Solid State Communications, Vol.57,No.l, pp.21-25, ]986. 0038-1098/86 $3.00 + .00 Printed in Great Britain. Pergamon Press Ltd.

THE MAGN~.TOCOULOMB LEVELS IN THE S~IICONDUCTORB WITH K~/~E'B DISPERSION LAW

A.A. Zhukov Institute of Radioengineering, Electronics and Automation, 117~5@, Moscow, USBR

(Received 25 September 1985 by G.S. Zhdanov)

It is found that the position of shallow magnetocoulomb levels for semiconductors with Kane's dispersion law can be obtained analy- tically on the basis of known solution for unrelativistic hydrogen atom by using dependent on magnetic field "relativistic" value of effective mass at the bottom of corresponding Landau subband. The "quantum" correction essential for ground state due to noncommutati- vity of impulse and coulomb potential operators is found. It is shown that for two-level Kane model in strong magnetic field the parameter ~*and the anysotropy of bound state wave function have an upper limit and the energy of all levels varies as~~- . The results obtained are applied to relativistic hydrogen atom.

The electron energy levels in the presence of magnetic field and cou- lomb potential of nucleus (the magne- tocoulomb levels) for the unrelativis- tic atom of hydrogen and for the semi- conductors with parabolic dispersion law h~ve been studied by many authors (e.g.'). For most semiconductors with narrow bands the dispersion law is nonparabolic and ca~ be well described using Kane's model. <

For the f~st time the magnetocoulomb levels in ~-InSb bas@don Kane's model were found by Lateen. ~ Afterwards si- nilar calculations were carried out in ~-6. The main shortcoming of this oalculations is that they are valid only

for certain parameters of InSb disper- sion law and can't be applied for other compounds and for other dispersion law parameters. In this case it is necessa- ry to make out a new sophisticated variation calculation. In this work we have developed the analytic mgthod which was proposed earlier inF. This method reduces the problem of magneto- coulomb levels in semiconductors with Kane's dispersion law to parabolic one.

The magnetocoulomb levels equation for Kane's model can be obtained on the basis of Hamiltonian for "pure" magne- tic problem 8 by including the coulomb impurity potential. For ~lIZ this equation is

~_ -E-~-~2 V 0

~ + 0 - ~ V

-~P+ 0 0

0 0 ~K~J z

O 0 0

~KI~ z O 0

~ z 0 0

o

0 0

0 ~KP z

0 K~ +

0 -~K~_

0 ~KP_

°

O 0 0

0 0 0

0 0 O

- ~ V O 0

0 - ~ V 0

o o -~A+v

f l

f~

f~

f,.. t

f .

f~

fE

f~

-0

(1 )

21

22

where ~+ : (Px± iP~) /1 / -2 , g ~ g 2

P : ~ h g + -e-A, V : - g t ,

K is interband matrix element in Kane'r~ model, ~ is the energy gap, ~ - spin- orbit splitting of valence band; energy is measured from the middle of forbidden band.

By solving the equation (1) with res- pect to ~ and ~a , the following sys- tem of equations was obtained.

THE SEMICONDUCTORS WITH KANE'S DISPERSION LAW

£~MV/E~ g terms it is possible to deduce

the following equation

i {~,.~)~P+F'_ + P_P+ +Pf- ~2@] , -

* v ÷v~ + re} i , ~ : ~ ~ a~v 5],~ (5)

Vol. 57, No. 1

where

"' { p , P-} f,_ -,'- n~r~ , +

1 g a

- (~. [,~+ ,,-v)~J.((P+v.l-~_ - {~_v./- P.,.)t'i + ~ ,~'~" t ../

• EF~ 9 - v ) ~ - #~. 9-~ ,~-v) ~]({~-gp~-;P/}P--):g ( 2 )

