IL NUOVO CIYIENTO Vor.. L V B , N. 1 11 Maggio 1968

Polaron Energy Spectrum in a Magnetic Field (*)C).

K. K. BAJAJ

Department o/ .Physics and Astronomy, University o] Kansas - Lawrence, Kans.

(ricevuto il 29 Novembre 1967)

S u m m a r y . - - A study is made of the energy spectrum of a polaron in a magnetic field for the case of weak and intermediate couplings. The results obtained are compared with those of a numerical calculation of the self-energy of an electron in a magnetic field interacting with the optical phonons of a polar crystal. The method used is based on the approach first given by Onsager. The results obtained are compared with those of previous variational calculations. We also investigate the numerical change of the polaron mass with magnetic field and compare it with tha t due to the nonparabolicity of the conduction band in GaAs and CdTe. In CdTe the firs~ effect produces a change which is about three times larger than the one produced by the second, whereas in GaAs it is the latter effect which is more important.

1 . - I n t r o d u c t i o n .

Studies of the ene rgy s p e c t r u m of a n e lec t ron m o v i n g slowly in the con-

d u c t i o n b a n d of a po la r c rys ta l a n d i n t e r a c t i n g w i th the opt ica l p h o n o n s of

t he m a t e r i a l h a v e b e e n car r ied ou t b y severa l au tho r s (1-6). Such a s y s t e m

(*) Par t of this work was done while the author was at Purdue University and was reported in Semiconductor Research Semi-annual Report October 1, 1965 to March 31, 1966 of Purdue University.

(**) Supported in part by the Advanced Research Projects Agency and the U.S. Army Research Office, Durham and by the University of Kansas.

(1) H. FROrrLICE: Adv. in Phys., 3, 325 (1954). Reference to earlier work can be found in this work.

(~) R. P. F~33:~cIAN: Phys. l~ev., 97, 660 (1955). (8) E. H. LI~]~ and K. YA1KAZAKI: Phys. J~ev., 1 t l , 728 (1958). (~) D. S. FALK: Phys. l~ev., 115, 1074 (1959). (5) T. D. Sel~ULTZ: Phys. Rev., 116, 526 (1959). (6) An especia]ly useful reference is the book Polarons and Excitons, edited by

C. G. KuPa~R and G. D. Wr[I~FIELD (Edinburgh, 1963).

POLA/~ON EIWEI~GT SPECTRIJM IN A MAGNETIC FIELD 245

where the electron cont inuously emits and reabsorbs v i r tua l phonons is called a polaron. Calculations have been made for the case ill which the coupling be tween the electron and polar izat ion field of the la t t ice is weak and for t h a t in which it is reasonably strong. The first s i tuat ion is realized in m a n y of the I I I - V and I I - V I compounds while the second is appl icable to the alkali halides.

Inves t iga t ions of the energy spec t rum of a polaron in the presence of a uni form magne t ic field have been m a d e b y TuLtln (7), HELLWARTH and PLATZ~A~ (s), and b y LAI~SE~ (9). TULUn (7) finds t h a t the polaron effective mass for weak magne t ic fields is g iven b y t ha t of the field-free mass plus a t e r m propor t iona l to (o~/(o) ~. Here ~o~ is the cyclotron resonance f requency of the bare electron and co is the opt ieal -phonon frequency. Al though derived for the case of in te rmedia te coupling the effective mass he obtains does not approach the weak coupling resul t to order ~, where ~ is a constant t ha t meas- ures the s t rength of the coupling be tween the electron and the optical phonons. This is in d i sagreement wi th the resul ts of the presen t work. ~[ELLWARTH and I°LATZMA:N (s) c o n c e r n themselves wi th the free energy of polarons in a magnet ic field bu t not wi th the energy spect rum. Thus the comparison is not immedia t e ly possible. Using a var ia t ional me thod re la ted closely to the in- t e rmed ia t e coupling theory of Lv.E, LOW and PINES (~o), LAICSE~ (~) has ob- ta ined the ground s ta te and low-lying exci ted s ta tes of a polaron in a magnet ic field. The model I t ami l t on i an is t h a t used b y FR6I~LIOH (1). The results we obtain are quite s imi lar to those obta ined b y LARSEN. Our procedure consists in calculat ing the energy spec t rum of a polaron in a magnet ic field using Onsager 's theory (~) and compar ing the results with those obta ined direct ly using second-order pe r tu rba t ion theory on the Landau s ta tes (1~) of an elec- t ron i n a magnet ic field.

