Electron-electron interaction in effcetive mass equation

Volume 12, rmmb~r 3

PHYSICS LETTERS

ELECTRON-ELECTRON

INTERACTION

1 October 1964

IN E F F E C T I V E

MASS

EQUATION

A, W E ~ S S M A N ~ and L S T A N Faculty of Physics, Babes-Bo~y~t University, Cluj, Roumania Received:4 S~a~mb~r 1964

In this paper we try to recalculate the well-known effective-mass equation, give, by Kohn !) for a semi-conductor in a many=electron ease, taken into account the long-range electron-electron interaction which is not possible to be neglected for sufflclentlyshort wave-le*gths and sufficientlyhigh frequencies 2), W e cenaider an .'impuritycentre in a semi-conductor, with a simple hand, having the minimum at k = 0. The Hamilteate, of the system in the usual notation is I): "

Zee2

1 ~

e2

Using the condition: v

v

(~)

(P~ - i,k o.~)~ ffio, we obtain with ~e. well-known techs!que 3) for the Hamiltonian (i) the following expression: H =//o + Hosc + H s r + ~ ,

(3)

where J,r~p.2

X o = /D= I LI-7--..v(~,)|, zm • J

(4)

i s the Hamiltoniau of the free electrons~

%sc -i D - 2 h
V .~V V2 ~V ~ V . (P~% +~ %%h

(s)

is the Hamiltoltiml of the plasmon field 2)~ Q/zV and Pk v are the field variables satisfying the conditions: Ql~'~v = .Rkv; pk*V = .p_kV wv is the frequency o£ the collective oscillations of the beund valence-elee. irons, which satisfy the dispersion relation ~) a~v2 = w 2p + k 2 , (6) 1

COp,= ( 4 ~ 2 / m ) ~ being the frequency of the b a r e electron oscillations, k the wave vector, (v) the m e a n velocity of "the electrons and h D = 1/;~D, where ~D i s the Dehye Wave-longth. 1

x-,

~k 2 --%

v

where ¢ (k, CO)i s the dielectric constant in the collective aDproximation 3). F o r the about mentioned f r e quencies we have the foIlowiog conditions: ~/o ~ Wl << cop;

COzo ~" ~2 >> ~ p

and

w i << ~ U v << ~2 ,

(8)

where l, :~, ~L, ,~ a r e the qtmntum numbers which :, cpresent the energy s t a t e s in the valence band {l, r) r e spectively conduction band (~, v). 196

Volmne 12, nnmber 3

PHYSICS L E T T E R S

1 October 1964

For solving the SchrC~divger equation: ( t t - ~ g , = O,

(9)

where H is given by (3) we expand the wave function:

V: = ~

f dl~ An,N,(K' ) gOn,N,K, ,

(10)

in t e r m s of the complete s y s t e m of functions:


(11)

tpN(Q)= k<~ tPNk(Qk) of Hose,

N h being the quantum nmu~her

of the oecillator with Qk coordinates. The eigenfunetlons of H o from (4) are Slatar determinants £orme~i with B1och-fuaetions 5). The ~ n N K functions satisfy the orthonormallty condition:

(¢'n#K, %'~'t¢) =f ~i(K~-K)' ~ r ~"*n o#~~.n. .'.o WN'*""~N' dr

d Q = 5nn, 5NN, 8(I~ ~K) .

(12)

Because the plasmous a r e longitudinal pho~ns we must take k •K, and we obtain alter simple calculations, the effective-mass equation: [~N{-IV) ÷ Ueff(r)] Fn]~r) =

EF~A~r) ,

(I$)

where Fn~(r) are rite~rausformed fanetions 5), Ueff the screened potential ~md perturbatinn energy, and E ~ - i v ) is the expansion of E R N up to the second order in k~ 'l~akinginto account (6) and (8), alter the elimination of the first degree interbund terms, we obtain fr~)m (13): Eoo(0 ) + ½/~a~p[~ + J ~ ;

+1) + A ~ ] 6 N N ,

+ ½kotk/3{l+ ~(v2>

k~h~ +__~_iD(va)M(v~)~n] m o Ln'

"~V~F

[

~

+ #~ml%v,}~ ~

(1~)

=, ,.~ lla2S'oN(k')~ ~,

'J- = ~ ' ~ '

"~ " - / k = o

"~

~

o,

since (~3) in absence of a potential is satlsfied only in this case. (14)

:

~o~(O) = Eoo(O) + ½ ~ p [ ~ )

+,~/~~]~N~,

(~)

gives the shift of the fundamental level due to the electron-electron interaction. The.f-sum rule for the electron-electron interaction is : .

2t~N x-, (Va)nn ' (V~n'n m 2 n' *n

02EnN(k)

En~ - En

with 1 I + (v2)~ .. . . . . ,-¢-~-~,. ~ - ~ = - ~ - S-~p t 4 , ~ + ~ v ~ . + ~ +4.~.loN.A,, .

(l';~

F r o m our investigations we obtain the result that for exclte~lstates with sufficientlyhigh i~t,,qu¢~¢l~, the effective muss depends on the screened e!ectron-electron, respectively electroa plasmon interae~ t~ons. On the other hand, electron-electron 10rig-range interaction is the cause of the shHt ol the ~erit~~ isvel shift which depends on the plasmon-energy~ mean velocity and quart,turnlevel of the co~lectlve ~wctilations. The shift of the fuudamontal energy level and the variation of.ths effective m~ss at the bottt~, of the band appears also for N = O, due to the vacuum interactions.

IH