2 Z~ +~ (E+_~_V)(E,_}.~+a_v) {p_p+}5~. Ke[_} z

-(£-, ~, . a-v)eJ({P-g}P,-{P~ vJP-)f~ ~ ~ E 2"

where { P V } : P V - V P i s a

commutator of the operators P and V . For arbitrary binding energy of im-

purity levels it is very difficult to obtain the solution of this system. Essential simplification may be reached assuming that the levels are shallow, i.e. the difference between the enmgy of the magnetocoulomb level ENHv~ and the energy of the bottom of the corres- ponding Landau subband L "5 c~ - . r--~ r'_~ N O#MV- -r_~M v - 1-~/ is stroll cgmpared to the energy ~#', (where ~ = ~q/2 is the I, spin, N,~ Iv, V is the quantum numbers ). Taking into account only first order of

Ao~S (g +~ -,- ~.~ -,- .s ) - f F '} p V..I. =

(4) ~Sn 2 ( F ~ * ~ + # ; A ~ .... ~ .o .....

V2= "3ko 'LN 2~ " : /~ '2° '~J

A#

Vol. 57, No. ] THE SEMICONDUCTORS WITH

= is magnetic length, ~-

for 91 , and ~ - - ~ for ~2 -

The terms VI and V2 arise due to the noncommutativity of operators P and V . For the quasiclassical ~p~ro- ach these terms can be neglected ~,~. Then the equation (~) reduces to the corresponding equation for magnetocou- lomb levels in the parabolic disper- sion law. The only difference is that the quantity Dq~ (H), which depends on H, is uaed in th~s relation, instead of a constant mass m*. Hence the prob- lem of magnetocoulomb levels for Kane's dispersion law is determined by the solution known for unrelativistic hydro- gen atom ~HMV /~ = ~ (~) with

and ~9 being replaced by renormalized values y = s

and R~= r n # ( H ) e ~ / z ~ = a ~ , res- pectively.

The terms V 1 and Vp were taken into account in two ways. ~In the first case the solution of equation (5) was obtain- ed by variational method with probe function'. In the second case the varia- tional procedure was made without terms Vl and V 9. The influence of these terms was considered on the basis of the first order perturbation theory. This second approach is preferable because it per- mits for arbitrary dispersion law para- meters to obtain binding energy analyti- cally on the basis of known dependences for reduced matrix elements A =

: <N vI( VJ a n d S -

The values of A and B for different ~% are given in the table. We have found that both methods give similar results for ~Mv within I% accuracy.

It is obvious that ground level posi- tion is the most sensitive to the influ- ence of "quantum correction" V 1 + V 2 . In this case in extremely high mag- netic fields the binding energy is decreased by 15-20%. For the excited

TA~E Matrix elements A=<NMvI {~V} ~ JNMv> ~/R~B 2 and B= <NMvl {P_V} P+- ~P+V} P_ 1NM~R~ B2

for different values of parameter ~.

~/M~ ~* 25 : 50 : 7~ 100: 1~O: 200 000 A 1.87 2.20 2.41 2.56 2.80 2.97

B 1.57 1.91 2.1~ 2.~0 2.~z~ 2.7~ 0~O A 0 0 O 0 0 0

-]~100 7.68 8.4-,5 8.71 8.81 8.8~ 8.76 001 A O O O 0 O 0

Bx'100 6.08 >.02 z~.58 4.>6 #.18 4%16

KANE'S DISPERSION LAW 23

levels the influence of V I + V 9 is smal- ler and it makes up about several per- cents. The scalculated dependences of the energies gNM v for main InSb donor magnetocoulomb levels are shown in fig.1.

