Effective Mass ApproXimation in the Band Theory

Toshinosuke MUTO

Department of Physics, Nihon Universit)-

(February 17, 1968)

I. Introduction

As is well-known, various observed facts in transport phenomena, optical properties*

magneto-optical and magneto-acoustic properties a=nd so forth of me.nifold solids can be

well understood on the basis of the band theory_ , at le.ast semi-quantitatively at present-

Then, in the analysis of the experimental results, it has become a very important target

to make accurate determin"-tion of the various band parameters which include effective

mass together with band gap, band width and Fermi surface. From the theoretical view-

point the effective mass may be regarded as a renormalized electron mass due to the-

inter.".ction of crystal electron with the periodic potential field of solid in the similar way

to the case of mass renormalization in quantum electrodynamics. However, it must bc

noticed that the effective mass concept in the band theory is shown to be of approximate

nature and is subjected to various limitations in its validity since only a part of the-

interaction with the periodic field may be amalgamated into an electron mass.

Now the various physical properties of semiconductors are known to be often governed

by relatively small concentrations of carriers with energies lying close to e~ band edge.

in the vicinity of which the effective mass is most useful for representing the energy

band structure. In the simplest case of isotropic effective mass, this means that the-

electron energy close to the band edge can be written approximately as

~~ (k) = :,~ (O)+fi2k2/2 m* ( I ) where e~(O) denotes the energy of a band edge and m* the effective mass. Equation (1),

apart from ~~ (O), represents the energy-k relation for a free particle with effective mass

m*, Ivhich can be positive or negative according to the band structure. It, then, follows.

that, as long as the carriers have energies close to the band edge, the relationship between_

their energies and wave number vectors is determined completely by the effective mass,

at the band edge, although the corresponding wave function is represented by modified

plane wave, being different from pure, free electron case.

In the discussion of effective mass in the pre-~'ent article, we shall use as a basis the~

-1-

one electron approximation for the motion of electrons in solids as usually done in the

band theory. Therefore, we shall make no attempt to discuss the effective mass concept

from a many-electron point of view, although this has been done notably by W. Kohn

and his associates.1)

II. Effective Mass by Perturbation Theory

,(2. 1) Simple Derivation of Effective Mass at a Band Edge

We shall begin by developing a.n expression for the effective mass at a band edge by

~perturbation theory from the one electron Schrddinger equation.

As is well known, the Schr6dinger equation of the periodic problem is written as

H' cn,'~(r) =e,, (h) cn,~'(r) ( 2 )

where

H=-P_+Vj,(r) p ~ V (3) 2m

~V;' (r) describes a periodic potential, m the true electron mass and n the band number,

The wave function which satisfies (2), i.e , the Bloch orbital may by shown to become

u'n,~(r)=eik'ru k(r) (4) where the normalization factor is included into u;t,~ (r) which is periodic function with the

_lattice period.

The substitution of (4) into (2) Ieads to

- - ~1 - -(p+hh)2+ Vp {r)J un,~ (r)=a~ (k) un k (r) 2_m

'o r

(H+Hf)ttn,~(,-) = -,,(h) li2k2 ( 5 )' ~ ( -* -~ ) = , un,~ (r) 2 m

in which

H' fik.pL ( 6 ) ~ m '

Being interested in s~(k) for small values of k, we shall consider k to be small and

treat H! as a perturbation on the states un,o(r) which satisfies the unperturbed

Schrodinger equation

-2-

,

,

H' un"o (r)=en' (O) un"o (r) . ( 7 )

Such a perturbation-theoretic procedure is called usually the "klp approxirnation "2>'

in the literature. The set of functions un"o(r) obtained by letting n! run over all band~~

forms a complete set for describing any function with the lattice period_

Thus, when the band edge lies at k=0 and is non-degenerate, the usual perturbation-

theoretic procedure gives

- li2k2 (k)- 2 m ~en (O)+(nlH I n)+ ' (n_IH'_i_n. ~(n'.I~T'I_~) + ... ( 8 ), ! ~ en (O) - sn' (O)

n'

and

un'? (r) = un'o (r) + ' (n'l HI n) un"o ('~'r)+ "" " , ( 9 ). - ~ ~ ~~ (O) - en' (O)

n'

where

~ ~2 k r) un o(r) (nIHln'):='Jod u ~o(r) t' ( ~' ', n im

hk ' (n I p(O)_~' ) (10) m '

and

, = J ( li r)un"o 1

(nip(O)In) 1･ dt'un'o =r (11) .r~

In the above computation the fundamental volume of~the cyclic condition is taken tc~

be unit volunie and -o. denotes a cell volume. Furthermore, the second term on the right

hand side of (8) must be zero, since the band is flat at k=0 and the band structure i~'

shown generally to become an even function of k.

Then, it follows, from (8),

~2k2__ + ~i f [_k ' (n_I p (O) I ~i)~_[~i.(_n~p_(_O)_In)l s,,(k~~')=en (O)+ ~ en(O) en' (O) (12)

2m m - ' which is correct to a second order of approximation in k.

Then we define in general an inverse effective mass tensor I ~ a ~=x, y, z. ( , , m* )~,~

for the n-th band by

1 1 2 / (ni.pi(O)!n')(~!lpP(O)In) ( = 2~ )

~a'~ + = ( 1 3), m * m m e~ (O) L en' (O) ~, p

n'

-3-

so that (19_) becomes

x'y'z

(k~) ,,(O)+ ~" I (14) ~ ~~ ( ^, s ) = e ~ -- 3 k, k**

2 m* ~,s

In the energy region close to the band edge, therefore, the shape of the energy

band is completely described by the components of the inverse effective mass tensor

defined by (13)

When we carry out a transformation to the principal axes of the inverse effective

m_ass tensor, the non-di~gonal component~ will vanish and then the energy band will

take the following simple expression.

h2 j(_ 1 1 1 k t (k) = en (O) + =.- } ~ '11 ) ( z2 * ( ) ) kx2 + ky2 + (14)' 1l~* m* J ~s ' ~/'/ *','.

The abave results (13) and (14) are easily generalized to a non-degenerate band edge

~occurring at some arbitrary value ko in the k-spoce The similar ."_nalysis to the above

shows that, for small value.s of (k=-~o)' 1;ve get

x'y'e

h2 1 ;., (~) ~~ e,, (~o) + ~~~ - ( - (15) ). ~ (k,3 - ko,~ ) _ (k~ ko")

m~, 9 ,*,J

=~nd

1 ' I f (nip"(k:)[n') (n'ip;(k~o)fn) (16) ( =-'n* /^,3 o'(1';+ Ini ~ s,,(~o)~en' (k'~'r') -- 1 n l

tt '

where

=_ J i ( l ~' fi ~ _ n]p(ko)1,2/) b (11)! d u,i ko ' i ~r/ u~e"k*i - !'

When the bands nl, n2' n3"""n^ all have the same band edge at k=0, say, so that

Tve have r-fold degenerate band edge e~(O), the interband coupling energy, to a second

~)rder of approximation in k, may be given by

A n n li2 / [k ' (n l[p (O)1 n~Llk ' (n" J p (O)1 ~~ ')] (17) , ,=- -~ e m ~ (O) - ell" (O)

which is easily derived through a slight generalization of the last term on the right hand

side of (1_2).

According to perturbation theory of degenerate states, we can derive the secular

-4-

,

r

1

'equation which determines the energy values required.

(O)+ h2ki+Anl nl f A A nl,n. ni,n2 2m

h2k2 j '4n2'nl An2'n. l e,* (O)+ 2 In: +An2 n2

=0. (18)

_/1 An.,n2 (O)+ li2k2 +An.,n.~e n*,nl 9- In

Unfortunately in this case, the energy expression as a function of k can not be

written down explicitly but only be evaluated numerically.

The effective mass tensor derived above ((13) and (16)) is produced through interband

'coupling, which are induced by the momentum matrix elements (nlp(O)]je/) or (njp(ko)In').

Since the energy band involves the effect of periodic potential field, the interaction of

an electron with the periodic potential may be considered to be amalgamated into the

inverse effective mass tensor to a second order of approximation in k, which corresponds

to the mass renormalization procedure.

Furthermore, the interband coupling between distant band edges will make relatively

small contributions to each other's effective mass tensor as a result of the energy de-

nominator of (13) or (16) For the sake of illustration, we shall here consider a simple

.example of two-band model in which w~ need only take into account their mutual

interaction to account for their effective mass tensors, the effect of the other bands on

their effective masses being neglected approximately because of the larger energy de-

nominators. Suppose we denote two bands of this type by el(O) and 52(O), assuming that

,el (O)>e2(O). For the sake of simplicity, we shall assume for the non-diagonal components

'of the effective llLass tensor to become zero. Then we have, from (13),

1 1 2 (1!p~(O)12) (2lp"(O)]1) (19) ( = ) + ml* m2 S1 (O) -e2 (O) m ","

and

(.__ _) = I ~_ (~Ip"_(q)ll~_~~lp"~P):2)_ (20) :* l' +~ e2 (O) -*-1 (O) m2 1 n (r,(~ In

which lead to a very simple relation between two effective masses of the bands I and 2,

~ts follows.

( ~ ~~ (~1 ! ) = l 1 1 1 ) =-Inl* + ~ (21) *.. ~f ~,~ In f n '''2 ' ~,~

-;)

Furthe more, if we rewrite (19) and (20) as

l I A I A ( - = , (-.- ) = IL,* -~ ) 7'11* ~ + ~ In 1 n 'n 1 n m. "," ","

with

A= 2 (2P"(O) 1) 2 >0 In sl (O) -s2 (O)

then it follows,

In 'n (Inl*) =i~~!f ' (In2 ) - 1-A ~ (22) * ^,^ ","

For the case of weak interband coupling (J<<1), we get (ml*)* a<(m *) while, for ","

the case of strong interband coupling (A>>1), we have ('nl*)~,*>0 and (m2*)^,~<0, which

shows the relation between the effective mass tensors and the interband coupling strength

in the mentioned simple case of two-band model

(2. 2) The Accurary of the Inverse Effective Mass Tensor (13)

As clearly seen in (8), the perturbation series is essentially a power senes m a

quantity of

S- (nlH'In') (23) = e^(O)-en'(O) '

and we have entirely neglected terms of S2 and higher order in the computation of the

effective mass tensor. Therefore, our effective mass approximation is seen to be valid

to the extent to which S is so small quantity that terms of order S2 can be neglected

Using (10), we get

S-~k･p (23)f tndE '

where p is a typical momentum matrix element between ba_nds and AE a typical inter-

band separation. In order to get some approximate estimate of S, the sustitution of p-

~i/a and AE-li:2/2rna2 (band width in the Peierls-Brillouin approximation) into (23)' Ieads

to

S a k-~f (23)" ~, '

where a denotes lattice constant and t, the de Broglie ~1'ave length of a crystal electron.