Cyclotron resonance exper iments in several I I I - V compounds have been reviewed b y PALIK, TEITLER and WALLIS (13). A small change in the cyclotron effective mass of an electron as a funct ion of the magnet ic field is observed. This change is a t t r ibu ted to the nonparabol ie i ty of the conduct ion band. In the presen t work we show that , in some cases, the var ia t ion of the polaron effective mass with magnet ic field is of the same order of magni tude as the

effect observed.

(~) A. V. TULUB: 2urn. J~ksp. Teor. Fiz., 36, 565 (1959). (English translation: Soy. Phys., JETP, 9, 392 (1959).)

(s) R. W. HELLWARTH and P. M. PLATZMA~: Phys. Rev., 128, 1599 (1962). (9) D. M. LARSEN: Phys. l~ev., 135, A419 (1964).

(lo) T. D. LEE, F. E. Low and D. PINES: Phys. Rev., 90, 297 (1953). (11) L. 0NSAGE~: Phil. Mag., 43, 1006 (1952). (12) L. D. LANDAU: Zeits: Phys., 64, 629 (1930). (13) E. D. PLAIK, S. TEITLER and R. F. W'ALLIS: Journ. Appl. Phys. Suppl., 32,

2132 (1961). See also references to previous work mentioned in this article.

2 4 6 K . K . BAJAJ

2 . - T h e o r y .

We consider a single electron of effective mass m and charge - - e moving slowly in the conduction band of a polar semiconductor . We suppose fu r the r t h a t the energy surfaces possess m i n i m a at the center (k = 0) of the ]~ril- louin zone and tha t they are spherical. These assumptions are satisfied to a considerable degree b y m a n y of the I I I - V compounds such GaAs and I n P

and b y some of I I - V I compounds such as CdTe. The dynamica l proper t ies of the sys tem we envisage are specified b y the

t t ami l ton i an opera tor (~)

(1) p2 • t He = ~ + ~ [vo~o exp [iq-~] + v~ a~ exp [ - ~q. ~]] + ~ ( ~ . o + ~). q q

The first t e r m is the band energy of the electron in the absence of any interac- t ion with the phonon field. The th i rd t e r m is the energy of the longi tudinal optical phonons. He re a~ and a~ are des t ruct ion and creat ion opera tors for a p h o n o n of wave vec tor q. We m a k e the assumpt ion t h a t the optical phonon f requency to is independent of q, i.e. there is no dispersion in the opt ical b ranch of the phonon spectrum. The second t e r m in eq. (1) gives the in te rac t ion t t ami l ton i an of the electron at the posit ion r and the opt ical -phonon field. We have excluded the t rasverse optical phonons since the i r effect is small as compared with t ha t of the longitudinal optical phonons. The case of the elec- t ron in te rac t ing with the longi tudinal acoustic phonons is t r ea t ed in the la te r pa r t of this paper . The quan t i t y V~ stands for

(2) vo- iso~ [ ~ ~ [ 4 ~

where g is the dimensionless coupling cons tant

(3)

first in t roduced b y F~5~.[LIOm I n this expression so is the h igh-f requency (optical) dielectric constant of the host lat t ice and e its s tat ic dielectric con- stunt.

We now calculate the energy of the sys tem described b y eq. (1) as a func- t ion of a wave vector k of the electron. We consider two examples . F i r s t we take the ease in which ~ << 1 so t h a t second-order pe r tu rba t ion theory is sufficiently accurate. Treat ing the second t e r m in eq. (1) as a pe r tu rba t ion H',

POLARON ENERGY SPECTRUM IN A IKAGNETIC FIELD 247

we have, to second order at t empera ture / ' = 0 °K,

(4) S,[<k--q; ~qtH'[k', Oq>l ~ Eo(k) = ~(k) + ~ ~(k) -- e(k -- q) -- t~co '

where e ( k ) = ~%~/2m and [k; nq> designates a state in which the electron is

in a state with wave vector k and there are n phonons of wave vector q. If q~ = (2m~o/~)l << qv the Debye wave vector and fur ther if (~4) k < qr then

(5) Eo(k) = e(k) - - ~/~o(qv//c) arc sin ( k [~) .

When we consider the electron energies such tha t k << q~ an expansion of eq. (5) in powers of k/qr yields

(6) Eo(k) = -- zc]&o -~ 2m ~ 160 m 2 ~ . . . .

