE„(k}—E]B,(k) + Ð UZ, B„(k) =0, — where (1.2.7)
atk’
" | * iŒ'—k}+
unk, = T Pda uy EO OG) dr, (1.2.8)
From the qualitative argument of the preceding section it is natural to assume
that B,(k) is nonvanishing only in a small region near the Brillouin zone center Considering the matrix element (1.2.8) at small k and k’, we can let
Unk F Ung and uqty 2 tạo The integrand in (1.2.8) contains a rapidly
oscillating term uz unto with the periodicity of the lattice, and a slowly
varying factor U(r) exp[(k'—k)-rl This allows us to write (1.2.8) as a product of two integrals
* 1 dew
UẠR, = Sura tua dt fe® tu ar, (1.2.9)
with the first integral taken over the volume of a unit cell From the orthonormality of the Bloch functions at k = 0 it follows that
Sul ovtnro dr = Sant (1.2.10)
Trang 166 1 The Strueture of Isolated Impurity States
Thus we arrive at the equation for B, (k) in a closed form:
[E, (kK) -E 1B, (kK) + SUkKK)B, (kK) = 0, where £ (2.11) ' 1 i(k'10-r k) = — Uk’) Va fe U(r) dr Are? =————r - kV o(k-k')? (1.2.12)
Substituting (1.2.4) and (1.2.12) into (1.2.11) we find
ek? 2m — E|B,(k) - (k) 4e? xử 2 ek? B,(k’) =0 — 1 )=0 (1.2.13) 2
Strictly speaking, in (1.2.13) we must sum only over k’ within the first
Brillouin zone In practice, however, the function B,(k) which satisfies
(1.2.13) is vanishingly small outside a small neighborhood of &k = 0 Therefore with a good accuracy, we can extend the summation in (1.2.13) to infinity This allows us to change to the coordinate representation of the wave function, 1 vị? F(t) = D Br wel , : (1.2.14) k
where the summation is over all k Multiplying (1.2.13) by exp (ik-r) and
summing over k we obtain
# | re) = EFO (1.2.15)
Equation (1.2.15) coincides with the Schrédinger equation for a hydrogen
atom, but with the effective mass m and the dielectric constant x The spectrum of bound states is given by the usual expression E,=- 4 — V1 m1 nh (1.2.16) and the ground-state wave function is of the form F(r) = (ma3)~!U2e”7!2 where (1.2.17) a = fi'x/me? ~ (2.18)
is an effective Bohr radius which determines the characteristic dimensions of the wave function Because of the large values of the dielectric constant and
Trang 17the small values of the effective mass, the Bohr radii in semiconductors turn out to be quite large For example, in GaAs, where «= 12.6 and
m — 0.066 mo, one finds @ = 100A The effective-mass method is useful for this reason Calculating B,(k) for the ground state from (1.2.14) and
(1.2.17) we find
8m12 1
vg? a$2 (k?+a 202
whence it is seen that the function B,{k) indeed falls rapidly for k > a7!
For the typical semiconductors, a~! is of order 10°? of the reciprocal lattice constant, which justifies our approximations
The full wave function of an impurity center is determined by (1.2.6) in which we can let By: = SyyB, Hence we have B, (k) = (1.2.19) ro WO) = DB, Won , k 1 k ~ ra DB (uy xe, (1.2.20) Neglecting, as above, the dependence of Unk, ON k we find 1 ier
V(r) = Tạ Uno) 3% (ke? = „a(r)FƑ Œ) (1.2.21)
Thus, the wave function of an impurity state is a Bloch wave function at the bottom of the band, modulated by a large-scale hydrogenlike function
Let us conclude our consideration of the case of a single extremum at the
zone center by presenting Table 1.1 [1.2], in which the theoretical and
experimental values of donor levels are compared Discrepancies are due to the fact that the effective-mass method poorly describes the potential at short distances from the center
Table 1.1, Dielectric permittivities, effective masses, and theoretical and experimental values of the
Trang 188 1 The Structure of Isolated Impurily States
1.2.2 Several Equivalent Extrema
The conduction band of such semiconductors as germanium and silicon
possesses several extrema located at equivalent points of the Brillouin zone
The conduction band minimum in silicon is located at k2 = 0.85 (2x/ag) in
the [100] direction, ao being the lattice constant The constant energy
surfaces represent ellipsoids of revolution around [100] The cubic symmetry of the crystal requires the existence of identical ellipsoids in each equivalent
direction Thus, there must be six equivalent minima in silicon Choosing the
z axis along [100], we find the following expression for the energy near a
[100] extremum:
pe Kune 2m, 7 7 + Im œ2 + kÐ 7 7” (1.2.22) “
The longitudinal (m,) and the transverse (m,) effective masses in silicon have the following values [1.2]: mg = 0.916 mo and mt, = 0.190 mọ, where mạ is the free-electron mass
The extrema of the conduction band of germanium lie exactly on the Brillouin zone boundary in the direction of the cube’s body diagonals Each
extremum is therefore shared by two zones, so that the number of extrema
equals half the number of the equivalent directions, ie., equals 4 The
constant energy surfaces in k space are ellipsoids of revolution with their axes
along the body diagonals The effective masses are [1.2]: m1, = 1.58 m9 and
m, = 0.08 my
The effective-mass method can easily be generalized to the case where several extrema are present In complete analogy with (1.2.21) the solution
of (1.2.5) near an extremum at k = k; is of the form
xy) = Fy); () , (1.2.23)
where ¢; is a Bloch function with k = ky normalized to the unit cell, and F;
is a smooth envelope function satisfying the equation Fj) = EF; (), (1.2.24) e? E,C¡V) - — Kr
where the function E;(k) is the conduction-band energy in the vicinity of k,
The energy eigenvalues £ are, of course, the same for all equivalent extrema Because of this, the impurity levels in the effective-mass approximation have an additional N-fold degeneracy (N being the number of equivalent extrema) In germanium and silicon this degeneracy is partially lifted because of corrections to the effective-mass method These corrections are
due to the fact that for small r the potential cannot be described by an
expression of the form (1.1.1) since the macroscopic dielectric constant is
meaningless at short distances One has to introduce an additional short-
Trang 19with & as in (1.2.12) but remain finite at large k These matrix elements
connect wave functions belonging to different extrema This leads to a splitting of the impurity states which were degenerate within the effective- mass approximation, as well as to a shift of energy levels which is usually called the chemical shift The form of such splitting can be predicted on the basis of symmetry alone, without a detailed knowledge of the short-range potential
We shall discuss only the ground-state splitting The correct wave
functions are linear combinations of the form
@ = > af?x;, i=1L.N (1.2.25)
=
The form of the splitting, as well as the coefficients al? for germanium and silicon, are determined from the symmetry properties with respect to transformations of the tetrahedral group Ty It turns out that in silicon the sixfold degenerate ground state is split into one singlet, one doublet, and one triplet The lowest energy state is the singlet Its wave function is of the form
= ew (1.2.26)
In germanium the fourfold degenerate ground state is split into one singlet and one triplet, The ground state is singlet, and its wave function is of the
form
4_ 1 ( )
$'= DX: 1.2.27
In the effective-mass approximation, the energy of an impurity state is determined by solving (1.2.24) Substituting into this equation the ellipsoidal
spectrum (1.2.22) we find
F=0 (1.2.28)
+ dx? ôy?