Thus, as long as k is much smaller than the width of the first Brillouin zone or the de

-6-

1

J

(

(

F

Broglie wave length is much larger than the lattice constant, S may be taken to be small

quantity and our approximation adopted in the derivation of the inverse effective mass

tensor will be reasonably valid.

(2. 3) Temperature Dependence of the Effective Mass Tensor

Through allowing for the interaction of crystal electrons with lattice vibrations, the

Bloch orbital may be shown to be modified by the temperature-dependent, additional

terms, the resulting redistribution of crystal electrons being realized3) and, furthermore,

the band edge is found to have temperature-dependent shift allowing for the addition of

the self energy term4) in the similar way to the case of quantum electrodynamics. The

former effect has been actually observed in the temperature change of Knight shift of

alkali metals and the latter one in the temperature change of band gap in sorne semi-

conductors. Thus, the modified Bloch orbital will be expected to give rise to the tempera-

ture effect of momentum matrix elements in the numeraters of (13), while, the temperature

change of energy gap leads to the temperature effect of the denominators of (13). The

combined effect mentioned above, therefore, will result into the temperature-dependence

of the inverse effective mass tensor.

However, an apparent, temperature-dependence is often seen due to the electrons

being associated with different band states at different temperatures; namely, as the

temperature is raised, electrons will fill up higher states in the band where our effective

mass approximation breaks down.

(2. 4) Inclusion of Spin-Orbit Interaction

When the spin-orbit interaction of crystal electron is included in addition to the

periodic potential, the Schr6dinger equation may be written as5)

[ ~ J~ +V (r)+ ~i ~ (24) p (c x VVp(~'r)) ･p] ~'f ac 2 m 4 m2c2

where c describes Pauli spin operator.

Since the spin-orbit interaction is seen to have translation symmetry, the correspond-

ing wave function is easily shown to have the standard form.

cn,~(r, (F)=eik' r un,~(r, o) (25)

where (T denotes spin variable.

Putting (25) into (24), we get, through a simple procedure as in the case of spinless

-7-

electron.

(H+H/) un.1(r, o~)=e,, (k) un k (' (T) (26)

~~'here

rl~ ~i_~(' x rV (r)) p (27) H=-P___ + V1' (t)+

2 m 4 7n2(~ a nd

'=- ~.(;+ 2 ~ p~) H ~i:_ Ii_ (ax rV (r) (28) 171 4 rnc

The k-p approximation procedure c*"n be proceeded actually in the similar way as

(~ _ ii --before except that the matrix elements of p~s now replaced by those of p+- . - (c

~ x rVp(r)) and the basis set is defined by un,o(r, o) instead of un,o(r). An actual estimate

by Kane5) has shown that the difference between both matrix elements is of the order'

^s I (" ~1 170

(2. 5) Another Expression of the Effective Mass Tensor

By making use of the well-known sum rule6) in the band theory, i e ,

2 ,n a25~(k) / (n l p" (k) j n/) (n! I p~ (k) I nl _ 6~ ~ (2_9) ~ -- = -111 " ~ ~~2 ak^ak~ ' n ' 'n' (k) - 5n (k")

the inverse effective mass tensor (13) may be easily shown to have another expression as-

l ( a2~.,(k) ~ ( = - )*. ~~ ~･~ ak^akp /k=0 1 n*

~,j

The above expression can be easily generalized to the definition of the effective mass,.

tensor at k state of an energy band instead of the band edge

1 / a2sn('~'k) ( == ) 1 ~ , . '~･¥. ak~ak,3~ (30)! In~ (k' )., ,

,~

~

III. Effective Mass by Canonica] Transforlnation

In the previous treatnlent, we have made use of the ordinary perturbation theory for'

deriving the effective mass tensor at the band edge and in the present chapter the

procedure by canonical tr.".nsformation wlll be adopted to get the essentially same result

as before, for the reason that the present procedure will be shown to be useful for the

~8-

~

,

perturbed periodic problem dealt with in the later chapter.

Since u,e,o(r), which satisfy (7), form a complete, orthogonal set of periodic functions.,

the solutlon of (5) may be expanded into the mentioned orthogonal functions as follows.-

un,~' (r)= ~ An'un',o(r) . (31)' n

Substituting (31) into (5) and multiplying by un:o(r) to the left hand side, we get,.

after performing the integration, -

(O)+ ~:2k2 ~k･(n [ p (O) i n!)_ An' = ~~ (k'~') A~ , (32)-I ~ e~ J A~ + 2m m which is the simultaneous equation to be solved by canonical transformation.

Nolv we shall write (32) in the matrix equation such F3._s

H' A = g~ (k) A, (33) where A represents a column matrix with elements A~ and H has the matnx elements~

of

(n IH: n') = (n [Hoi n') + (n ~Hlf n' ) (34)

in which

(n IHOJ n') = (O) + fi2k2 (35) ( o g^ ) n.11 2 m

and

(n H , n')= -~k ･(n f p (O) i n'=,=) (36)

m ' The equation (3-2) is easily shown to be equal to (33) by using the well-known la,w~

of ma-trix-multiplication, i.e.,

HA~ = ~ (n fHI n') A,e' ' (37) n*

Now. A,e' in (31) are regarded a-s the conlponents of wave function un.'~'k(r) in a

representation in which un',o(r) are taken to be the basic set. In this representation the

original Schrddinger equation has been reduced into the equation (32) or (33) for An"

which contains the ternl of H1 which couples the equations for different A,e' and is prd-

portional to k. The existence of such coupling terms is a result of our choice of basic

set of representation. Then, we c.an choose a different set of basic functions such that.

there appears no interband coupling terxns proportional to k in the new representation-

-9~

What we do is to find a canonical transformation of the required character ~vhich can

~relate the new representation with the old one.

We, therefore, put

A = T' B (38) 'or

A - vq' D (38Y '*- L i n,n*Dn"

1~lhere T rs a unrtar~ matnx (T-In,n'=T*n',n)' so that we get, from (33),

H･ B= s., (k) B . (39) where

HF T- IHT (40) When we put

T=es and T-1=e~s . (41) we have

H=e~S HeS

= (1-S+S2/2! -... ) H(1+S+S l"' +･････')

=H+[H. S]+'r [[H, Sl. S]+-"" (42)

The substitution of (34) into (.42) Ieads to

1 -- 1 (43) H=H0+H1+[Ho, S]+[HI S]+15T~[[Ho. S] S]+=[[HI S], S]+'-""-' :1. ' 2! ' In order to eliminate the interband-coupling terms of the first order in k we put

Hl+[Ho. S]=0 , ' (44) from which the unitary matrix of a canonical transformation and the corresponding new

basic set are obtained-

The matrix elements of (44) are

(n H In )+ ~: {(n]Holn") (n"IS,n ) (n Sln") (n"iHolnr)J =0 . (44)!

By using (35) and (36) together with the unitary character of T=eS. (44)' gives, for

the explicit expression of the matrix elements of S.

(nl Sln )= - ~k･(n:p(O)fn') (1-6n n ) (45) nt ( a~(O) - sn'(O))

- ro -

~

,

which is the

Putting

first order quantity in

(44) into (43), we get

k_

H=H0+~ [HI ' sl+}[[Hl' s.,], sl+'

of which the second and third terms become the second order and the third

tities in k, respectively.

Ne*"lecting the third order quant

we find, using (35), (36) and (45),

'ties and the higher ones with respect

(n~H[n ) (nfH1 [n )+~ (niHl[n") (n"[S[n!)_(n[S;n") (n"fHl;n/)} ' o ' ~{ 1

'e"

= a'~(O)+ )' ! (k'(n'p(O)In")) (k'(n'l~p(O)In!~ ~2k2 + ~2 ( , 2 ~ { ---27n on'n ~n' (O) - an" (O) 2m n"

(k '(n Ji P (O) i n")) (k ' (n"I p (O) I n'))

+ ~ 5~ (O) - en" (O)

" ( 8 e (O)+ ~i_ I l { - )^. =~ ~ kp r8n'n' +(n [ A{ n/) (1 -an'n') ' k^

2 m* ~' p

Where

order

(46)

quan--

to k of (46).

(47>

(n JA I n') ~i ' J (k .(n l p (O) f n")) (k '(n"I p (O) I n')) = 2~ - ~ 2 l 2 m ~' (O) - en" (O) n"

+ (k'(nj p (O) il2")) (k.(n";p (O) j n')) I (48) E~ (O) -e." (O) ~~ f '

The interband coupling terms of the order of k2, i.e., (48) will produce the effect of~

higher order than k2 on any particular band when they are eliminated. This is actually

seen in the sirnple case of two interacting bands as follows.

Taking account of the fact that (1 A 2) produces the interband coupling between

the bands el(O) and e2(O), the energies in the two bands are given by the eigenvalues of

the rrlatrix;

fi2k2

5 (o)+-~;;~+(11Ail)

(2 fA f 1) e. (o) +

(1 fA[ 2)

~2k2 --- + (2 f Af 2) 2 m

(49>

- 11 -

Then the mentioned eigenvalues are easil_~" seen to be

h2k2 (1 A 1)+(_2 A_2)_ ~ (k'~')=~-; (51 (O) + e_. (O))+ - 2 v_m +

_{ el 2 - }; 1 + Tr (O) (O~+(1 A 1) () A ')] +(1 A 2)(? A 1) (50)

which shows that the interband coupling terms (1 [A 2) and (2!A,fl) give rise to the term

of order of k4 in the energy s (k) provided that the bands I and 2 are not degenerate

with each other at k=0, i e , sl(O)~52 (O)

Thus, ('1.,'H:,1!) be.comes diagonal to a second order of ~Lpproximation in k and yields.

using (47),

-- fi2 k I (51) ~ ( . .3 , ~~ )-s~ (k) = 3 ,, (O) + -- k ~ m* ~,j

~. j

which is iust the result obtained b_v the perturb.,~~tion theory in the preceding chapter.

Furthermore, a closer examination shows that the convergence criterion for the validity of

(51) is exactly the same as (23) whlch we deduced from the perturbation theory.

Next we shall go on to discuss the corresponding w.".ve function. Using (45), we get

( ! ( I [n )] '' 21_1 S + -"' ) (n/ T n")=~n ~l+S+ - 2

1 - (n/ S21 n")+ on n +(n/lS n")+ ~1

=6n','1"~ hk'(n/p(O) n") +..

'n ( n (O)-5n"(O})

,and

An'=~ (n T n")Bn n*'

Bn ~ ' hk'(n'=p(O) 'nt') (52) = , ~- - l. -- Bn"+""' In (En' (O) n' (O))

n"

The substitution of (52) into (31) Ieads to

- Ir ' hk'(n/jp(O)fn//) Bn un"o(=~'r)+""" , - ~~ Lin,~; (r) = LBn un',o (;) "7~..･n;~(5)~ Sn" (O)i "

n' n' ~" When we disregard the interband coupling in the zero order approximation, it follows.