It is worth to study in details the two level approximation of Kane' s model, which is valid for A>>$$ . Prom the dependence DQ~ {H)=2 E~ ~*/Eq it follows that in high magnetic fields limit (wMICL), such that ~w~=eHB/nl*g~, the parameter ~ tends to the value 6

I (5) -

g2 where ~ = --~. is the Born parameter B ~ n n n in Kane's model ~--~-~is effective g-factor ~7

Thus semicoi i ors with dispersion law the model parameter of magnetoculomb problem ~* is limited. Furthermore the value ~ does not de- pend on 6~ . This nontrivial result leads to some interesting consequences-"

(i) in HMFL the value of binding ener~ for all magnetocoulomb levels with E; << A is proportional to ~H- ; ~(ii) the anisotropy of bound states

wave function is also limited and in HMFL the characteristic scale of wave function exponential decrease in longitudinal di- rection is proportional to the magnetic

len~thtiii)~fo~ high ~ the values of ~ become small and impurity states loose their quasibound character, they merge with continuous states;

(iv) in H~ dependence of binding energy 66~~ changes the dependence of hopping conductivity on magnetic field.

E4

/

/ooo) / ~ ~ /

I~/~/~'~'~-"O 0 I) + i~iO H, kOe2~O 300

Fig.1. The binding energy for some InSb donor magnetocoulomb levels. Solid line represents full calculation, the dash-dotted line is the result of neg- lecting by V I + V 2 term and the dashed

line show the parabolic approach

(~(H)-- = ~Q~). For (0~0) and (OOI)

states solid and dash-dotted lines are me~ed. The parameters~ ~ = 16,45,

- O,O139 mo, ~ = 236 meV, ~ = - 805 meV Were used in this calculations.

24 THE SEMICONDUCTORS WITH

The described analysis is based on the assumptiom that the bound states are shallo~ ~or single charged centers this assumption is correct when parameter

= 2

is much smaller than unity. The two level Kane's model is des-

cribed by the expression similar to Dirac relativistic equation for hyd- rogen atom. Consequently the results obtained earlier are applicable to the last case. For example, by the same way we can find that for hydrogen atom in relativistic magnetic field the parame- ter ~* is also limited by the value

~c~ ~ ~ 1 (7)

2 w h e r e ~ = ~ i s the f i n e s t r u c t u r e cons tan t and 9 ~ 2 is ~ -factor of free electron. The ekperimental~energy values of main impurity magnetooptical transition in D - InSb ~u and our calculated dependences are shown in fig.2. The small divergence bet- ween measured and calculated data @~ems to be caused by the "chemical shift". T M

The most appropriate objects for expe- rimental verification of our results for two level Kane's model are solid solutions

KANE'S DISPERSION LAW Vol. 57, No. !

(Ref. 12) we may obtain that for all quan- tum numbers /V , M ~nd ~ parameter is smaller than 6-IO -<. Then for ground state the calculations give ~ -- 150 and ( ~u*/ O~ ) - 5,6. Hence in Hg~ ~Cd~Te compounds magnetic field cannot p~fuc$ anysotropy of the bound state wave func- tion greater than 5,6. The condition ~ ~7~°~ = £ for x = 0,18 is fulfilled for H - 8 kOe.~

In fig.5 experimental values of activa- tion energy E A for temperature dependences of resisitivi~y in the regionj~f impurity conduction for n-Hg^ 8Cdo ~Te ~ are shown as a function of ~i~2~: In ~c~cordance with the above results these data are fitted well by the straight line.

In n-IriSh the metal-nonmetal transition in high magnetic fie~ ~ studied usi~ both galvanomagnetic ~' and optical "° methods.

Our analysis has shown that the electron concentration in conduction band Ilo = ~/~17 ~/~ is related to cri- tical values

(8) all =#2 m~ £~

by the expression similar to Mott's one

= ~ =r , '/e where F)q 6 fT] (H) a nd g~ ~'000

of compounds AIIB VI near the gapless state are renormalized effective mass and for example~n-Hg~_~CdTe alloys (O,16 ' binding energy for ground state, respec- g x ~ 0,2). ~ere ~u~$ pQblished experi- tively (fig.4). mental data available for these alloys. For Hg1_xCdxTe (0,13 ~ X ~ 0,19) in

Using known parameters for Hg1_xCdxTe study 18 it was proposed unusual crite-

~ = 8,~-I0 -8 eV. cm (Ref.11) and ~ = 18 rich of such transition h~ ~ ~ = 0,37.