This equation contains one dimensionless parameter, which is usually chosen to be the ratio of the transverse and longitudinal masses: y = m,/m
Let us recast (1.2.28) in dimensionless form, using atomic units corresponding to the transverse mass, viz r'= r/a,, E'= E/E,, a, =
fi'x/e?m,, and E, = h?/2m,a? We obtain 2 2 2 oe +2 tot tế Qz ôx2 say? * £=0 (1.2.29)
Trang 2010 1 The SIructure of Isolated Impurity States
latter to the two-dimensional Coulomb problem Naturally, one would like to know the ground-state energy as a function of the parameter y in (1.2.23)
This problem was solved by Kohn and Luttinger [1.4] by a variational method
which proved to be very successful The trial function was chosen in the form (x+y?) 4 Zz 2 F = (rai ay) 'expi— 5 at ail 1/2 › (1.2.30)
and the two parameters a, and a, were found by minimizing the energy For = 1 the variational procedure yields the exact result: a, = ay, and the wave function is identical to that of the hydrogen atom ground state A less trivial fact is that even for y = 0 the variational energy differs from the exact one by
less than 8%,
The dependence E'(y) obtained by the variational method is shown in
Fig 1.2 Using the above values for the effective masses and the dielectric constants of 11.40 for silicon and 15.36 for germanium, we find that the effective-mass approximation of the ground-state energy is 31.27 meV for
silicon and 9.81 meV for germanium [1.2]
ˆ
4 Exact value
Fig 1.2 Ground-state energy in
units of E, calculated with the
trial function (1.2.30) [1.4] and 2 32 ñ4 nã 28 Tp 7? plotted as a function of y!⁄3
The excited-state energies are calculated in a similar way Equation (1.2.29) is axially symmetric and hence the good quantum numbers are the magnetic quantum number m and the parity Levels corresponding to the same magnitude of m but different signs are degenerate It is customary
to label the excited states of (1.2.29) by the corresponding hydrogenic states
at y= 1 However, for arbitrary y one has additional level splittings For example, the 2p state of hydrogen is replaced by one state 2p9 with m =0 and two degenerate states 2p with m = +1
We conclude this section by presenting Table 1.2 [1.2], which shows both theoretical and experimental values of energy for the ground and excited
states in silicon and germanium As is clear from the table, the effective-emass
Trang 21Table 1,2 Donor ionization energies in Ge and Si Theoretical values were calculated in the effec- tive-mass approximation Experimental values correspond to the impurities indicaled in paren- theses Energies of all s-state tevels, split due to the chemical shift, are listed separately Material E\;, [meV] Ex, [meV] Si (theor) 31.27 11.51 Si (P) 45.5 33.9 32.6 11.45 Si (As) 53.7 32.6 31.2 11.49 Si (Sb) 42.7 32.9 30.6 11.52 Ge (theor) 9.81 4.74 Ge (P) 12.9 9.9 4.75 Ge (As) 14.17 100 4.75 Ge (Sb) 10.32 10.0 4.74
by an incorrect treatment of the short-range part of the potential As discussed above, including this correction gives rise to a splitting of the ground level and a substantial shift which increases the ground-state binding energy The effective-mass method works much better for p states The reason is that wave functions of these states vanish at the origin, so that their energy is not sensitive to the exact form of the potential near the impurity
center
1.3 Impurity Levels Near a Point of Band Degeneracy
Let us suppose that eigenstates of the Hamiltonian (1.2.1) are degenerate at a
certain point in k space This means that there exist ¢ > 1 Bloch functions oi satisfying the equation
Hobdy = En, Wojn, f= 12.40 (1.3.1)
In what follows we shall consider the case when the degenerate point coincides with the center of the Brillouin zone, k = 0 Such is the case for the valence band of typical semiconductors This can be simply viewed as a consequence of splitting the atomic p orbital At k = 0 ‘the band is sixfold degenerate, spin included For finite & the degeneracy is partially lifted
(Fig 1.3)
The effective-mass method is easily generalized to include this case [1.5]
Trang 2212 1 The Structure of Isolated [Impurity States
Fig 1.3a,b Band diagram of typicat semiconductors (@) Spin-orbit cou- pling is absent and the valence band is sixfold degenerate at k= 0 (b) Spin-orbit coupling splits a subband of total angular momentum j = 1/2 a) satisfy an equation analogous to (1.2.15), viz t 3 5| 3 Hƒ! 0.ô; + U0)ä¡; J’ Lagat p=-ihV (1.3.3)
The difference is that here the envelope function constitutes a r-component
column vector, so that the He form a matrix not only with respect to the Cartesian indices a,6 but also with respect to the ¢ components of F;
This result provides a quite natural generalization of the effective-mass
method discussed in the preceding section For a detailed derivation of (1.3.3) the reader is advised to consult the original paper by Luttinger and
Kohn [1.5] In that work the matrix H is expressed in terms of the matrix elements of momentum, taken between Bloch functions Another possible
approach to the problem is simply to write down the matrix H from symmetry considerations This approach is chosen below
So far we have not been taking into account effects due to spin-orbit coupling To include these effects one must add to the ideal-lattice electron Hamiltonian H a term of the form
Hy) = t+ lox VVI6, 4ml c? 434)
which represents a relativistic correction, as follows from the Dirac equation Here a is the spin operator (Pauli matrices), V(r) is the potential energy of
an electron in a periodic field which appears in (1.2.1), and c is the speed of light
The valence-band structure is pronouncedly affected by the spin-orbit
interaction, In the presence of such interaction the spin and the orbital