Bn ' = 3~,,n

- 12 -

and then

/ h~･(n/j;(O)fn) ~ ~ ~~ u.,~ (r) = u~,o (r) + ~,(.-"(O~ je" (O jj~~ u~',o (r) + (53)

which is exactly the same expression as in the perturbation approach, i.e., (9).

IV. Perturbed Periodic Problem by the Effective Mass Approximation

=(4. 1) Introduction

In the present chapter the problem of an electron motion in perturbed periodic lattice

containing some point defect will be dealt with by the effective mass approximation.

The lattice defects are produced actually by embedded foreign atoms, Iacking or displaced

atoms from the normal lattice sites and the other kinds of point defects. Then the

electronic wave function satisfies the Schr6dinger equation

J PL+ VI' ('~'r)+ U(r)}ip(1 ) (v(r) (54) ( 2m

where U(r) is the additional, Iocalized potential associated with point defects. The

_potential field in (54) has no longer lattice periodicity so that we cannot associate with

electron's motion a well defined wave vector k. Generally, we have two kinds of the

approximate methods of solution for (54), one of which is to use the expansion of ~'r (r)

into Wannier functions (Wannier representation) due to Slater and Koster7) and another

one of which is to adopt the expansion into Bloch orbitals (Bloch representation) accord-

ing to Luttinger and Kohn.8) In the following the latter approach will be described for

deriving the effective mass equation of the perturbed periodic problem.

In the effective mass approximation the Schr6dinger equation (54) is shown to be

eventually reduced to

f p2 (55) + U (1l~) } F(r~~') = ~F('~>r)

l-2 m*

and

9'! (r) = F(r) u^,o (1~) (56)

for a simple case of the isotropic effective mass. The equation (55) is formally similar

~to the Schr~dinger equation of an electron with effective mass m* in the potential field

~U(r) alone, which is easily solvable colnpared. with (54).

- 13 -

(4. 2) Expansion Procedure due to Luttinger and Kohn

The wave function ,v' (~i of (54) may be expanded into the orthogonal set of Bloch'_

orbitals in the following way.

c(1i)= ~Id~='/k B^,(k/)u^' k (r)elk ' (57)

Substituting (31) into the above equation, we get

~ r - - -･ - -･ --ik"' r c (r) = ~ ~ j dk!B., (k!) A.,,^,, (k') u~"', o (1~) e

", ",,

r- - - --ik " r (58) = ~~, j dkfA .,' (k!) u^' .o (r) e ,

~vhere

A~,' (k') = ~ B., (k!) A~,.~,' (k) . (58)'

Putting (58) into (54) and multiplying by u^,o(r) e~ik'r to the left side, then it follows, .

after making integration,

{ ~~ (O)+ ~2k2 ) - ~k' (n ! p (O) ; n') A., (k'=') + L~ J[~ dk'~'! (n~' Uf n"~'k!) An' (k~~'/) ~. A~(k)+ 2m f 7n

= 5A ~ (k) , (59) l~'hich is the simultaneus equation for different A,*(k) and

- - r - -- - - -(nk I U' n'k') = Jdeei'k' ~k:,'r U (r) u.,o (r) u^',o (r) . (60)

Apart from the last term dependent on U, (59) is exactly the same as (32) and only ~

the U-dependent term induces the intra-band coupling which gives rise to the interaction _

among states with different k values.

We could now attempt to proceed as before by finding a canonical transformation

which removes the interband and intra-band coupling terms of (59). In actual fact, such

procedure could be made, but the approximate operator S would depend on the last two '

terms on the left hand side of (59) and, comparing this with our previous analysis of

the last chapter, we should find an effective mass which depends upon U. The effective

mass concept introduced in the last chapter is closely connected with the band structure

alone and, therefore, this is what we do not want. The only way to avoid the U- -

dependence of the effective mass would be to find if there is any circumsta_nce under which

(n, ~iUJn', k:,) becomes approximately diagonal with respect to the band number n and _

- 14 -

n'. If such circumstances exist, we may be able to use our normal effectrve mass If

it is not the case, then we certainly cannot use our normal effective mass. In the follow-

ing we shall discuss, before making actually a canonical transformation, the mentioned

circumstances which enable us to introduce our normal effective mass concept.

Since u~o(r)un"o(r) is periodic function with the lattice period, it may be expanded

into the Fourier series.

u o(r)u o(r) ~Bnm'n eil(~'r (61) m

where K~ represents translation vector in the reciprocal lattice, i.e.. K17~=mlbl+'n2b2+

In3b3 with bi as primitive translation vectors in reciprocal lattice and m~=positive and

negative integers.

Using (61), (60) becomes

,f (n, ~:Uj n!, k~~>! )= ~B"m"' J dTe- ii~'k-k'~"-K~~,~)･~*'r U(~~'r)

m

= (2 ,-,)3 ~ B~"' W (~-~, -K~) , (62) ;'t

where we have defined the Fourier transform of U(r) as

~ = I -~ ~ 1

W (k) (2 sT)3 (63) dte~ ik'r U(r)

Now, the orthonormalization relation of Bloch orbitals is

~ ~ r ･ - ~ ---a~..,a (k - k/) = J d r u~,~ (r) u~',k" ( 1-) e~ i{k ~ k')'r

,-- f -- - -= ~B~" (k, k') J dte ~ i(k - k' - '~ ~)'r

m = (2 ~)3~ B~"' (~, k'~>/) ~ (~_k~*' - K~~>~) , (64)

m using the relations of

un'~ (~~'r) u~', ~~>'k (~~>r) = ~B"m'"' (~ k~'!) eiK~~>~'r (64y

m and

J ~ - (64)" dee- ik' r= (2 1~)3 ~ (k'~') .

,Corresponding to the circumstances mentioned before, the localized potential U(1~~'-) is

- 15 -

assumed to be slowly varying function within a unit cell, which leads to the result that

appreciable values of W(k) in (63) come from rather small values of k in comparison

with the first Brillouin zone boundary. Furthermore, k-k/ _K~ I will be restricted to,

rather small values for appreciable W(k-k' -Km) of (6_,!), in which case we should take

K,,*=0, allowing for the fact that only small values of k and k/ make dominant con-

tributions, as mentioned above

Substituting K,^=0 into (64), we have

(2 7c)3 Bo"'"' (k, k!) = ~ . (65) ".",

The right hand side of (65) is seen to be independent of ~ and k and therefore

we obtain

o ' (O, O) (2 IT)3 Bo"'"' (k, k/) = o~ ~ "'" (2 I~)3 B ",",-

~,^ (66) ~~(2 IT)3 Bo ' '

Using (66), then It follows from (62)

(n k U: n/, ~7)= W(~-k'~'!)a , (67) ",",

which is valid only for slowly varying function of U(1-) within a unit cell.

Substituting (67) ino (59), we get finally

t 2 In J ~ ~- A~, (~)+ Jdk! W(~-~/) A~ (~l) f ~2k2 1 lik . (n : p (O) , nl) 50 (O) + A,, (k) + ,n

,, ,

= ~A~(k) . (68) (4. 3) Application of Canonical Transformation to (68)

In order to eliminate the interband coupling term of the first order in k, i.e , the-

second term on the left hand side of (68), the similar method of procedure as in the-

chapter 111 will be adopted. Before carrying out such procedure, we must rewrite (68) in_

the matrix form.

Then, we have

H' A= 5A , (69) , where A represents one colum matrix whose conrponents are A.,(k/) and

(n~ IHj n 'k'~'!) = (11~:'Ho" nlk') + (n~fHlf n'k'~'/) + (n~fH2: n'k'~'!)

- 16 -

~

with

(nk f Hol n/k') = (e~ (O) + ~2k2/2 m) ~,,,., o~ (k - k') , (70)

(nk H 'n'k') ~k (njP(O)_n')_ ~(~-~) , (70)/ 1 n

and

(nk j H2' n'k') = W(k- k') 6.,~, . (70)" The matrix equation (69) is readily shown to reduce into the component equation*-.

(68) by making use of the following forrriula.

(H' A)~ (k) = ~A~ (k)

or

~J ~, - , = dk (nklH n!k') 4 (~')=~A.,(~).

",

As usual, a canonic8.1 transforma.tion is set up as follo~v~

A=es B (71)= The equation (69) may be transformed into

H' B= 5B . (72) , where

H= e ~ sHes

1 =H+[H, S] +-= [[H S], Sl+ ' " ' '

2! ' = H0+ Hl +H." + [HO ' S] +[HI ' Sl +[H2 ' Sl+

1 1 l +1:~~ [[Ho'S] S +- H1'S] S]+~T[[H2 Sl S]+ (73) , 1 2T [[ ,

In order to eliminate the interbcnd coupling term of the first order in k, we may pht.

H1 + rHO ' S] =0 , (74) whose component equations are easily found to become

lik'(npl (O);n/) fi2k2 ~2k'2 1 -6 (~-'~',k ) + { 5~ (O) -5^' (O) + - (n~[ SI n'~/)=0 . (75),

1lt 2ln ~ -2rn f - 17 -

The solution of (75) is

(nk~':SI n/~!)= hk'(n p (O) n ) (76) 'n ~~ ~(~-~k/) (1-'n'n ' o ')

which are seen to become of the first order in k as expected.

The substitution of (74) into (73) Ieads to

H=H0+H.~.++ [HI , Sl+[Hz ' S J+ "' ' ' (77)

Taking account of (70), (70)!, (70)" and (76), a closer examination of each term of

'(77) shows that the fourth and higher terms become of the higher order small quantities

in comparison with the first three terms of (77) as far as the magnitude of k is restricted

to small values in the first Brillouin zone (fundamental assumption).

Then, it follows,

(n'~'k H= n/k')~; (nk':Ho! n'k')+(n~[H21 n/~/)+~ (n'~'k [HI , S] f nl~') , (78)

where

(nk f rHl , S] I n'k!) W(k-'~'k/) ~(k'~'-k'~'~:)･(n [p'(O) 'tl/)_

,l,~n (6i'n' (di) '

:and the other matrix elements of (78) are given by (70) and (70) ,,

The matrix elements of (7_2) are

~ d~'k'(nk~!H n'k~~") B,~'(k~~~'/)=sB,,(~),

into which (70), (70)// and (78) wlll be substltuted

Then, we obtain

fi2 1 1 -- ( -' - ~ - -{a~ ~ ( (t ) (O) + T k (80) ---- k; JBn (k) + ~dk' W(k - k/) B~ (k!)= aB~ (k) , m* ^3 . ~~

in which the interband coupling terms of the second and higher order in k have been

neglected. The equation (80) is actually the effective mass equation in k-space of the

perturbed periodic problem. Next, we shall proceed to derive the effective mass equation

in coordinate space. For that purpose, we shall introduce the Fourier transform of B~(k) .