We now calculate the energy spectrum of a polaron in the presence of a magnetic field using the theory given by O~SAG~,R (~) and Ln~smTz and

KOS~VrCH (~5) independently. We shall briefly review here the formulat ion of

this theory as presented by the lat ter two authors and then use it to s tudy the energy spectrum of the polaron.

Let us suppose tha t we are interested in the s tudy of the motion of a charged quasi-particle under the general dispersion law

(7) Eo = Eo(p~, p~, pz) ,

in the presence of a magnetic field. If the magnet ic field B is directed along the z-axis~ the Hamil tonian of such a particle in the presence of a magnet ic

field is obtained, formally, by replacing in eq. (7) the momen tum component p , by P~, such tha t

(S) [pv, _p~] --. i]geB ; 0

[P~, g ] = [ix, Po] = 0.

The relation between P~ and P, corresponds to the adjustable relation between

(~4) This implies s(k)< ~w, i.e we assume that the energy of the electron above the bottom of the conduction band is less than the optieal-phonon energy so that no real phonon can be emitted.

(15) I. M. LIFS~ITz and A. M. KosEvic~" Zurn. Eksp. Teor. Fiz., 29, 730 (1955). (English translation: Soy. Phys. JETP, 2, 636 (1956).)

248 ~. I~. BAJAJ

the general ized co-ordinate and generalized m o m e n t u m

(9) [P~, Q~] : - - i~ .

The role of the generalized co-ordinate opera tor is p layed here b y the opera tor (--v/eB) P~. Therefore the semi-classicM quant iza t ion condition

(10) 1

m a y be wr i t t en as

(11) e 2~eB P ~ d P ~ : ( n + y ) h ( 0 < 7 < 1 ) .

(For the case of ~ quadrat ic dispersion law 7 = ½; in general 7 differs f rom ½.) The in tegra l ~ P . d P . defines the area bounded b y p lane closed curve

(12) [ Eo(~, ~ , P~) = const,

I P~ = po = eons t .

This allows us to wri te (11) in a more symmet r i ca l fo rm

C "

Here 2(Eo, P~) is the are a in te rcep ted on the surface of cons tant energy

(14) ~ o ( ~ , P~, i,~) - E0

b y a plane perpendicular to the direct ion of the magnet ic field. E q u a t i o n (13) gives us the implici t dependence of the energy of the quasi-part icle in the magne t ic field upon the q u a n t u m n u m b e r n

Eo = E~(p~ ; B ) .

The foregoing procedure of calculating the energy spec t rum is valid provided the surface (14) does not intersect itself~ and each of the curves (12) is located within one of the cells of the reciprocal latt ice. I n addit ion the radius of cur- va tu r e of the electron t ra jec to ry mus t be assumed to be much larger t han the la t t ice constant of the mater ia l .

POLA/~ON :ENERGY S P E C T R U M I N .4. M A G N E T I C FIELD 249

We now calculate the energy spec t rum of our sys tem described by eq. (6) in the presence of a magnet ic field using eq. (13). Af ter calculat ing the area

of cross-section S(Eo, k~) in a s t ra igh t forward fashion we find t h a t

(15) ~2 2 hk~

E~(ko) = - - ~ + ~ ( 1 - - ~/6)(n + ~) + ~ (1 - - ~/6) - -

We are now in a posit ion to eva lua te the cyclotron effective mass and the longi tudinal effective mass of the polaron in a magnet ic field. The cyclotron effective mass m~(n, B, ]4) is defined b y the condition

?ieB (16 ) m*,c- - E~+~(ko) - - ~ ( k ~ ) .

Simple subs t i tu t ion o~ cq. (16) into eq. (15) gives

(17) m 3~o~ , = 1 - - ~ / 6 - - n + m± 200) \ ~' + ~ ÷ 2too)J"

The longitudinal effective mass is defined as

( ~ s ) * - ~ ~k~ m L

and is given in this simple case b y

(19) m ~ = 1 - - ~/6 -- ~ n ÷ Y + 2m~oj .

I n the case of in te rmedia te coupling we use the result (~o)

h~W 2 3 h3ztW ~ (20) • o ( w ) = - ~o~ + ~ (1 + ~/6) -1 - (1 + ~/6)-~.