Trang 23break into two groups of different energy One group contains four states and corresponds to the total momentum 3/2, while the other contains two states of momentum 1/2
A similar splitting occurs in the erystal When spin-orbit coupling is taken into account, the valence band, originally sixfold degenerate at k = 0, splits into two bands, which are fourfold and twofold degenerate (Fig 1.3b) The twofold degenerate band is shifted down by the amount A, which is the energy of the spin-orbit interaction (values of A for Ge and Si are given in
Table 1.3)
Table 1.3 Parameters of the Luttinger Hamiltonian, spin-orbil splitting energy A, and dielectric constants « for Si and Ge [1.2] Material 1 *% % A K Si 4.22 0.39 144 0.04 11.4 Ge 13.35 4.25 569 0.29 15.4
The general form of the matrix H in the presence of spin-orbit coupling is rather complicated and we shall not write it down Considerable simplification is achieved in two limiting cases The first case corresponds to a weak interaction when the spin and coordinate variables are decoupled and
the matrix H°?A,Hg is, in fact, a 3 x 3 matrix The second case is when the
interaction is strong, and the band of total momentum 1/2 is far removed in energy, so that its influence can be ignored in the vicinity of the valence-band edge In this case only the four top bands need be considered in (1.3.2), and
the matrix H°? 6p reduces to a 4 x 4 matrix
In relation to the structure of an impurity state, we can use the first approximation when the spin-orbit splitting A is small compared to the impurity ionization energy, and the second approximation in the opposite case It will be seen below that the strong spin-orbit coupling approximation is valid for germanium and for a number of other semiconductors Let us begin with this approximation
We have to construct the matrix H“’,f, in the neighborhood of a
fourfold degenerate band The Hamiltonian must satisfy the requirements of crystal symmetry under rotations of the coordinate system The wave function is represented by a column of four coordinate functions Quantum- mechanically this implies that an equation for a spin-3/2 particle is needed In order to write such an equation, one usually introduces a pseudovector J which represents the spin momentum operator Its components are fourth- rank matrices whose meaning is analogous to the Pauli matrices for spin-1/2
Trang 2414 1 The Structure of Isotated Impurity States
There exist only two invariants which are quadratic in fp, namely ÿŸ and
(f- D?, under arbitrary rotations of the coordinate axis A rotationally
invariant Hamiltonian would then contain only two independent parameters, and the matrix H could be written in the form
L | pe 5
H=— mẹ 2 T¡ + 2
— yp: | (1.3.5)
However, cubic symmetry is less restrictive than full spherical symmetry In
a cubic crystal there is one more invariant which is quadratic in ô, namely p? J2 + py J} + 6À J2 Therefore the full Hamiltonian can be written in the form I § Be 2.2 H=—— mạ |! +2 2| 2 = ~›tộ - 1) lp: + Œa—+) (82 J2 + 8272 + 0222| (1.3.6)
This Hamiltonian is named after Luttinger, who was the first to derive it
[1.6] The parameters yj, +; and +y; are well known for many
semiconductors Their values for Ge and Si are given in Table 1.3
It should be noted that (1.3.6) represents the complete Hamiltonian only for crystal lattices possessing a center of inversion For crystals without inversion symmetry, the Hamiltonian contains not only quadratic but also linear terms in p For example, in crystals with a zinc-blende lattice (InSb,
GaAs, etc.) the Hamiltonian may include the following term:
Bu Wx, J} — 2D} + By, U2 - JD) + 6,02 ~ 2D), (1.3.7)
where the braces stand for an anticommutator, {a,b} = ab + ba However,
the linear terms are very small in all extensively studied semiconductors, and can usually be neglected For this reason we shall not consider them further
We have obtained a system of four second-order equations (1.3.3) which
determine the energy and the wave functions of an impurity center Before
embarking on its solution let us consider the system (1.3.3) for free particles
by setting U(r) = 0 The Hamiltonian (1.3.6) describes motion of a spin-3/2
particle Its first term represents the kinetic energy operator, and the
remaining terms describe a peculiar spin-orbit interaction Inasmuch as the
Hamiltonian does not explicitly depend on the coordinates, one should seek a
solution to the Schrédinger equation in the form:
F,-A, m] er (1.3.8)
Trang 25
Hy obtained from (1.3.6) by substituting Ak for the operator p By definition, one has k_ |k| ~⁄ Dị k (1.3.9) D> Ai; ij
Consider first the Hamiltonian (1.3.5), which is a scalar Its eigenvalues do
not depend on the direction of k with respect to the crystallographic axes We can therefore direct k along the z axis and use the representation in
which the J? matrix is diagonal The system (1.3.9) then breaks down into
four independent equations with two different energy eigenvalues? :
y1727
E¿= —— Wk, EB, = —— PR é 219 4 2mo (1.3.10)
If both energies are positive, ie., y, + 2y > O and y, — 2y > 0, then the obtained branches of the spectrum are called the light and the heavy holes The hole masses m, and m, are defined in terms of the parameters y, and y:
imo m9
> mM, =
yitey xi2
mạ (13.11)
(assuming y > 0) The valence band with positive mg and m, is shown in
Fig 1.3b For other values of y, and y it is possible for the energies Ey and
E;, given by (1.3.10) to be of different sign Such is the case in the gapless semiconductors (a—Sn or "grey tin", HgTe, HgSe) In this situation one of the branches (1.3.10) corresponds to an empty conduction band and the other
to a filled valence band Both bands converge at k =0, ie., there is no forbidden gap The theory of impurity states in gapless semiconductors is rather unique, and will not be described here (the interested reader should