~ C -･ ~~ ~ (81) F~ (r)= jdk eik' r B~(k),

- 18 -

,

where the integral, as usual, extends over the first Brilloum zone

Noting that

eik' r k,xkp B~ (k)= -VaVP eik' r B~(k) ,

and multiplying by eik'r to the left side of (80), we obtain, after making the mtegration

over the first Brillouin zone in k-space,

li~ I I - -- C ~r ~ ~ - - ~-jdk'~'{e (O) n L( - ) - VaVp f Bn (k) etk' r+ j d k j dk! W(k - k!) Bn (k!) etk' r 2 rn* aP

a's

= sF(r) . (82) , With the definition of W(k) in (63), the second term on the left hand side becomes.

ldkjdk J - f 3 ~ ~' d~! U(;,)e-i':k~~'-k'~")'~'r' Bn(k/) eik'r = Jdr! U(r!) F~(r') A(r-r!). (2lr)3

where

-== J - --1

A (r) (2l~)~~'3 dk eik'r (84)

The integral region of (84) is restricted to the first Brillouin zone so that A(r) isl

not a pure ~-function of Dirac in which the k-integration extends over all k-space. How-

ever. A(1-) is a function which has a peak at r:;:O but spreads over the region whose'

width is the order of lattice constant, and, furthermore, satisfies

J -dT A(r)=1 ~

Thus, provided that, in integrals, A(r-r/) is multiplied by functions which vary slowly~

over a distance of lattice constant, it will act as a 6-function. Now we have restricted

U(r) to a gentle function and we shall assume that F,~(r) is likewise. (In the end our

results are shown to be consistent with this assumption). Then (83) will reduce into'

U(r) Fn(r) .

Finally, we get, from (82) ,

~2 1 { en ~( ~' Aa ' A~ }F.,('~'r) + U(~~>r) F~(~~r) = aF~(~'r) - ) (O) T m (85) '~,p (x'fi

which represents the effective mass equation in coordinate space of the perturbed periodicl

problem.

For a simple case of the isotropic effective mass m*. (85) reduces to

t2 r2Fn(~~'r) ~ U(11:) FT.(;) ::: (e - en(O)) Fn(~='r) . ~ 2m* (86>

- 19 -

which is iust the Schr6dinger equation for an electron lvith mass m* moving in a

potential field U(r) .

Next we shall examine how the solutions of (86) are related to the actual wave

function of an electron.

From (71), we have

I = =~J In(k) dk/ (nk 1 1 + S+ ' S2 + . . . . . . ! n!k)) Bn'(~/)

~B~(k) to the lo~vest order

Then, it follows, from (58),

~ ~J ~ ~ c(r)= dk/Bn"(k!)un",o(~'1) eik~~"'7 to the lowest order,

~n" "" I ,,, l': = F (,)un o(1)

using (81) W'hen ~ve are restricted to an electron in the n-band, we obtain

~ ( ,') = Fn(;) un'o{'~'~, ) . (87)

When U(1-) is an attrc~ctive potential of suitable magnitude, (86) will yield a number

'of bound states and the corresponding eigenfunction F~(r) becomes more or less localized

function in a similar way to the case of a_tcmic functions However, the slowly varying

'character of U(r) will reflect on the rather spreaded and slowly varying function for

_Fl~(1"), which qualitative behaviour~) are described in Fig. 1.

p(F)

~~--'./. Fn(r) un,o(~~r)

t

On the other hand,

according to Wigner

the

and

Bloch

Seitz

Fig 1

orbital will have an approximate form

approximation, which is compared with

- 20 -

of eik'run'o(r),

(87). As seen

~

from (86), the bound states are found to appear, below the band bottom, in the forbidden

_region of the band structure. .

'(4. 4) Case of Degenerate Bande at the Baud Edge

The preceding procedure in (4. 3) will be easily extended to r degenerate bands at

~the band edge (~=0). , Then, we have

nl(O)=enz(O)=" " " =En'(O)

*and the corresponding Bloch orbitals are

unl'o(r), un2'o(r), -""', un',o(r),

which are described by unj,o(r), (j~: ~P , """ r) for the sake of simplicity and the remain-1, ~,

ing Bloch orbitals corresponding to the other bands at ~=0 are written as uni'o(r~'), i~~j

As in (58), the wave function which satisfies (54) may be expanded as follows.

-･ ~J ~ ~ = d k/An'(k/) un',o ('~'r) eik~"r . ~;! (r) (54)! Since the Bloch orbitals corresponding to degenerate bands have the same parity as

*shown by group theory, it follows,

(nji p (O) I nj')=0 , (88)

which shows that the interband coupling terms between respective degenerate bands

Thus, there exist interband coupling terms between unj'o(r) and uni'o(r), which first

order terms in k can be eliminated by canonical transformation as in the case of non-

'degenerate bands. Since the procedure is quite similar to that of the preceding section,

the resulting equation is written immediately as

- ~~ ~nj (O) Bnj (k) + (kaDj~~, k,9) Bnj' (k) + J W(~ -~/ ) Bnj (~~'k/ ) d~,

j'= I ~.P

BnJ (k) ( j= 1, 2, . . . (89) -' , r)

where

Dj~,~ I o~j,j o ~+ (nj[p(lr_(O)ni) (nilP"~O).nl J") (90) . 1 , ~, ~------ (O) - ani(O) "~ 2m ~;

- 21 -

The simultaneous equations of (89) are the effective mass equations in k-space for-

the case of degenerate bands. Using the Fourier transform of B~j(k) given by (81), (89) ,

may be transformed into the equation in coordinate space as follows.

r ^~ j' = o ^.p ~ irfi) F.j (~~>r) HT, i U(~'~'r) nj,j'F.j, (r'~') = (5 -5.j (O)) F.j (r) .

~- ~ (-ir~)Dj,j' ( (91) (j=1,2, ･････, ' r)

The leading term in the corresponding wave function becomes

c ('~'r) = iF^i (/11) u.i,o (r'~') . (92) j=1

(4. 5) Applications of the Effective Mass Equation to Ge and Si

For illustration, we shall show briefly an application of the effective mass equation _

for germanium and silicon crystals. As is well-known, the conduction band structure of

silicon9) is shown to have six equivalent minima at the points (ko .0,0), (-ko ,0,0), ......

(0,0, -k~~'o) in ~-space. The exact magnitude of ko is not yet known, but there are indica-

tions that the conduction band minima are about three-quarters of the way between k=0 '

and the zone boundaries.10) Near one of these minima, say the one on the (0,0,1) axis,

the band energy is given by an expression of the form

- h2 h2 5, (k)- 2lnt 2mt (k*2+ky2) _ (93) (k. - ko)2 +

Here the zero of energy is taken at the bottom of conduction band and the longitudinal _

~_r.d transverse masses ml and Int have the following values ;

ml = (0.98 :!: 0.02) m,

mt = (0.19 d: 0.01) m. (94)

For the case of germanium, it is known with certainty from the behaviour of the

cyclotron resonance pattern that the conduction band minima lie on the (1, 1,1) axis and

four equivalent axes There is other strong evidencell) that the conduction band minima

occur at the zone boundary such as (1,1,1), (-1, -1, -1) and so on. The energy band

structure in the vicinity of the minima is again given by the expression of the form (93),

taking a new axis along the (1,1,1) direction. The effective masses have the following:

valuesl2) in germanium :

'n, = (1 .60 :t 0.008) m,

7nt = (0.0813 i 0.002) In. (95) ,

- 22 -

1

r

The effective mass equation corresponding to (93) may be written as

li2 az li2 02 { ( ' ai ay2 )+U(1-)}F(r) eF(r) ~ 2ml az2 2 m.t ax2 + (96) in which the energy 5 is measured from the conduction band minima.

If now one of the atoms of the perfect lattice is replaced by a positive, singly

charged ion, the localized, additive potential U(r) may be supposed to be given by

U(r) = - e21J~l- (97) at large distances from the impurity ion, ,~ being the static dielectric constant. The

reason for using the static dielectric constant is that the frequency of the. impurity electron

will turn out to be much lower than that of other electrons in the crystal, whose polariza-

tion gives rise to /i~ in homopolar material. In actual fact, the true U(r) will be rather

complicated potential in the vicinity of the impurity ion, but we shall suppose that the

region does not play an important role and we shall use (97) for all values of r- The

static dielectric constants are i~=16 for Ge and ~:=12 for Si, respectively.

To a very good approximation the normalized wave function of the ground state can

be represented byi3)

- 1 F(r)= ~~a2bjrii e~[(x"+y )/a2+.2/b2]l!2 ' (98)

This functional form, which is exact in the limit of equal effective masses, was found to

be very accurate in the limit mt/mt-=-

A variational calculation gives the following values for a and b.

a=25 O X 10 cm, b=,14.2 X 10-8cm for Sl

* and

a=645xl0-8em, b=22.7xl0-3cm fo Ge

respectively.

The corresponding energies of a ground state are given by

E (ground state)= -0.0298 ev for Si ,

* and

E (ground state)= -0.00905 ev for Ge,

which are compared with the observed values of -0.039 ev for P irnpurity and -0.049

-ev for As impurity in Si crystal, while, in Ge crystal, -0.012 ev for P impurity and

-0.013 ev for As impurity. The comparison with the experimental results seems to be

- 23 ~

good allowing for the various kinds of approximations involved in the effective mass.

equation. Finally, it should be noted that the radius of the ground state orbit becomes.

very large compared with the Bohr radius, i.e.. 7-25 x 10-scm (Si) and ~-64 5XIO-scm_

(Ge), which seem to justify the use of (97~-

V. Effect of External Forces on Crystal Electron by the Effective

Mass Approximation

(5. 1) Effect of Electrostatic Field on Crystal Electron

When an electric field L' is applied along x axis, the corresponding potential energy~

of a crystal electron becomes -eEx, which takes very large magnitude for unbounded

crystals and becomes larger compared with the level distance within an energy band.

These situations will make the ordinary perturbation theory inapplicable to such problem.

In order to avoid the mentioned difHculty we have to construct a wave packet whose

spatial extension is very small compared with the fundamental volume in the cyclic con-

dition of unbounded crystal, and investigate the effect of electrostatic field on such wave=

packet of crystal electron

For example, the wave packet can be described b_~~

k + d k _ J- = c(k!)un,~~"k('~~'r)eik"'~'r d'~'k/

c (1[~) = (99) k - dk

As is shown in quantum mechanics, the motion of wave packet is subjected to'

Ne~~;'ton's equation of motion in a short time interval compared with the decay time of

wave packet as far as the external forces vary very slowly within the spatial extension

o{ wave packet. Then the classical law of energy conservation is valid for such case.