160 m2~o

for the case W<< qv. The results for m~ and m* are obta ined in an analogous fashion and need not be explici t ly displayed here. I n calculat ing the expres- sion for the energy levels of a quasi-part icle in the presence of a magnet ic field use has been made of the Bohr-Sommerfe ld quant iza t ion rule, which has been derived b y W K B (semi-classical) approximat ion . This is an a sympto t i c ap- p rox imat imi which is valid for large q u a n t u m numbers . Following A~G¥~ES we have shown (~) t h a t in the case of a polaron the energy spec t rum in the

(lS) K. K. BAJAJ: to be published.

2 5 0 K. ] i . BAJA,I

presence of a magnet ic field is indeed given by eq. (15) even for small quan tum numbers like n z 0, 1, 2, etc.

Another way of calculating the energy spectrum of a polaron in the pres- ence of a magnetic field is by the use of the effective-mass ~ploroximation which

has been derived, in our case, by using Wannier functions. The method is briefly outlined in the Appendix A.

I n order to test the val idi ty of our procedure we have calculated m[ from

eq. (17) and directly using second'-order 1perturbation theory on the Landau states (12) of an electron in a d.c. magnetic field. The equations used are given in the Appendix B but the results are summarized in Table I. The numerical

s tudy was carried out for the ease of GaAs using (~7) (m/mo)= 0.070, ~ = 0.06 and /to~ : 5.903.10 -1~ erg. We see tha t the agreement between the two cal- culations is ra ther good. The effect in GaAs is small and was used here only

as an illustration. However, in CdTe the change in the 1oolaron effective mass with magnet ic field will be much more pronounced, l~or CdTe we have (is)

m = 0 . 0 9 0 m o and ~ = 0 . 4 . Using ~ c o = 0 . 0 2 1 3 e V we find at B-----I00kG,

m z ----0.10rap which differs from the bare mass by more than 10 ~ and from

the zero-field 1oolaron mass b y about 4 %.

TABL]~ I. -- Values o] (m~/mo) computed ]rom perturbation theory on the Landau levels and using eq. (17).

B (gauss)

6 • 1 0 4

8 • 1 0 4

1 • 105 1.2.105

n ~ (erg)

1.582" 10 -14 2.110.10 -14 2.638.10 -14 3.165.10 - 14

Calculated using perturbation theory

0.07083 0.07097 0.071 12 0.07130

m~/mo eq. (17)

0.07086 0.07092 0.07098 0.07104

We have also compared this result with the change in the effective mass

with magnetic field due to the nonloarabolieity of the conduction band. For

this purpose we have used Kane ' s theory (19). Using the values A = 0.8 e¥

and E a-- 1.6 eV for the spin-orbit splitting and the energy gap in CdTe, we

find tha t the change in effective mass is about 1.5 % at 100 kG. In the case

of GaAs the change in the effective mass due to the nonloarabolicity of the conduction band is of the order of 2 ~o at 100 kG. This is to be compared with

a corresponding variat ion in the 1oolaron mass of about 0.4 %.

(17) O. MAD]~LUI~G: Physics of I [ I -V Compounds (New York, 1964), p. 92, 101. (ls) K. K. KANAZAWA and 1~. C. BROWN: Phys. t~ev., 135, h 1757 (1964). (19) E. 0. KAN]~: Journ. Phys. Chem. Solids, 1, 249 (1957).

POLAI~ON :ENERGY SIP:EeTI~IXNI IN A I~AGN:ETIC ?FIELD 251

We shall now s tudy the energy spec t rum of an electron in terac t ing with the longitudinal acoustic phonons in the crys ta l lat t ice in the presence of a uniform magnet ic field. Here again we assume tha t the electron moves in the conduct ion band of a semiconductor whose m i n i m u m lies a t k ~-- [0, 0, 0]. I t is fu r the r supposed t h a t the mate r i a l is isotropic and the energy surfaces are sphereical. The t t ami l t on i an of our sys tem is then given b y (2o)

(21) H o = - ~ m ÷ i E ~ [a~exp[iq.r]--a~exp[--iq.r]]÷~h~o~(a~a~4-½).q

Here E~ is the deformat ion po ten t ia l cons tant of the matcrial~ ~ is the mass density, s is the speed of sound and V is the volume of the mater ia l . As be- fore aq and a~ are the dest ruct ion a n d creat ion operators of a longitudinal acoustic phonon of wave vec tor q. As the in terac t ion be tween the electron and the acoustic phonon field is general ly weak, we use~ as before, the second- order pe r tu rba t ion theory to calculate the energy spec t rum of our sys tem and have

q~o

E~h ~" q~ dq h 2 k ~ 1. - - | s i n 0 dO (22) Eo(k) 2m ~ es j (~/2m)q -- (t~/m)k cos0 ~-/~s

l) 0

In teg ra t ing over 0 and q and assuming t h a t the group veloci ty of the electron is less t h a n the veloci ty of sound, i.e.