consult [1.7])
Consider now the properties of the Hamiltonian (1.3.6) Its eigenvalues do depend on the direction of k with respect to the crystallographic axes The energy spectrum can be found directly from (1.3.1) using the explicit form of the matrices J The resulting spectrum is of the form
RP
Een = 3p |» + [443 k Hg
1/2
+12|x‡ ~ x] [kẺ k + kệ kệ + kệ «| | (1.3.12)
It contains two twofold degenerate branches which are also usually called the light-hole and the heavy-hole branches The plus sign in (1.3.12) corresponds to light holes The constant energy surfaces described by (1.3.12) are no
Trang 2616 1 The Structure of Isolated Impurity States
longer spherically symmetrical, but become somewhat crimped (raspberry- like) For a number of semiconductors, including germanium, the anisotropy of the isoenergetic surface is rather weak In these cases a sufficiently good approximation (sometimes referred to as the spherical model) is obtained by
restricting the Hamiltonian to the form (1.3.5) One can show that this form gives the best approximation to (1.3.6) if one lets
vy = By3 + 2y2)/5 (1.3.13)
From (1.3.11), (1.3.13) and Table 1.3 we find that in the case of germanium
me = 0.042 mo and m, = 0.32 myo
Let us now return to the impurity-state theory Early calculations of the acceptor binding energies were carried out variationally using the exact Hamiltonian (1.3.6) and choosing the trial wave functions on grounds of symmetry [1.1], More recently, a number of important advances have been
made with the help of the spherical model, i-e., using the Hamiltonian (1.3.5)
[1.2] As discussed above, neglecting the ripples on isoenergetic surfaces is a good approximation for many semiconductors At the same time it simplifies the variational calculations considerably and also yields a number of exact results
This advantage is due to the fact that the spherically symmetric Hamiltonian (1.3.5) commutes with the total angular momentum operator j
[under the Hamiltonian (1.3.6) j is not conserved] The operator j can be represented as a sum j= J +L of the spin momentum J and the orbital
momentum L = #”![r x pl, neither of which commutes with the Hamiltonian
separately It should be recalled that the term “spin” is used here only to stress the mathematical analogy discussed above In reality this "spin" is related to an orbital splitting of the atomic levels which constitute the valence
band The orbital momentum L characterizes only a "macroscopic" orbital
motion described by the effective-mass method Nevertheless, this formal analogy will permit us to use the angular momentum coupling rules familiar from quantum mechanics and obtain meaningful results
As discussed above, the particle spin in our case is 3/2 It follows that the
eigenvalues of the total angular momentum j can take only half-integer values not less than 1/2 From the addition theorem for angular momenta it also follows that for a given j 2 3/2 the orbital quantum number ¢ can take
four values which are integers in the interval ƒ—3/2) S @ < (j+3/2) For
Trang 27Here W¿„ is a spherical harmonic; x, is an eigenvector of the matrix J,
corresponding to the eigenvalue » (ie J,x, = #x,), with « taking on the
3/2 ÿ mp —M
M =m+u The allowed values of @ for a given j were considered above The expression (1.3.14) is to be substituted into the Schrddinger equation values —3/2, —1/2, 1/2, 3/2; is the Wigner 3—j symbol, with
[H+ UO) Fig = EjFin (1.3.15)
in which H is given by (1.3.5) and U by (1.1.1) For j > 1/2 one finds a
system of ordinary differential equations connecting the four radial functions R;,¢ corresponding to a given j This system is further reduced to two systems of two equations, one involving functions with @ = j+t/2 and
€ = j-3/2, and the other functions with @= j—1/2 and @ = j+3/2 This
important property is a consequence of the conservation of parity The only radial functions which are connected are those corresponding to spherical harmonics of the same parity The ground state corresponds to a value of j such that the expansion (1.3.14) contains a term with = 0 In accordance with the above discussion, this means that the ground state must have the total momentum j = 3/2 The corresponding radial functions Ro = R39 and R2 = R32 obey the following coupled equations: 4đ 1||4 3 d 1| 4Ro (+8) dx | dx x R+a 54 xị] dx +28|—e+ 2| 8; =0, x 4 2] dRo 3 2|| 4 a+8) dx! x dx + 0-8) dx x dx x Ro 2 + 28|-e++] Ry = 0, (1.3.16) x
where 8 = (y\—2y)/ (y,+2y) = me/m,, and the dimensionless atomic units x =rm,e2/kh? and ¢ = — E242B2/e mạ are based on the heavy-hole mass
Thus, in the spherical model, the system of four coupled partial differential equations (1.3.3) reduces to a system of two ordinary differential
equations containing one dimensionless parameter 6, which is the ratio of masses of light and heavy holes Computer calculations by Gel’mont and
Trang 2818 1 The Structure of Isolated Impurity States
Fig 1.4 Dependence of the energy of the acceptor ground state (divided by the heavy-hole Bohr energy) on the ratio of light and heavy hole masses [1.9]
0 05 r3
For 8= ! one has c= l by deñnition Gelmont and D’yakonov found that for Ø >0 the energy c= 4/9 Thus, in the entire range of 6, the ground-state energy varies approximately by a factor of 2, and its order of magnitude agrees with the Bohr energy of a heavy hole Note that in germanium @ = 0.13, and the quantity (4/9) m,e4/2«h? = 8.1 meV gives a reasonable estimate of the ground-state energy (cf Table 1.4) In InSb one
has 8 = 0.03 and 4 m,e4/18 «2A? = 8 meV, while the experimental value of
the ground-state energy for such acceptors as Zn and Cd is 7.5 meV We see that for a number of semiconductors it is a valid approximation to take the hole mass ratio 8 = 0