Thus the increase in energy of the wave packet within 6t is given by

~h'= aen(~) ak (100), akl I '

and the work done bv an external force L' becomes

6E= - eEv*o*t . (101), The classical la¥v of energy conservation requires

a~~(k) ik = (102> -eEv!it akl l

- 24 ~

1

)

On the other hand, the average velocity is given b.v

v I ae~(~)_ (103). "=T ~k'='~~1 '

which is the well-known theoreml4) in the band theory

The substitution of (103) into (102) Ieads to

~kl = ~ eElliot

or

kl = ~ eE/li (l04)', For an application of electrostatic field E along arbitrary direction, we get

k = - eE/~ (105)-The three dimensional generalization of (103) is given by

~.1 = I~~ rk･^~(k) (106> v-

and the time-differentiation of (l06) becomes

x yz

d~ I d - IYl("; df v=T ~T rke,,(k)=TL ak~ Vk5~(k) ke

p

Putting (105) into the above equation, we get flnally

d l~ ~ ~1'( = ( -eFt'5) ' (107)> dt v m* )~p

where

1 ( a2a~(k) ( * = ) l (108) , ) m ~~ ~'~ ak~akp '

which is the inverse effective mass tensor at a ~-level of the energy band a~(~~>k). The-

equation (107) shows the motion of an electron with effective mass in an electrostatic

field alone, the effect of the periodic field being amalgamated into the effective mass

tensor.

In the above discussion we have neglected the possibility of the electron being excited

from one band to another by the electric field. The possibility is of importance only for~

large fields for suflicient length of time, which can be realized actually in the phenomena_

of electrical breakdown and hot electrons in semiconductors.

- 25 -

<(5. 2) Effect of Magnetic Field on Crystal Electron

In the present section we shall deal with the motion of an electron near the band

edge under the influence of magnetic field.

The magnetic field H is supposed to be applied along z axis and then the correspond-

-ing vector potential A is given by

.4*=-Ily -4y=0. A= O (109) The Schrddinger equation is

= cr, (110) H~'r ;. .

llvhere

H ?_1 (--__ce --)2 ~ = - s s2 (111) y2 'n~ p A +Vp(r) H0+~yp*+i,n ' 'n.

Ho = ~ P_ + V1'(' ) (112) ?- 'n

s=eH,'c, and e is an electrofi's charge.

In the similar way to the case of the preceding chapter, the wave funtion ~'! (r) may

be expanded into the orthogonal set of eik"~u.,,o(1~) -

- r ･ - - -･ - (113) ,') (r) = ~^, jdk!A ~'(k!) eik"r u~',o(1~) .

Substituting (113) into (110), the similar procedure as in the last chapter leads to the

'simultaneous equations for A,,(k).

f = --, , = = -~, j d k' (nk j H j nfk!) A,*'(k!) = aA~(k) . (114)

where

(nklH n'k/) (nk Ho nl~1)+~, (nk lyp. n'k/) +__il_ (n~~'k ly2 1 n/~/) . (.115) 2 In

Similarly to the case of the perturbed periodic problem, we get

h2k2 ~~.~.6(~-~/)+~~lk (.n p(O)[n ) o(k k ) 1~~} (11; ~Hof n'k~~>/) = { ~ ~(O) + (116)

m In Then.

(nk lypxi n/k/) li a ~ ~ I ~~ ~ - (eik~+" ru^ o('))dt = e~ik' r u~o(1~)y--i ~r'x

- 26 -

,

F

fi a ei('~'k'~kl,.ru~o('~~'r)y(~k./+ _ i ' ax u^',o(r)d =1 - ) = t =1 '~J ~' =- -1 o ~k:~ et k -k)' r u.,o(r) (tk./ +p")u~',o(~'r)d~

-1 a i ~~~~'(fik 6.~+(nfp'(O)fn'))a(k k )

taking account of the periodic nature of u.,o(r) , and

1 a ~ (~kxo~~ +(nlp (O)[n/))'_Ta~'~'k'y ~(k k )

1 a ~ J(n~:yp"In!k'~'!) T ak'~! ~(k k/¥ (117> = I (fik*a..,e'+ (n l p*(O) I n')) -

m m

Next, we shall calculate (n,k{y2;n/,k!) in a similar way as above.

- r (nk fy2, n'k~~") = J y2ei(k~~"~k'~')'r u^,o('~'r) u~',o(r)

= y )2] -. ~ ~ 1 a et(k ~k)'r u~o(r) u^',o('~'r) d~ ( , ~' i ak

0'26 (k - k!) ak'~'2 ~6~,^,

s2 ~ s2 a2a(k-k') ~ 2m (nk fy'l n/k') - - 2 m ak'y2 (118> ".", .

Substituting (116), (117) and (118) into (115), it follows

(nk;Hln/k/)= li2k'- a(k-k/)+ ~k. I a6(k k ) s2 a2~(k-k/) a.,~ ~ ~ {(a.,(O)+ 277e } ' ~ ) ~ - s' In i ak'y ~ ak/ 2 2m

+ ~ j(~k (n p(O) n )6(k k)+s(nlp'(O)Inf) I a~(k-k!) (119)' }

~i ak/y '

where the expression on the second line of (119) represents the interband coupling terms-

The matrix elements of (119) may be written, for later convenience, as

H= Ho + H* + Hb + H' , (120) where

~ ~ ( ~2k2 ) - - , (nk IHo; n!k/) = ~e~(O) + ~n'~' 6 (k - k!) (121)' 2m

-fik. I a~~~-k!) 6 (121)r (n~lH~f n'~!) = __f ^,^, , 1 n i 6k! y

- 27 -

(n'~'k!Hb nf~!) s2 rj2i(k-k!) . (121)!/ = - ------ o "'n' 2 m. r"'kf ~'2 '

and

1 1 a6(k-k!) ~ ~ = { ~ ~ I ~ !y } (nk Hln/k') ,~ hk'(n!p(O)fn!)6(~-~/)+s(n p'(O)In/)1= ' (121)1!! ak

The simultaneous equation (114) can be rewritten in the matrix equation as before.

HA = ~A (12')_) Bv a canonical transformation of

A = eSB (123) '(1_?_~) may by transformed into

HB= aB , (1 24) ~1;vhere

k= e ~ SHes . (125) Using (1_90), we get

k=H0+H~+H'+[Ho ' S]+Hb+[H,, , S]+[Hb , S]+[H S]+ (126)

In order to eliminate the interband couplinba term of the frst order in kl we may put

H +[H , S]=0 , (127) from which the matrix elements of S can be obtained as follows.

_ (nk:H/ n!kl) (n~fSln!~l) _ for n~n' , ~ 5 ,, (O) - en'(O) (128) =0 for n=n'.

The substitution of (127) into (126) Ieads to

H= H0+Ha' +Hr' + [Ha ' S] + [Hb , S] +~ [H!. S] + " " ' (129)

The matrix elements of [Ha ' S] and [Hr' , S] are easily shown to vanish, using (128)_

Then we have

~ ~J ~ > 1 ~ I d k!r {(nk fH/]' n//~//) (n/rk~~'!/1 S I n~~'k/) _ (n~jf S f n!!k~~'//) (n!!k~~'!!lH/j nk~~'/)} - -(nk;[H',S]lnk!)=

~Q 2 n' ~ 28 -

(

1

r

In

+ '(hk ~ i

= dk!/ (nklH In k )(n!/k!!IH/Jnk!) ~/J --- -e~ ~ / ll!l (O) - a~"(O)

"',

1 ' (~k ' (n lp(O) f n//)) (~k ' (n!/J p (O) [ n)) -= { ~ --- ~ (k - k~~[) m2 E~(O) - ~n"(O)

",,

(n lp (O) n//)) (n/!Ip"(O) In)+ (~k! ' (n//jp (O)1 n)) (n lp"(O) In//) _s

'n(O) - 5n"(O)

n"

(n 'px(O)! n!!) (n/!fPa'(O)Jn) s2 a"~6(k-k!) 1 - ~, ~~(O) - sn"(O) . ak/' 2 1 '

n'

which the relation such as

~

a6 (k - k')

ak! y

(130)

~and

J ---a6(k-k//) a6(k!/_k/) _ a2 _ aky/! dk'/ rJ:1kfy2 6(k-~/) ak!,,

the other well-known formula involving o~-function are made use of.

Thus, the matrix elements of H are

(n~ltl n~!) = (n~~'klHof nk~~'/) + (n~fH'.i nk'~'!) + (n~fHb f nk'='!) + 11' (n~~kl [Hr. S] I n~'k!)

+ higher order terms

li2k2 2 a20~ (k~- k~~/) = (O)+ 6(k k)+ -~k* I a6(~-k'/_.__)_ s (a~ ) ~-~' s ' ~~ akly 2 1?1 ak/~2 ~'- m 1 n

+ ~! (hk ' (n f p (O) I n!/)) (hk ' (n!/j p (O) I n)) -1 -- o~ (k - k~'!) ,~' e~(O) - a~"(O)

' (n l p (O) I nl!)) (n/!j p.(O) I n) + (tk" (nl/l p (O) I n)) (n J p'(O) I n// ) + ~'(fik 1

~~' s a~ (k - k,)

n**

s2 / (n fPx(O) !n//)~n//J pa'(O) f n) - ~, --171~ n(O) - sn"(O) m

n'

s~(O)-~n"(O)

a-"a (k - k')

i

ak/ y2

ak!~!

making use of (121), (121)',

Then, the first and the

(121)'/ and (130).

fourth terms of (131) gives

h' ( * en (O) + = ( k~ 1 ) , ~ ) ~ * k 2 m ~~

~3

- 29 -

(131)

(132)

and the third and sixth terms are reduced into

7( 1'} a26(~-k~'!) - ) * xx o'k!y2 using the definitiOn of the effective mass tensor'

Finally. the second and fifth terms become

s a6(k-k/) r ~k^ I ~ {~L 2 ( m~ I (kx~k!x)} tk!~ / 1 ) ) x"J+ 2 'n + 2 ~.m T ak/~! "x

a *) J s a~(k-k!) ik~ I fik/ ~[ 2 ( In* ( I . )

=i ak/2/ + 2 m x" (rx

using the formula;

(ki~~k/x) o (~j(~-~!)=0' ak/2/

Allowing for the symmetry character of many crystal lattice' we have

( = * ( l 1 _ )~x )xa * In m

Then' the above equatiOn reduces to

s a~(k-k/) h l i ak/y ~~( In* (ka+k!~) )

"x

s_ a6(k-k!) 1 1 ~[ ~( ) ( ) l 1 fik ~~ ak/~! * ~x+~ ~(k!^_ka~ * ^x

m m

lik ~x s a6(k-k/1+~( n;* is6(k-k/) = (- ) l ) ~~, ~a ~ i ~ ak!y yx 1 n

using the relation.