~k

~n

we get the following expression for Eo(k)

4z2es~ + k~ -~ + ~ • - - 3 \2ms]]J 15 ~esh\2ms] kd"

Here we have re ta ined the leading t e rms in the coefficients of k °, k ~ and k ~. To calculate the energy spec t rum of this sys tem in the presence of a magne t ic field we use Onsager~s theory as outlined earl ier and get

I

(20) C. ]~ITTEL: Quantum Theory o] Solids (New York, 1963), 1o. 23, 132,

2 5 2 x . K . BAJXZ

Defining as before

and pu t t i ng

?ieB m~.e -- E,~+~(k~) -- E~(k~)

/V~m - - ~ % , y = ½, 8~2@sh

we get for the cyclotron mass

(25) m~ = m 1 q- - ~ - C0a in \2ms]

Similar ly the longi tudinal effective mass m* is given by

m ~ - - ' m l + - ~ m w o 1 - - - ~ l n \ 2 m s ] ] 15 s ~ V \ 2 m ] ] "

I t is always energetical ly possible for an electron in a given Landau level to make a t ransi t ion to a lower level b y emi t t ing an acoustic phonon and there- fore the only Landau level which is comple te ly well defined is n = 0 even at the absolute zero t empera tu re . Therefore the approach we have outl ined here is fa i r ly reliable only when the magnet ic field is so weak t h a t tirol~ms 2 is less t han uni ty . Subst i tu t ing the values of the var ious physical quant i t ies for a typ ica l semiconductor , say GaAs, in eq. (25)~ we find t h a t the difference be tween m± and m for weak magne t ic fields is m u c h smaller t h a n the cor- responding difference in the case of e lect ron-opt ical phonon coupling.

The au thor wishes to t h a n k Prof. S. t~ODRm~rEZ for suggesting the p rob lem and for his continued guidance, encouragement and suppor t during the course of this work. Useful discussions with Profs. P. GOLD~A~:~:E~: J . W. C~LV~HOIrSE and G. ASCAR~,,LLI, and Dr. 1~. LuzzI are grateful ly acknowledged. The au thor also wishes to t h a n k Dr. l~. E. STO~En for help with the numer ica l calculations.

A P r ~ D I X A

We shall now briefly outl ine ano ther m e thod of calculat ing the energy levels of a polaron in a magne t ic field. This m e thod is based on the effect ive-mass approx imat ion . The to ta l m o m e n t u m W of our sys t em in the absence of a

POLAI~ON ENEI~GY SPECTI~UM IN ~ MAGNETIC FIELD 253

magnet ic field is given as

(A.~) IV= ~k -4- y a~a~q. q

I t is easy to ver i fy t h a t W commutes wi th Ho. Thus W is a good q u a n t u m number and is a cons tan t of the mot ion . We know t h a t in the absence of an electron-phonon in te rac t ion the c rys ta l m o m e n t u m ~k commutes wi th the electron I t a m i l t o n i a n and hence this Hami l t on i an is i nva r i an t under t ransla- tions of la t t ice vectors R which are canonical ly conjugate to the c rys ta l mo- menta . As Ho commutes wi th IV, there exist , a t least formally , d isp lacement vectors ~, canonical ly conjugate to W such t h a t Ho remains i nva r i an t under t ransla t ions of these d i sp lacement vectors , i.e.

(A.2) Ho(r) : Ho(r ~- f~) .

As the e lect ron-phonon in te rac t ion H a m i l t o n i a n is a slowly va ry ing funct ion o f r , eq. (1) can be wr i t t en as (31)

~2 (A.3) Ho = + V' ( r ) ,

2too where

(A.4) V'(r)= V(r) ~- ~[V~a~exp[ iq . r ] ~- V q ~* a~t exp [-- iq . r]] -[- ~hco(a~a~ ~- ½) q q

and V(r) is the periodic po ten t i a l t h a t the electron experiences in the absence of e lee t ron-phonon in terac t ion .

F r o m (A.2) i t follows t h a t

(A.5) V'(r) = V r ( r + o ) .