We have discussed the spherical model for the case when the Hamiltonian reduces to a fourth-rank matrix, which corresponds to a strong spin-orbit interaction The spherical approximation is also sufficiently good in the opposite limit of small A It again simplifies the problem by permitting a
separation of variables
In the work of Baldereschi and Lipari [1.10] the acceptor-level energies were calculated variationally in both limiting cases The values obtained for the ground-state energy of germanium and silicon are listed in Table 1.4
Trang 29For silicon Baldereschi and Lipari gave an interpolation between the limits of strong and weak spin-orbit interaction They also calculated the ground
and excited states for a number of other semiconductors
1.4 Asymptotic Behavior of the Impurity-State Wave Functions
In developing the theory of hopping conductivity in lightly doped semiconductors it is important to know the behavior of the electron wave function far from impurity centers The wave functions of localized states
decrease exponentially at large distances, and for many important problems it
is sufficient to know the exponential factor
In this section we restrict ourselves to calculating the exponential part of the wave function at large distances from the center We shall be using the
effective-mass method, which works well at large distances provided the
characteristic decay length q~! is large compared to the lattice spacing We begin with the case of a nondegenerate band The effective-mass
equation is of the form
[E@+UMIF=EF, a4.)
where E(p) is the dependence of energy on the crystal momentum for the
band under consideration, and p = - i#V
The asymptotic behavior of the wave function can be determined in the quasi-classical approximation, in which we seek a solution in the form
Fo = GSO (1.4.2)
We are interested in the behavior of the wave function deep under the barrier
In this classically inaccessible region the function is exponentially small, i.e., the imaginary part of the action S is large This ensures the applicability of the quasi-classical method, which is valid when
d ii
—— | << 1 1.4.3
dr |dS/dr| Ị ( )
What makes our discussion below somewhat unusual is that the quasi- classical method is rarely used in solving three-dimensional problems with nonseparable variables
Substitute (1.4.2) into (1.4.1) and, following the usual rules, keep only the
lowest-order terms in the Planck constant A This means that only the first
derivatives of S' should be kept, so that
E@F=E(VS)F (1.4.4)
Trang 3020 1 The Structure of Isolated Impurity States
r Fig 1.5 A quasi-classical trajectory passing through the
point r
E(VS)+U=E (1.4.5)
Suppose we know the wave function F on a certain surface o (Fig 1.5)
That surface is assumed to be much closer to the impurity center than the
point r, at which we are evaluating the function F(r), but still far enough
away for the condition (1.4.3) to be satisfied on the entire surface At each
point of the surface « we can then consider both the action S(r,) and its gradient VS [the normal component of the gradient is determined by (1.4.5)]
to be known
Through each point of the surface o we draw a trajectory (ray) which satisfies the classical equations of motion and has the momentum p= VS on
the surface c Then at an arbitrary point r, the solution to (1.4.5) can be
expressed as an integral along the trajectory that passes through r:
r
Sứ) =.SŒ) + Ƒp-dr, (1.4.6)
Ue
where r, is the point at which the trajectory intersects c The equation of the trajectory is determined from Hamilton’s equations
ðE ,_ _ðU
ap’ P or
Let us now choose the surface o at a distance from the center such that one can neglect the value of the potential U(r) In this case the trajectory which
enters (1.4.6) is a straight line on which p = po = const By definition, the value of the action S(r,) is small compared to S(r), while the value of the integral (1.4.6) is determined by its upper limit We thus obtain the simple
result: +
(1.4.7)
Sứ) =pạg-r (1.4.8)
As seen from Fig 1.5, on each trajectory which passes through the surface o and a point r far removed from a, the velocity must be close in direction to the radius vector r (the impurity center is assumed to be located at the
origin) Hence the vector pp in (1.4.8) must satisfy the conditions
ðE /|ðE
ðp op B= Po = n› (1.4.9)
Trang 31where n=r/r These two conditions must be complemented by one which follows from (1.4.5): E(p)) =E (1.4.10) Thus, the exponential factor of the wave function at large distances is of the form re (14.11)
with pg given by (1.4.9) and (1.4.10)
To illustrate the method, we first consider the trivial case E(p) = p/m In this case the velocity 8£/dp coincides with the direction of p, and hence in
(1.4.11) one must set Po parallel to n Then one has
FQ) =e with ạ= wenn (1.4.12)
If we substitute for E the hydrogenlike spectrum (1.2.16) obtained in the
effective-mass approximation, we find that the exponent in (1.4.12) coincides, as it should, with that of the exact solutions For example, the ground state
has g = a7!, where a = fi?x/me? is the effective Bohr radius It is important
to realize, however, that the expression (1.4.12) is of a more general nature than (1.2.16), and its validity in some sense extends beyond the effective-mass approximation
Indeed, energies obtained in the effective-mass approximation differ from true energies of the impurity levels, mainly because of the inadequacy of the potential (1.1.1) at short distances from the impurity This is seen, for
example, from the fact that the effective-mass method gives much better
results for p-state energies than for s states (cf Table 1.2) In contrast, our
derivation of (1.4.12) made no use at all of the Schrédinger equation at short distances Therefore the energy E in (1.4.12) should not be taken as an