(k -k'a) a6(k-k!)_~(_'k k) for a y o:]k/ ~

=0 for a=x'z' summing up (132). (133) and (134), we get finally

t7 1 j. - = (n~iEfn~/)=:{5?2(O)+ ~ ~( ~ ~ ) 2 k ^ ~ k~ ,6 (k - k!) + m*

^ p

- 30 -

(133)

(134~,,

1

1 s a6(k-k/)+h 1 +~hk ( y ~ ( m is6(k-k!) ) ~~ (~ (~ ) m* (~x7 ak/ ~- ~ yx

a2~ (k - k!) s22 ( ~* xx ak/y2 ' (135) - ) The matrix element of (124) is

1 ~ ~ ~ ~ (136) (nk H nk!) Bn(k/)d~'k/::=aBn(k)

and, substituting (135) into the above equation, we obtain

~2 aBn(k) { an (O) + a k p } Bn(k~~>) + isl J~lik,~ ( ) 1 * ( ) 1* ~ k 2 ax aky m* In a ~ (r p

h ( I oa2Bn(~~>k) (137) ) ~ ~(~~:* ) ~ ~ - ~ 1 - BI~(k) - - :=eBn(k'~>) +is l~ m* yx xx ak~!2 making use of the well-known formula;

J y -~L~) . a o(k k ) B (k )dk/:=_aBn(k)

al]k! aky J --~f . ot~2 a(k k ) B (k )dk'~>!=a2B~~(~)_

ak!y2 a~y2 ' The equation (137) is the effective mass equation in k-space of a crystal electron in

the magnetic field. Next, we shall go on to derive the effective mass equation in coor-

- dinate space by making use of the Fourier transform of Bn(k) as before.

First, the Fourier tranSform of the first term on the left hand side of (137) is shown

~:to become

l - ( ) J li2 1 ~ '~~ -l* ( ~ (xp en(O)+ 2 a ~ kpJBn(k)etk'rdk k

m (( p

fi2 a 2 = {5n(O) 2 ap axaax Fn(k) (138) - ~( ) * ~} ~ l

m a p

Secondly, the Fourier transform of the second and third terms of (137) is found to be

1 a * ~( 1 1 aBn(k) - - l Jlts~lik ( etk r +ts~~ m* ~ ) ) ~! Bn(~'k) eik' r Jdk'~>

m (rx aky yx (~x ~ r yFn(~=>r)+1~ m* fi( 1 :;=s{~a:( * T (r } ) l iF,1'(r) ) ~

m yx

- 31 -

2 * (y~Var+ I V~y)F?1(~~'r), =i ~( ) l fi (139), 1 n "x

using the partial integration and Bn(k)::(O for large k.

Finally, the Fourier transform of the fourth term of (137) is shown to become

a2B~(k) -' -)J -=i * ) - ' - i( ~ ak~!~"~ eik' r dk 2 (140)' (--1_ y2F~(1-)

m xx x" ' bv_ taking twice the partial integration.

Summing up (138), (139) and (140), (137) is transformed into the effective mass equa--

tion in coordinate space as follows.

s~ a2 g (O) 2 *) ( ) 1 { ^ ~( n ~( * ^ ' ~ ) l 1 yl' V(~ +~= V(ry +T (1'p ox^oxi3 'n m "x

s2 ) + 2 (~ ~ 1:' y2}FT,('='il)=aF.,(11~) . (141) xx

The above equation can be derived formally through the following simple procedure.

The band structure, i.e., the solution of the periodic problem

fi t {- ~ A+VI'(1~~~)r9)'nk~'=a,](~)cnk

2ln

may be written as

~" 1 n ( * '9 , ^,~ " ) =5 ' ~,. "p e (k) (O)+T m (142> k k

in the vicinity of the band edge to the effective mass approximation as shown before-

Then, the following substitution is carried out

1 e ka -~ -- rn ~~~A~

in (142) and the resulting operator is applied on FT"(r), which leads to

5ls(T ~ 1 r--~c A)Fr'(r~)=5F~('~'r)

or

~' I e { JIL~( * ) ,^( 4fi }F ('~'-7)--F (~~') T "~ ^ )(T l 1 i~ ' ') n -= n ' e en(O) + r IE' 4~ r 9 fic (143~ 2 1 n ^'s ^, p

~ 32 -

which is easily shown to reduce to (141) for a special gauge of vector potential (109).

The equation (143) is actually th.e standard form of the effective mass equation of a

crystal electron under the influence of magnetic field.

Special Case of Isotropic Effective Mass.

In the present case we have

( ;k 1 l -) , -- ( =: ::: ) 1 O for a:~:p m* a~r m m* ap

Then, (143) reduces to

fi2 1 . _ _A) F~(1'~~-)=(~^,-~^(O))F~(;), ~' (. e 2 m V fic (1 44) which is equivalent to the Schrddinger equation which describes an electron with effective~

mass 171* in the magnetic field A alone.

For the case of a special gauge of A,.=A==0 and Ay=Hx, i.e., H along z-axis, the

solution of (144) is easily found to become

1 ~ eliH ~2k32 (145> (O) + (N+ --/ + 2 m*c 2 m*

and

-* ( , ' ) F~('='r) = eik ~/y + ikzz e x - xo (146) ~x-xo)2/2L2 Hili

L

where N=0, 1,2, ･･.･ (quantum numbers of Landau levels) L (di/eH)~, x0=L2ky and

Hili(x) denotes Hermitic polynomial of order N. ky and kz represent y and z components

of the wave number vector, and xo corresponds to a coordinate of the center of the-

circular orbit around the magnetic field. Therefore, when the depth of crystal along ~

axis is l, we have to put

-1<xo < I ,

which leads to

- l/L2 < ky < l/L2 .

Since (145) does not involve ky, the stationary states with different ky values ar~

seen to become degenerate with each other. Finally, it shou.Id be stressed that ky and

k. are restricted to small magnitudes and H.N((x-xo)/L) should be slowly varying functiorL

as far as the effective mass approximation does have reasonable validity.

- 33 ~

VI．Exciton　Problem　by　the　E鉦ective　Mas8A即roximation

く6．　1〉　　Introd［uction

　　　The　exciton17）is　＆　quantum　of　electronic　excitation　energy　traveling　through　a

periodic　stmcture，whose　motion　is　characterized　by　a　wave　vector。Experimentally　the

exciton　has　been　actually　observed　iu　the　optical　properties　of　non－metaUic　crystals．As

＆matter　of　facts，in　a　wide　variety　of　materials　there　exists　charαcteristic　structure　in

optical　absorption　and　reflecting　spectra　which　cannot　be　explained　by　the　band　theory

based　upon　the　one　electron　approximation　but　can　most　reasonably　be　explained　on　the

basis　of　exciton　states　peculiar　to　the　many－electron　systems．

　　　In　the　present　chapter　it　is　not　the　purpose　to　develop　a　comprehens重ve　survey　of　the

exciton　problem　but　we　shall　describe　only　a　useful　application　of　the　effective　mass

approximatiQn　to　the　exclton　problem，on　account　of　which　a　very　simp玉e　picture　of　an

exciton　is　successfully　obtained．

＜6．2）　Fundamental　Equation　of　an　Exciton

　　　We　shall　suppose　the　crystal　to　be　composed　of　IV　atoms　each　of　which　has　two

valence　electrons　outside　the　closed　shelL　The　Hamiltonian　for2』Velectrons　in　the　crystal

lS

　　　　　　　　　　　　　　　　　　　丑一慌Σム＋Σ㈲＋二；チ、睾，i

　　　　　　　　　　　　　　　　　　　　　　　　　　　　’　　　　　∫　　　　　　’＞ノ

・and　the　Schr6dinger　equation　is

πψ0＝Eψ’

（147）

（148）

where　the『五rst　term　of（147）is　kinetic　energy　of　the　electrons，the　second　one　valence

electron’s　interaction　with　the　nucle五and　closed　shell　electrons　together　with　the　spin－

orbit　interaction　and　the　last　one　mutual　Coulomb　interactlon　of　valence　electrons．We

assume　that　all　the　nuclei　are　at　their　equilibrium　positions．

　　　Generally　speαking，the　Schr6dinger　equation　of　many－electron　systems　is　reduced　into

that　of　one　electron　system，introducing　an　approximate　average　potential丘eld，wh圭ch　is

usually　called　the　Hartree　approximation，Unfortunately，however，the　one　electron　system

to　Hartree　approximation　is　not　always　unique　for　electrons　in　the　crystal．The　next　step

for　treating　many－electron　systems　is　to　constmct　an　antisymmetric　or　Slater’s　deter－

minanta豆function　corresponding　to　a　de丘nite　electron　configuration　of2ノ〉electrons　over

the　one　electron　energy　levels，through　which　the　exchange　interactions　among　electrons

一34一

r

are explicitly taken into account. Then, when we want to proceed to get the solutions:

of stlll higher approximation to the many-electron systems, configuration-mixing method

is taken into consideration, in which the total wave function may be expanded into the=

orthogonal set of Slater's determinantal functions corresponding to different electron con-

flgurations and thus the correlation effect among electrons turns out to be allowed for

approximately. Except for some particular problems the method of configuration-mixing

is too hard to be carried through on account of the extremely high degree of the cor-

responding secular equation, but the exciton problem is shown to be a typical one of the

solvable problems in the framework of the method of configuration-mixing.

(a) Ground State of the 2 N Electron Systems

" Bloch representation "

In this case the one electron orbitals, of which Slater's determinantal function i~

constructed, are Bloch orbitals which satisfy

~2 { ~ 2m d + Vp(~='r) }c~k~'=g~(k~~') c~k--'. (149),

Then, the Slater's determinantal function corresponding to a filled valence band may'

be lvritten as

ep I . (150) 0= ~ ･ ~,~ opPpvk (1) (~vkl p (2) ･･.･.. cvk~~',~"p (2N) (2 1¥T!) I /~~~

P

where we have

(~vk'~'1'" = f!"vk"=; (r) (r_12((T) '

~5v~1,p=cvk'~1(r)o; ~((i)'

and

(rm'(c) is spin eigenfunction.

The ot-h~r notations in (150) have the usual meanings. The corresponding energy of th~

ground state is found to be

E (c Hep) 2LT1{e(k)+J(5vk(1)lU Vplcvk~'(~~'r)d~}

~ ~{J 1 + '! cv~'k',' (rl)cvk~,,.,(~~'rz) JILe cvk~;.(~~rl)cv~~"k "' (r2)dfldr2

rl 2

k,. ~~..'