40f , . I ,

/

I •

| m i I

H,kOe ~g.2. The experimental data for two

low energy magnetooptical tr~sition between donor states in InSb vs magnetic field and the calculated dependences with the parameters of Ref. 9.

, , ]

>

d~

21o T f,4 2.8 Fig.5. The activation energy E A of

the temperature dependence of the resi~auce in the region of impurity conductivity for Hg~ ~Cd noTe as a function of H I/2. u,v v,~

Vol. 57, No. l THE SEMICONDUCTORS WITH KANE'S DISPERSION LAW

o - 1 e - 2

I I I

t01~ 10 ~5 1016

n~,cm-3

Fig.4. The experimental data for cri- tical magnetic field value He for metal- nonmetal transition in n-InSb vs concent- ration n o = jV~ -iV a (References: 1-15, 2-14, 3-16). The solid line is cal- culated using the expressions (8) and (9) with the parameters of Ref.9.

25

The peculiar feature of this relation is that it is valid for samples with diffe- rent x. Such result becomes clear on the basis of our analysis that gives in HMFL

~H=~).~(~) (~(~)~j)independent of ~ . Using known parameters of

Hgl_xCdxTe one can find from expression

(9) in HMFL the relation Da %' 2 = 0,32,

I would like to thank S.D.Beneslavsky and S.M.Chudinov for fruitful discussions.

~EFERENCES

1. Wallis R.F., Bowlden H.J.J.Phys.Chem.Solids, 7, 78 (1958) 2. Kane E. Semiconductors and Semimetals. Acad.Press, N.Y., v.1,1966,

p.75 3- Larsen D.M.J.Phrs.Chem.Sol., 29, 271 (1968) @. Ortenberg M. J.~hys.Chem.Bol., ~_~, 597 (1973) 5. Lin-Chung P.J., Henvis B.W. Phys.Rev., B12, 630 (1975) 6. Zawadzki W., Wlasak J. Theor.Asp. New De-~lop.Magn.-Opt., ed.

I.T. Devreese, N.Y., 1980, p.347 7. Zhukov A.A. ~is'ma Zh. Exper. Theor.Fiz., @0, 380 (198@)

JETP Lett., @0, 1193 (1985) 8. Bowers k., Yapet Y. ~hys.Rev., 115, lq65 (1959) 9. Raymond A.~ Robert J.L., Zawadzki W., Wlasak J. J.Phys., CIF,

2381 (1984) 10. Kaplan R. Phys.Rev., 181, 1154 (1~69) 11. Bovina L.A., Brandt N.-~., Dolbanov S.V., ~seev V.V., St~feev V.I.,

~onomarev Ya.G., Zh.Eksp.Theor.Fiz., 8@, 1453 (1983) Soy.Phys. JE'2~ 57, 845 ('1983)

12. Halbo-~. ~hys.Stat.Sol. (b), 59, }87 (1973) 13. Vas G., Herlach F. Proc.Int. Conf. Appl. High Magn.Fields Semicond.

Phys., Grenoble, 1982, p.378 14. Ishida S., Otsuka E. J.Phys.Soc. Jap., @_~, 124 (1977) q5. Robert J.L., Raymond A., Aulombard R.L.,Bousquet C. Phil.Mag.,

16. D,.~ ~,, = (,.no,-,~aa,y:VO~a~l~;,,,~. J .Phys. Soc. Jap . , 2~8 674, 1.970 17, Shklovskii B.I. Fiz. Techn.Polupr., _8,416 (1973)

Sov.Phys.-Semicond., 8, 268, 1974 18. ~" ikov A.B., BaranskTi P.I., Zhidkov A.V. Pis~ma Zh.Eksp.Teor.

~iz., 35, 464 (1982) JETP Lett., 35, 574 (1982)