effective-mass approximation for energy It makes sense to substitute into this
formula the experimental values for energy levels which are different for
different impurities One should remember, of course, that one of the assumptions which went into its derivation was the validity of an effective mass at large distances, i.e, the condition gay << 1, where apg is the lattice period
Trang 322 1 The Structure of Isolated Impurity States
Pox = Mxd, Poy = Myd, Por = md/y , (1.4.14)
where d = (pg, + pi + y’pd,)'? and y = m,/mty Substituting (1.4.14) into
(1.4.10) for a negative energy E we have 2 2m, |z| đˆ=—————— (1.4.15) ne tng + ably Substituting (1.4.15) and (1.4.14) into (1.4.11) we obtain F(t) = e0, (1.4.16) where
gin) = & "2m, |E| (2 + Hộ + n2/y)]2, (1.4.17)
We see that the asymptotic shape of the wave function, described by
surfaces of constant exponential factor, is also ellipsoidal
Let us make estimates for germanium The wave function corresponding to one of the ellipsoids falls off in the direction of its revolution axis
(n, = n, = 0) with a characteristic decay length gz! = hQm El)? = 13.8A,
and in the transverse direction with the length
gi! = hQm, |E))~? = 614A
(We have used |£| = 12.9 meV, which corresponds to the ground state of a phosphorus impurity.)
As pointed out in Sect 1.2, in semiconductors with several equivalent
ellipsoids (Ge, Si) the wave function represents a linear combination of functions [cf (1.2.25 - 27)] corresponding to different ellipsoids To each ellipsoid corresponds its own function g,;(n) which takes into account the
ellipsoid’s orientation in k space
However, at large distances from the center in any direction in coordinate
space, some of the modulating functions will become much larger than others
The largest modulating function for a given direction is determined by
(1.4.16) with
q(n) = min{g;@)} (1.4.18)
Trang 33with d/dx and with 26lel in (1.3.16), and adding and subtracting the two equations, we find d*(Ro+Ro) 2 = BelRy+ Ro) , dx d?(Ry— Ro) a = Ry Ro) dx (1.4.19)
For small values of 8 = n2t¢/m,, the sum of the functions falls off much more slowly than their difference Therefore at large x,
R,= Roe exp (—x Ble = exp | —r v/2m;]E]/R] (1.4.20)
We see that in the case of disparate hole masses the asymptotic behavior of wave functions at large distances is governed by the light mass, while the binding energy is mainly determined by the heavy mass
This important conclusion remains, of course, valid beyond the spherical model, i.e., when crimped isoenergetic surfaces are taken into account In order to consider this case we generalize the above quasi-classical method to include the case of degenerate bands
We shall seek a solution to (1.3.3) in the form
F,= 7 =A i SOM | (1.4.21)
Ak=VS,
where the A; satisfy (1.3.9), i.e., represent components of an eigenvector of
the matrix obtained from the Hamiltonian (1.3.6) by substituting a c number for the operator p Substituting (1.4.21) into (1.3.3) and neglecting all terms containing second derivatives of S(t), we have 3 3 as as 4 4 3 3 Myf pabsky= > MBS BH Š nụ dụ j=l a=) ap=1 Xa Xã io) = Ex, (SF, , (1.4.22)
where the function Ez,(VS) is obtained from (1.3.12) by substituting Ak for VS, and the plus or minus sign in (1.3.12) is chosen depending on which
branch of the spectrum corresponds to the eigenvector 4;
Thus, we obtain in place of (1.3.3) a Hamilton-Jacobi equation
E;„(VS) = E-U() (1.4.23)
Trang 3424 L The Structure of Isolated [mpurity States
heavy (S;,) holes The fact that these actions are decoupled, that is to say, Sy does not enter the equation for Sy and vice versa, means that in the quasi- classical approximation the motions corresponding to the light and the heavy holes can be regarded as independent At short distances from the center the quasi-classical approximation is not valid These motions are no longer independent and the wave functions are coupled Therefore in reality, at large distances the solution sought represents a superposition of the solutions
(1.4.21) which correspond to light and heavy holes It is easy to see that if S, and S, are such solutions of (1.4.23), then Im(S,) < Im(S,) Hence the
asymptotic behavior of F; is determined by the action Sy As in the simple
band case, the result is expressed by (1.4.9 - 11), where one should substitute E2(p) instead of £ (p)
The result can also be expressed in the form (1.4.16), although it is difficult to solve equations (1.4.9) and (1.4.10) for an arbitrary direction Therefore we confine ourselves to calculating g(n) in two directions: [100] (m = 1, my =n, = 0) and (111) G@, = ay = ny = 1/V3) It is readily seen
that the maximum variation of the quantity q(n) occurs between these two directions The calculation is simplified considerably by the fact that in both directions the group velocity O£,/8p is parallel to the momentum p Therefore the direction of pg in (1.4.11) coincides with n According to
(1.3.12) we have in the [100] direction 2 E,=— ly, + 2m (1.4.24) 2mo From (1.4.10) and (1.4.11) it follows that QmolE |)? =a (1.4.25 411001 ñŒ + 12 ) Similarly, in the [111] direction (2ma|E D12 = 1.4.26 đụ Aly + ryt ) The spherical model result (1.4.20) can also be put in a similar form: (2m |g|)1⁄2 sph = ————— Aly, + 20% + 2yz)] (1.4.27)
For germanium doped with gallium (JE| = 11 meV) we find giibo = 87A, qaiu = 92A, and đạn = 90 Ả We see that in this case the spherical
Trang 35In the preceding chapter we considered the structure of electron states in the vicinity of a single impurity center We now proceed to the most fundamental questions in the theory of doped semiconductors: how do impurity states belonging to different centers influence one another, and what is the resultant energy spectrum for a crystal containing a finite concentration of impurities?