S･- ･ - } ~ cvk~:..(rl)cv~~"k ." (1~2) ~'~~n2 cv" (r~'2)cv'~"k "' ('~1)dTldtz - (151>

- 35 -

“Wamier　representation”

We　shall　de五ne　the　Fourier　transfor皿of　Bloch　orbital，i．e．．

　　　　　　　　　　　　　ヌ　　　　　　　べ　　　　　　　

砺，75（r）＝N「7Σビ’た’Rφ齋’（r），

　　　　　　　　　　て

Wannier　function　by

（152）

　　　The　characteristic　properties　of　Wamier　function　are　shown　to　form　an　ortho－normal

set　of　functions　and，　furthermore，　to　become　more　Qr　less　localize（l　compεしred　with　the

extended　Bloch　orbital　over　the　whole　space　region　of　the　crystaL

　　　The　Slater’s　determinantal　fmction　constructed　by　Wamier　fmctions，corresponding

』to　the后11ed　valence　band，may　be　written　as

φ・㌦2病f／2Σδp煽1，α（・）砧否・・β（2）……幅・β（2N）・　（・53）

By　the　use　of（152）and

P

α罐1，α＝α品（ア’）α1（σ）etc，

　　　　　　　　　　　　　2

he　well・known　theorem　of　determinant　we　can　easily　show

φo＝φoπ1． （154）

Then，it　follows

E。匹（φ評，丑φ。｝γ）＝（ψ。，πφ。）＝E。， （155）

which　means　that　the　ground　state　energy　of22〉valence　electrons　in　the丘lled　valence

band　is　Eo　in　both　representations．

（b）　丁猛e　First　Excited　States　of　the2N　ElectrQn　Systems

　　　We　shall　here　consider　the　excited　electron　con五gurations　in　which　an　electron　with

wave　mmber　vectorゐ1むand　spin　of一σin　the　valence　band　is　removed　to茸state　of　the

conduction　band　with　wave　number　vectorた，and　spin　ofσノ，thus　a　positive　hole　in　the

valence　band　and　an　excited　electron　in　the　conduction　band　being　realized　in　the　excited

con五guration．

　　　　‘Bloch　representation”

　　　The　antisymmetric　wave　function　corresponding　to　theαbove　excited　configuration

is　giVen　by

φ甑忍）r2詰戸Σδppφ嗣・）φ謁

　　　　　　　　　　　　　　　　P

　　　　　　　　　　Xφ鵡，σ（づ）φ‘鶏，σ・（歪＋1）一・

　　　　　　　　　　　　　　　　一36一

，、窒（2）

φ鵡，β（2N）・ （156〉

f

Now, in the ground state corresponding to the filled valence band we have

2N 2,v k O and l~n (157) ~~,= ･ ~- ,= -2CF O

i=1 i=1 'since the electron configuration is governed by Pauli principle. Otl the other hand, in

~the mentioned excited electron configuration the total wave vector and the total spin are

_given by

li

A- k kn and S.=1,(a+(7 ) (158)

Therefore, in the similar way to the case of helium atom, we get

1 ~p 1/2 (c~c~ eplsv"c) (159) for a singlet configuration

*and

1 c)"" ' "=~='1/~~ (c"P+cp~) (160) c ~ p

vc v' ~c vc '

-for triplet cofiguration, respectively, allowing for the fact that the mentioned electron

-configuration is specified by one electron and one positive hole.

The Inatrix elements of total Hamiltonian, which can be colnputed by (156), (159)

;and (160), are shown to become

(vck*kh {Hj' vck.'k/~') = 6k~~>~'k~=>f" 6~='k,,k~~'., [EO + W.(ke) ~ W,J(kh)]

+~~~-~,,k'~',,-~,, I (cke'vkh lgck. ,vkh)+2iM(ck vkh lglvkh ck )] (161)

~where o~M=1 for singlet configuration. 6M=0 for triplet conflguration and

We(k) = a.(k) + (ck I U- Vpi ck) + ~ [2(ck, vk/j gf ck, vk/) _ (ck, vk!] gJ vk!, ck)] , (162)

W,,(k) = a~(k) + (vk I U- Vpf v~) + ~ [2 (vk, vkllg] vk, vk') -(vk, vk!lgl vk!, vk)] , (163)

- ~ ~ - r ･ ･ e2 (vk ck/Jglvk/',ck///)=jcv~~'k (1) cc~"k (2) ~ ipv~~"k ' (1) c k (2) daldr2 (164) ' rl2

~e(k) and a~(k) denote Bloch energies of k-states in the conduction and valence bands, i.e.,

,ck state and vk state, respectively. W.(k) represents the total energy of an additive

electron in a ~ state of the conduction band, interacting with all other electrons in the

~valence band, whereas, W*(k) the total energy of an electron in a k state of the valence

- 37 -

band，interacting　with　all　other　electrons　in　the　valence　band．

to　be　a　self－consistent丘eld，we　have

When　Vp（7）is　supPoseα1

　　　　　　　　　　　　　　　　　　　　　　　　　　　　　琳（た）＝εu（た）．

　　　As　clearly　seenゆ（161），the　total　Hamiltonian丑becomes　diagonal　matrix　with

respect　to　total　wave　number　vector　K二為召一島，i．e．，excited　con丘gurations　with　different．

K　values　become　linearly　independent　of　each　other．

　　　“Wamier　representation”

　　　When　we　remove　an　electron　from　a　localized　Wannier　state　aromd1～鬼of　the　valence・

band　to　another　Wamier　state　around　R，of　the　conduction　band，we　get　an　excited

connguratian　which　may　be　described　by

　　　　　　　φ麗（顧）一（2鳶扉Σδp臨・α（・）α癖（2）一一幅・・（乞）α6鳥ρ・（歪＋・）

　　　　　　　　　　　　　　　　　　　　　　　　P

　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　砺織，β（21〉），　　　（165）

童n　which　one　positive　hole　in　the　state　of　α”π厄，＿σ　and　an　excited　electron　in　the　state　of－

α。π、，σ’are　realized　and　the　distance　between　a　hole　and　an　electron　is　denoted　by　β＝

Rθ一＆．

　　　By　the　use　of（152）and　the　theorem　of　deteminant　the　following　relation　between．

（156）and（165）is　shown　to　exist．

　　　　　　　　　　　　　　　　φ冨（ゑ，青，）＝N－1ΣΣ4謁・乱一ゑ・π厄）φ器（戸ぬ，π，），　　　　　　　　（166〉

　　　　　　　　　　　　　　　　　　　　　　　　　　　　π，πん

which　is　unitary　transformation　between　both　wave　functions，

　　　The　matrix　elements　of　total　Hamiltonian丑P　with　respect　to（165），i・e．，

　　　　　　　　　　　　　　　　　　　　　　　　（び6，R飽，R8「別麗，ノ㍑，Rεノ）　　　　　　　　（167〉

are　easily　computed　in　the　similar　way　as　in（161）and　are　found　to　become　functions　o「

（R1らノーR，），β，β〆only　on　account　of　their　translation　symmetry，where　R8＝Rん十β，R♂＝

Rんノ＋β〆，

　　　Therefore，wave　functions（156）and（165）cannot　diagonalize　the　total　Hamiltonian

丑as　shown　by　the　existence　of　non・diagonal　matrix　elements（161）and（167），which

means　that　wave　function　corresponding　to　a　single　electron　config皿ation　is　not　correct

wave　function．Then，we　have　to　proceed　a　step　to　higher　approxim乱tion　by　taking．1．

con五guration　mixing　into　account　as　explained　before，

　　　“Exciton　representation”

　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　一38一

、

According to the mentioned considerations we shall next construct a linear combina-

~ tion of (156) or (165).

Namely, we have

ep""".(K p) N 1/2~e~'~L,3 ~c;~'(~'k-k,~) (168)

= N- I /2 ~:ei~.~ ap""".' (~ , ~ + j~-'3) (169)

R

where k denotes a wave number vector of an excited electron of the conduction band

and R a lattice point around which a positive hole of the valence band is localized. The

equation (168) can be shown to be equ~:1 to (169) by using (166), and the above linear

cornbinations are called exciton representation, which reason will be seen later.

The matrix elements of the total Hamiltonian H with respect to exciton representa-

tions may be easily computed with the following result.

(vcK, p IHI vcK. P') = ~~ p'~>'E0+ W.(cp, c,3') -eiK'(P- ~') W,~(v, O ; v, (~ - P!))

+2 6M(cP, vOf gf vo, cP/) _ (cP, vojg f cP', vo)

+~~eiK'R {2 6M(c/S. VR,IgJ vo, c (fi/ +R)) - (cp'. VR ; gJ c (R + ~') , vO)} , (170)

R~0

where

~ ~ J ･ -(~)'~, cR~~>'f gf vR", cR"') = e2 (171) a*~+R"(1) a.R~""(2) dfldt2 a~~ (1) a.~'(-2) rl 2

and

W(cR, cR') = (cR! -fi2J/2 m + U!cR') + ~ {2 ~M(CR, vR"Jg! cR/, vR'!)

R''

- (cR, vR" gf '~'R/', cR')} . (172)

The expression (171) represents Coulomb integral and (172) corresponds to the interac-

tion energy of the charge distribution with density a.~' a.~f,, with all other electrons in

the crystal.

Now, (170) involves various quantities computed by Wannier functions. However,

it will be convenient later to have the second and third terms on the right hand side of

(170) transformed into the quantities computed by Bloch orbitals. Using (152), (162), (163)

~'and (172), we find, after some process of manipulations,

- 39 ~

VV*(c P, c p') - eiK'(~ - ~') W~(1~'O, I~' ( ~ - i'j/))

= ~¥r-1 ~eik'(~- p') [ Wr*(k) - W*(k - K)]

= eiK'(fi - p')/2 1¥r-1 ~eik'( p - ~'j [ W.(k + K/2) - W~(k - K/2)1 ･ (173)~

Then, we get

(VCK p IHI vcK, p') = a~ p" Eo +eiK'(19 ~ p'),'2 N-1 ~_ eik'("'~ p') t W*(k +K/2) -

- W~(k - K/2)] +2 o'M(c*S, ~O jgj vO, c,3') - (cp, vO J gj c p', vO)

+ ~eiK'R {2 iM(c~, 1~'R J' gj vO, c ( p' + R)) - (c,3, ･VR ig j c (R + fi") ' vO)} . (174)=,

R+0

Since the non-diagonal elements with respect to K are shown to become zero, (170).'

shows that we have to further proceed to diagonalize (170) with respect to p. For that:

p.urpose, we shall construct the following linear combination of wave functions with_

different fi of the exciton representation, i.e., (168) or (169).

ep~..A--'-= ~ U~*.~ (f3~*) c""".' (k, p~=") (175) ""

p

which substitution into (147) is found, as usual, to result into the difference equation

such as

~ (vc, Kje!H= ~~,c, Kt3') U~..A--'･ ( P') = EU~. If ( ~) (176), p~="

and the secular equation

detf(vc Kp^'H]vc Ki"") ErJ~-',fi I=0 . (177),

The solutions of (177) give the energy values of the exciton states and the corre-"

sponding wave functions are given by (175) after substituting the solution of (176) into'

(175). Actually, the approximate solutions of (177) have been obtained for some simpler

cases but we shall not come into the detailed discussion here.