Thus the H a m i l t o n i a n given b y (A.3) is ident ica l to the one for the Bloch electron wi th W and ~ replacing ]~k and R respect ively. Therefore the eigen- functions ~vw(r) and the eigenva]ues Eo(W) are given b y the equat ion

(A.6) Ho~rr(r) = Eo(W)~vw(r) .

We assume t h a t we can solve this p roblem. L e t us cons t ruc t the Wann ie r localized a tomic funct ions a(r--Ok) where p~ is the vec tor to the k-th la t t ice poin t in the new la t t ice spanned by the set of vectors p which are canonically conjugate to the m o m e n t a IV. The new vectors ~ go over to the usual la t t ice vectors R when ~ goes to zero. The funct ions a ( r - ~k) are defined as

(A.7)

(21) T. P. ~IcLE~N: Proceedings S.I.~., Course X X I I (New York, 1964), p. 479.

254 K.K. BAJ/tJ

where the summat ion is over all the N levels of the band. In the presence of a magnet ic field the Hami l ton ian takes the form

2~-; P + A + V'(,-),

where A is the vec tor potent ia l . We want to solve the equat ion

(A.9) H~ = ih 8~ ~t"

Expand ing ~0 in te rms of the Wannier funct ions a ( r - Ok) and following the procedure given by LUTTINGER (~2) w e get

(A.10) Eo (-- iV + ~ e A ) ~ : Eo~p

for the t ime- independent pa r t of the Sehr6dinger wave equation. In the case of ~ polaron the eigenfunetions ~v are the same as the Lundau funct ions and the eigenvalues we get are ident ical to those obta ined from eq. (15) when ~-~½.

A P P E N D I X ]3

We can also obtain the values of m* numerical ly by calculating, wi th in the f ramework of the weak-coupling approximat ion, the energy of the polaron in a magnet ic field• We use second-order pe r tu rba t ion theory s tar t ing with a Hami l ton ian identical to t ha t of eq. (1) except t ha t we replace p~/2m by (2m)- l (p+(e /c )A) 2. We obtain

qJ9

k~ , ~ dq sin0d0- (B.]) E . ( k D = ~ o ( ~ + i ) + ~ m + ~ ( )~ 0 Q

I],~'.(q sinO)l ~ • ( n - - n')hco~ + (h~k~q eosO/m) -- (ti~q 2 cos20/2m) -- ~ "

The funct ions ] , , , are those used previously by QuINz~ and I~O~)~tGVEZ (2~). The quan t i ty U is defined as

(B.2)

(22) ft. M. LUTTINGV,~: .Phys. ~ev., 84, 814 (1951). (~3) j . j . QVlNZ~ and S. •ODRIGUEZ: .Phys. Rev., 128, 2487 (1962).

POLARON E N E R G Y SP]ECTI~UM IN A I~AGN:ETIC I~IELD 255

R I A S S U N T O (*)

Si s t u d i a lo spet~ro di energ ia di u n p o l a r o n e in u n c a m p o m a g n e t i c o pe r i casi di accoppiamen~o debole e i n t e rmed io . Si eonf ron~ano i r isul~at i o t t e n u t i con quel l i di un calcolo numer i co de] l ' au~oenergia di u n e l e ~ r o n e in u n e a m p o magne~ico i n t e r a g e n t e con i f o n o n i o t t i c i di u n cris~allo polare . I1 m e t o d o si b a s a su l l ' app ross imaz ione d a t a pe r p r i m o d a Onsager . Si eonf ron~ano i r i s u l t a t i con quel l i di p r e e e d e n t i calcoli va r ia - zionali. Si s t u d i a a n c h e ]a va r i az ione n u m e r i e a del la m a s s a del po ]a rone col e a m p o magne- ~ieo e la si eonh-on~a con quel la d o v u t a a l la n o n parabo l ic i t i t del la b a n d a di condu- zione in GaAs e CdTe. I n CdTe il p r i m o effet~o p roduce u n a va r i az ione che ~ circa i r e vol te magg io re di quel la p r o d o t t a dal s econdo , m e n , r e in GaAs ~ l ' u l t i m o effe~to che

pifl i m p o r t a n t e .

(*) Traduz, ione a cura del ia Re~az ione .

~HepFeTtiqeei~t4lfi etrleKTp lIO.rIf/poHoB B MalFHHTHOM llO.rIe.

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(*) Ilepeaec)eno pebaKttue~.