It is often said that in the case of a finite concentration of similar impurities the single degenerate impurity level is replaced by an impurity band of finite width in energy Although the term "impurity band" will be used in our book, we would like to warn the reader from the outset that this term should be applied with great caution The impurity band of a weakly doped semiconductor does not possess the most important property of a crystal band: an electron localized near one of the impurity centers does not spread in time over other centers constituting the band The wave function remains localized Nevertheless, we shall be using the term “impurity band" to denote the aggregate of energy levels arising due to impurity centers
The first characteristic of an impurity band is its density-of-states function It is defined as the number of states with energies falling into a small energy interval, taken per unit length of this interval and per unit volume of the system It should be realized that in a macroscopic system the density of states is a continuous function of energy, even if one speaks of an impurity band which represents a set of discrete levels Thus, the density-of- states function does not contain the information which can allow us to distinguish a true band from a set of discrete levels unrelated to each other and randomly scattered in the energy space
The present chapter is devoted to very general questions related to the localization of electron states We begin by studying the quantum broadening of energy levels Using the tight-binding method we shall discuss properties of the narrow bands formed by impurity centers arranged in a periodic pattern at large distances from one another This standard treatment leads to a band structure which is unstable Electron-electron interaction gives rise to a splitting of the band in which the filled and the empty levels become
separated by a gap (the Mott transition) The varying potential at different
Trang 3626 2 Localization of Electronic States
2.1 Narrow Bands and the Mott Transition
We begin this section by considering an auxiliary problem which will be important later Suppose impurities are not randomly distributed in the
crystal but form a regular crystallic lattice (impurity sublattice) having a
much larger period than the host lattice With the help of the tight-binding method [2.1] we can obtain the energy spectrum and the wave functions of electrons in the impurity band This model will allow us to develop the terminology to be used in subsequent sections It will also allow us to discuss an important question concerning electron-electron interaction under narrow- band conditions
The potential produced by the impurity sublattice is of the form
VQ) = 3U(r-—r,), (2.1.1)
i
where the sum extends over all sites of the impurity sublattice and U(r) is the
potential of a single impurity center We shall treat the problem in the effective-mass approximation Let us assume that we know the wave functions ¢, and the energy levels &, corresponding to the Schrédinger
equation with the single-impurity potential lề
|-#-2+ ut) 2m bn = Enda s (2.1.2)
where » is an effective mass
For simplicity, we restrict ourselves throughout this chapter to the case of a standard band, i.e., a nondegenerate band with an isotropic and parabolic spectrum We shall not consider transitions between states of different bands of the host crystal Therefore we can drop the Bloch factors in the wave functions and consider only the envelope functions
In what follows it is assumed that the impurity band width is much smaller than the separation between the levels E,, so that we can restrict ourselves to the vicinity of one of these levels, Eo
It is well known that a wave function corresponding to the potential (2.1.1) should be constructed in the form of a superposition of functions o(r—r;) satisfying (2.1.2): v= Yajor-r) , (2.1.3) i >lz;?=1 (2.1.4) 2
Trang 37function ¥(r) is large As seen from (2.1.3), these regions are close
neighborhoods of the impurity centers, i.e, they lie within the sphere of influence of an individual center where (2.1.2) is valid Therefore the
expression (2.1.3) must be close to a solution of the Schrédinger equation with the potential (2.1.1) The coefficients a; should be obtained by
minimizing the energy
Inasmuch as the wave functions $(r—r,;) corresponding to different sites are not orthogonal, the energy expectation value cannot be expressed as a quadratic form in the coefficients aj However, when a << bg the overlap between neighboring states is small In this case it is sufficient to use the first approximation in the expansion of the energy expectation value in aj 73m The part of energy depending on aj Gj+m is given by
E = 34; ajimi (mr) : (2.1.5) jm
The quantity /(m) is called the energy overlap integral (or simply the overlap
integral) In this chapter we shall not be interested in the explicit form of I(m) Note only that under our assumptions this function is very small, since
it contains a factor exp(—@bo/a) (for nearest neighbors) with @ being a
numerical coefficient
As is well known, the set a; satisfying Bloch’s theorem
WŒ+r,) = W()e* (2.1.6)
is of the form a; = N7"exp(ik-r)) where N is the total number of sites in
the impurity lattice Substituting this expression into (2.1.3) and (2.1.5) we find w= Mở zoey | dels , (2.1.7) » /(m)etm, (2.1.8) m0 E
where m is a vector connecting a given lattice site with other sites Since
I(m) falls exponentially with increasing m, we can restrict the summation in
(2.1.8) to nearest neighbors only The resulting expressions depend on the lattice type For example, for a simple cubic lattice we find
E = 2 (bo) [cos k,bo + cos kybo + cos kz bol , (2.1.9)
and for a simple square lattice
E = 21 (bo) [cos kb + cos ky bo) « (2.1.10)
Trang 3828 2 Localization of Electronic States
energy values For a simple square or cubic lattice the first Brillouin zone
corresponds to all k in the interval —/by < ka < m/bo (here a stands for Cartesian indices)
The energy width of the allowed band Vs, equals 12|/(bo)| for a simple
cubic and 8|/(49)| for a simple square lattice In these two examples one may note a rule that V, = 2Z|I(bo)|, where Z is the number of nearest