(6. 3) Application of Effective Mass Approximation to the Exciton Problem

At present it is very difficult to get accurate solution of (176) for the actual crystal:

and we shall here sh~w an approximate method of solution for (176), which is calied the,

effective mass approximation to the exciton problem.

- 40 -

l'

Separating the matrix elements of (174) into two parts, we get

(~"c. Kp jHl vc, Kp/) = HO~: ~~ (K) + Jfl ~~>'fi (K) , (178),

- -~~ T1 ~~~r (~ K ) Hol~~'s, ~~ (K) = a~ ~~"p Eo + eiK(~ - P':'/2 Ar-1~etk'(P - P')L W.~k + -2

k e - - )]- ~~4, W,,(~ = (179)" 2 6p s 15

- r +2 ~M(C~, vO IgJ 1"O, cir3') J ~ ? (K)= - L(c'3, ~,Olglci(2.~~~L)', 1"O) -(~)~~",e =~r] e2

+LJVletK R[20M(cp VR gfvO c(B+~ )) (cfi vRlgJc(~+p ) vO)], (180)

~~F o

where (179) is seen to make appreciable contributions for large values of p, i.e., for large-

distance between a positive hole and an excited electron of the exciton, whereas, (180)

does only for small p, that comes from the rapidly decreasing property of (180) with

increasing P.

In ~he following discussion we shall approximately disregard (180) and, therefore, the

method of solution described below may be considered to be reasonable in the range of

large P or for an exciton of large size, which is called Wannier exciton in the literature~

Thus, we have

(vc, K, p jH1 ,vc. K, ~/) c:Ho~ '-',p (K ) . (181)'

The substitution of (181) into the difference equation (176) Ieads to

- Ir ---~r (~ K A- UK(P'~~') e~Ulk(t~~B) ) W(~-)] r~ _ EoU/k(p)+N-1_L_etk'(p-~',LW*~k+= - v p 2 2

fs',k

= EU"-'-K ( p) . (182), where

U'k (r~~5) =e'ik p/2 U~. K ( ~) (183),

In the case of large p compared with the lattice constant, the difference equation

(182) is shown to reduce into the differential equation through regarding P to become

approxirnately continuous instead of being discrete. Before takin~" such procedure we~

shall sho~v the useful theorem due to Wannier in the following.

- 41 -

~loes

:and

According to the Wannier theorem, the relationship

N- I ~eik'T: ~ - '3'). f (k) g (,3/) = f ( - i' 3)g ( ~) -

j'.k

hold

Proof :

We shall introduce the Fourier transform of g(P!) such that

g (i3!) = N- I /2 ~e ~ i" '3 G (,c)

The substitution of (185) into the left hand side of (184) Ieads to

~ ~ ~L;;;, f (k'=')g(iS'~'/) = ~V-3 '2 ~eik~',',s^'~' ,'~~":'~e~ i"'3'~" G (¥~) N- I ~ eik'( ~

s',k 3',k =N-3j2~f(k) G(J~)eik'~~e~i':k+' ~

k. * ;' = N-3!2_~_f ('~'k) G (~)e~.,'~>SN a~~o

k, *

=~¥T-112~e~i"~ f(-'~)G(,c).

Now, the power series expansion of j'(k) reads

J~(k)= ~ ai',j"k' k.,i' k j k k i* j'k'

then we have

f(-~)= ~ ai',j"k' (-/~-')i' (-'Cy)j' (_'~.)k' .

i"j"k'

Putting (187) into the above equation, we get finally

~ ~ - ･-~ ~ , JV-1/2 e~i"fi at j' k'( -'~,^)i' (-'~y)j' (_t~i)k' G ('~'x)

i"j"k'

l a * I a ' I a k' = ( ) l~ ~ ~ ~ ~ -'t )( z * )( j ai',j"k i ap ~~ ~a/3y i rj',J' N-1"2 e~i"~ G(,~)

i"j"k'

(= ) =f li Vfi g(;)

Application of Wannier theorem to the second terms on the left hand side

- 42 -

(184)

(185)

(186)

(187)

of (182)

1

,

leads to

[ c ~ ~) ~ ~~] W (1. r~+ K )_ W ( I Vp K/ U/~(p)=(E-Eo) U/--"A(~), (188) e ~ e ~2 . v 2 p which may be considered to be reasonable in the range of p>>lattice constant. The-

differerential equation (188) is more tractable than the difference equation (176) in the

actual computation.

A closer examination of (162) and (163) shows that when Vp(r) is actually a self--

consistent potential,

Wc(k)~(~c(=k) and W1;(k=):=el'(k~~) . (189)

To the effective mass approximation the band structure near the band edge (k=0) of botkL

conduction and valence bands shall be supposed to be

W; ('~'k)~:~e('~'k)::::~c(O)+ li2k2 (190) 2me* '

~2k2 Wv('~='k) = s ~(~~'k) ::: sv(O) - 2mh* ' (191)

where Ine* and mh* are the effective masses of the conduction and valence bands near- '

the band ,edge, respectively. 5c(O) and g~'(O) are the bottom energy of conduction band

and the top energy of valence band, respectively.

Then, (188) is found to have a sirnple form as follows.

~2 ~ 1 ~ l [ * _ * ) /~~ ~2( 1 -=A -1L_ 2,s fi p me K.VpJUK(~) ~T~ mn

= (E- Eo ~ F.G fi2 (192), - K2) UT~(p~) , 8 p

where

7ne*mh* - . (reduced mass of electron and hole) (193), f~ me* + Inh*

and

EG=ec(O)-ev(O), (band gap between conduction an^d valence bands). (194),

In order to eliminate the term K.rp, we put

U;eA'-'.(p)=ei~K'pF(p) (195> -43-

with

_1__ (mei],_iiILL) (196) oi-2 (me*+m )

The substitution of (195) into (192) Ieads to

[ I == -- -2 J ~ , li2 e2 h2A' [ ･- ･ -~~, Vp~~7 (197) ----- F(i3) F(i3) -E Eo EG 2(me*+m.h*) J

which yields a simple picture of an exciton for large 13'

(197) represents the Schr6dinger equation of the relative motion of an electrv~n (me*)

in the conduction band with respect to a positive hole (mh*) in the valence band interact-

ing with each other through Coulornb attraction- Thus, the equation of many electr.on

systems has been shown to reduce approximately to that of two particle systems.

The solutions of (197) are easilV. found, in the similar way to the case of hydrogen

atom, to become

,tte4 li2K2 ~ _ , n=1, 2 ･････ (198) 1 E + J G t~li2n' +~~(n" 'n,' )

~where the third term represents internal energy of an exciton and the fourth one trans-

lational energy of - the center-of-mass of an exciton. As far as translational energy is

small, the exclton energy (198) Iies within the energy gap above the ground state energy.

In this connection, it should be remarked that (198) represents really the energy values

of many electron systems.

The corresponding wave function of an exciton is given by

"r""' - = - t 'n~'/(m~'+ m',':,;1'~'c!; F,,,!""(p'~'~) c"v"c'(k. :~3) , (199) ~e :r vc.K

1livhere Fn,r,m(i"j) is the solution of (197) and translation character of an exciton is specified

by the exponential function of (199).

When the band structures of both conduction and valence bands near the band edges

ko and ko/, to the effective mass approximation, are written as

li2 -+ (~- ) ~ l,..* -' ~~ ~ ~ In ~c(~~'k)=ar(ko)+ 2 (200) (k - ko)~ - (k - ko),9 ~,3

~,*i

~~nd

~2 ( ) 1,* =5 ~,/ _ ~ - -' ,~r,~ E (k) t'(kj) I~ ~ (~='k-~0/)p , (201) - (k-ko/)~ m r

(t, 3

- 44 -

1

-then we obtain' for the effective mass equation'

)I r li2 a _ a mh* - me* L~ - ~( * . afi +~ )( U'I h2 li_~ ~(~:u ,, :' ) - a2 -rne* + InrL

9 adlJoF ;!"' me*m'~* ap!1a/9~; p 4 me*In'~?P ll'~'

=:[ _ _ __ ____ )p.~; l h2 Inn* + m o~ J ~(- -- e~_ E E EG -9_ (202) ~~ ~;;x'~!' J Ull m'~*Ine*

!i~;

usmg

'~=K-k +k and U!/ e 'ko +ko) fi/2 UK(fi) (203)

When we want to go on to still higher approximation, perturbation theory in quantum

mechanics should be adopted through taking account of J,-'~,e'~~'(K) as a perturbation.

Summing up the above results, it may be concluded that an exciton characteristic to

many electron systems may be regarded, to the effective mass approximation, as an

electron-hole pair with mutual attraction as far as their mutual distance becomes large

_In comparison with the lattice constant.

l)

:2)

3)

4)

5)

6)

'7)

8)

9)

1 O)

11)

l 2)

_ 1 3)

Ref erences

W Kohn: Phys. Rev 105, 509 (1957); 110, 857 (1958)

A Klein: Phys. Rev 115, 1136 (1959).

M Cardona and F.H. Pollak: Phys. Rev, 142, 530 (1966).

T Muto et al : J. Phys. Chem. Solids, 23, 1303 (1962).

T lvluto and S. Oyama: Prog Theor. Phys. 5, 833 (1950); 6,

E O. Kane: J. Phys. Chem. Solids, 8, 38 (1959).

A.H. Wilson : The Theory of Metals, 2nd ed. (1953), p. 47.

J C. Slater and G. Koster: Phys. Rev, 65, 1167 (1954).

J C Slater and W. Shockley: Phys. Rev, 50, 705 (1936)

J.M. Luttinger and W Kohn: Phys. Rev. 97, 869 (1955).

W. Kohn: Solid State Physics, 5, 257 (1957)

W. Kohn: Ioc. cit.

F. Herman: Phys. Rev 95, 847 (1955), Physica, 20, 801 (1954).

G G. McFarlane and V. Roberts: Phys. Rev. 98, 1965 (1955).

W Kohn: Phys Rev. 98, 1561 (1955).

D.K Stevens et al.: Phys Rev. 100, 1084 (1955),

R C. Fletcher et al,: Phys Rev, 100, 747 (1955).

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W. Kohn and J M. Luttinger: Phys Rev. 97, 1721 (1955); 98,

- 45 -

14 (1951).

915 (1955).

14〉

15）

16）

17）

A．H．Wilsonl　Tん6Tん607ッげ漉凄α」3，2nd　ed，（1953），p．47．

W．Kohn　and　J・M．Luttinger：Phys・Rev・97，869（1955）・

L，D．Landau　and　F，M，Lifshitz＝g照π‘㍑規M60んα痂05（1956），φ，471，

R．S．Knox＝Th6Thθor：y　o∫Eπ琵oπ5，S妙μ伽飢ε5‘o　So1‘4S螂θP妙5乞05（1963〉・

D．L．Dexter　an（1R．S，Knox：Eππoπ5（1965），

46