neighbors It is often believed (wrongly) that this rule is valid for all simple
lattices That this is not true can be seen from the example of a two-
dimensional triangular lattice for which Z = 6 but V, = 9|/(bo)| For small k we find from (2.1.9)
cŒ) = —1(bq)k?hậ , (2.1.11)
where, by definition,
e(k) = E-61 (by) (2.1.12)
As seen from (2.1.11), the quantity 42/2b¢ plays the role of an electron mass
in the newly formed band
As the separation between nearest neighbors increases, the allowed band
gets narrower [exponentially, since (bo) « exp—(8bo/a), where 8 is a
numerical factor] At the same time the mass becomes exponentially large This suggests that when the impurity concentration is high, i.e., bo is large, the band properties may be in some sense fictitious Nevertheless, we see, formally, that the electron wave function remains a modulated plane wave for an arbitrarily narrow band, and electrons can still move without scattering
Bands formed by impurities are no more than half filled, since every impurity contributes (or takes away) one electron, and the band is twofold spin degenerate Thus it appears that if the impurities really were arranged periodically, the conductivity of impurity electrons would be of a metallic
nature, however small the impurity concentration But this conclusion is
incorrect, even if a periodic impurity configuration is granted The problem lies in the single-electron approximation used in the above derivation This approximation, though adequate when dealing with wide allowed bands of typical metals, breaks down in the case of a narrow band,
As seen from (2.1.7), in the vicinity of each site the electron wave
function is closely approximated by the site function #(r) If we estimate the
interaction energy Uo of two electrons of opposite spins located on the same site, we find Uy = e?/a When the magnitude of Up is small compared to the
allowed band width V,, the wave function is only little perturbed by the
Trang 39£ | Fig 2.1 Dependence of electron bands on impurity 4 | sublattice period by To the left of point A is an in-
sulator, to the right a metal Et, | ! I tạ | Ị I —————1_— —x+ A tủy
At a finite value of bo, both levels spread into bands whose width is of
order |/(bo)| (Fig 2.1) The number of positions in cach of these bands is
half that in the band (2.1.8) and equals the number of lattice sites (the
bottom band cannot contain a site occupied by two electrons) The lower
band will become filled and the upper band empty Thus, if |/(do)| << Uo
our material is dielectric With decreasing 59 a certain point A is reached at which the width of the forbidden gap vanishes and the system goes into a metallic state This transition is usually referred to as the Mott transition Its nature is not yet entirely clear, especially as regards the behavior of electrical conductivity at the transition point
Quantitatively, the Mott transition is usually studied with the help of Hubbard’s model In this extremely simplified model it is assumed that electrons repel each other only when they are located on the same site The Hubbard Hamiltonian is of the form:
H= Ð lỮỨn)4j, đit +2 D Ajottj-o » (2.1.13)
jm #0 jo
where njg = aj, đ;„ is the occupation number operator of states on site j with spin o This Hamiltonian is obtained from (2.1.5) by adding a term describing the repulsion of electrons located on the same site and having opposite spins
The Hubbard model admits of an exact solution only in the one-
dimensional case [2.2] The result contradicts the above qualitative
arguments In the one-dimensional case a gap remains in the spectrum for all values of I(b)/Uo, i.e., the one-dimensional model always describes an insulator Such behavior, however, is usually considered an exclusive property of one-dimensional systems, so that it does not cast a shadow over the above qualitative picture, so long as that picture refers to two- or hree-dimensional
systems
It is not our intention to review all theories developed in connection with the Mott transition A detailed discussion of these theories can be found in
Mott’s book (2.3] or in the review by Khomskii [2.4] Returning now to
Trang 4030 2 Localization of Electronic States
referring to Hubbard’s bottom band (cf Fig 2.1) Broadening of this band due either to the quantum overlap or the fluctuating classical field randomly shifting the levels will be assumed to be small compared to the distance Ug to the second band When studying the structure of impurity bands at zero temperature we can, therefore, assume that each site can contain no more than one electron It should be noted that this follows not from the Pauli principle, which allows two electrons in each orbital state, but is due rather to
the Coulomb interaction, which in this case turns out to be more restrictive
2.2 Anderson Transition
In this section we return to the single-electron approximation and continue studying the impurity band structure However, we shall no longer assume that impurities form an ideal lattice Ideality of the impurity sublattice can be violated in several ways The simplest way is to assume a random spatial configuration of impurities We shall return to this case later Right now we shall consider another practically important case of disorder which has been studied in more detail
Suppose the impurities are located on sites of a regular lattice but the
electron level on each site is different In other words, let us consider a
system of periodically arranged potential wells of varying depth (Fig 2.2)
LỊ R Fig, 2.2 Potential wells in the Anderson model
Denote by ¢; the deviation of an electron level on site j from the average value The system’s Hamiltonian in the site representation is obtained from (2.1.5) by adding a term representing the sum of electron energies on isolated sites:
H=Dgajat > lừm)47 ajam - (2.2.1)
7 Jun =0
The energies ¢; are assumed to be random and uncorrelated In other words, the probability that a particular site will have a given energy is independent of the other sites’ energies The energy distribution is assumed to be uniform
in some interval W; ie., the distribution function P (©) is of the form: U/W, |e < W/2
P() = (2.2.2)
0 ld > W/2