Possible beat effect of specific heat due to impurities in semi-metal in external magnetic field

Solid State Communications, Vol. 24, pp. 647—650, 1977.

Pergamon Press.

Printed in Great Britain

POSSIBLE BEAT EFFECT OF SPECIFIC HEAT DUE TO IMPURITIES IN SEMI-METAL IN EXTERNAL MAGNETIC FIELD* S. Kiama Ferromagnetics Laboratory, Institute of Molecular Physics of the Polish Academy of Sciences, Poznañ, Poland1 (Received 2May 1977 byJ.L. Olsen) The simple model of a single-band semi-metal or degenerate semiconductor with impurities in an external quantizing magnetic field is discussed. The susceptibility and specific heat are studied in their dependence on the field-induced quasi-local levels, due to the impurities. These levels are shown to cause additional oscillations in the thermodynamical quantities as functions of the magnetic field. Their contribution to the susceptibility and specific heat is comparable to the values for the undoped system if the impurity concentration c fulfils the condition c~= (Ce/4\/7T)(hW/EF)3’2, ‘ With Ce the electron concentration. 1. INTRODUCTION

H

VARIOUS theoretical papers [1—51have hitherto dealt with the effect of impurities on the structure of the electron energy spectrum of crystals in an external mag-

~.,

=

~1C0+V(r),

(1)

where: ~

(p

—

~-

A) + aPBH;

A

=

(2)

PBH(2n + 1) +

E~(pz,H)

+

UPBH,

a

±1,

2m n

=

0, 1,2

(3)

For a spherical potential well, related with a diamagnetic impurity not creating local levels at H = 0, the impurity potential V(r) can be approximated by a separable potential, permitting exact solution [5] of the problem of the effect of the impurity on the electron spectrum. It is found that, for a potential not creating bound states at H 0, the impurity acted on by the external field leads to the emergence of local levels and quasi-local levels, one of either in each Landau band and for each value of spin. The position E(n, a) of these levels (at n = 0, local levels, and at n = 1, 2, quasilocal levels) is given by the following expression [5]: E(n,a) = (n + y)hw e, (4) . . . ,

—

where: 2

2

I a 2 2 2 (5) 2~~l) h range of action of the potential, U Above, a is the 1”2 0 the depth of the potential well, and I = (h/mw) the magnetic radius. It is seen from equation (5) that e

____________

(0,0, H)

is the Hamiltonian of the ideal crystal, and V(r)is the potential energy of an electron in the field of impurities. The eigenvalues of ~C 0are of the form:

netic field. The deformation of the spectrum due to impurities is apparent in additional oscillations in the thermodynamical quantities of the system. With regard to their typically quantum nature, these oscillations are accessible to observation at low temperatures (kT~hw, with = IeJHImc cyclotron frequency), low impurity concentrations, and appropriately high magnetic fields ensuring that the quasi-local [5] levels do not lie in the region of diffluence of Landau levels due to thermal vibration and impurity, In the case of a single-band metal with low electron concentration (per atom) or a degenerate semiconductor, when the electron dispersion law is quadratic, the dynamics of the doped system in an external magnetic field can be studied in the effective mass (m) approximation. For an isotropic quadratic dispersion law, the Fermi surface at H = 0 is a sphere. The Hamiltonian of the system considered is of the forms: 1f

=

7r(a\(4ma2Uo\

‘~,

(0, Hx, 0),

supported Project MR-I.9. t* Research Address: IFM PAN, by 60-179 Poznari, ul. Smoluchowskiego 17/19.

e H2 and as H decreases the local arid quasi-local levels tend to the bottoms of the corresponding Landau bands and, at low H, can actually lie within the region of diffluence of the Landau level. Hence, e > kT is an

~ We shall be assumingg = 2 for simplicity. The extension to cases with g> 2 and an anisotropic dispersion law is trivial. 647

648

IMPURITIES IN SEMI-METAL IN EXTERNAL MAGNETIC FiELD

additional pre-condition for the observation of the levels in question.

~1(~,H,0)

=

~ir~

Vol. 24, No. 9

~(~_hw+c)2+r2

0

2. THERMODYNAMICAL POTENTIAL 2

E((~fh~.,j)—l] (n+1)hw

We of write the thermodyriamical potential system electrons in the well known form of the ~

T)

=

+2

—kT Jd~p(~)ln[l+ exP(~k;~)].(6)

j

~ n=1

~

nh~

+ l)ho.i +

~)2

+ F

The essential role with regard to the thermo- (13)

where integration over d ~ includes summation over spin parameter. For the present discussion, it is convenient to rewrite equation (6) as follows [61:

dynamical quantities belongs to the number of the levels lying in the Landau band adjacent to the Fermi surface and being induced by the magnetic field. From the physical viewpoint, the only case of interest is that of

$ d~fZCLI 4 ch2~/2 ~JCT,H,0)

quasi-local of small width the Lorentzians in (13) canlevels be approximated by F thewhen s-function putting

0

~

T)

=

+

(7)

F 2 + F2 (x —x0)

We now have: ~2(p,H, 0)

=

—

JdE~p(E~)(i.t —

~)O ~

I forx>0, (Oforx<0.

(8)

By equation (7), &~2(ji)atTr 0 is determined by ~2Oi) at T = 0. We thus proceed to calculate ~(j.z,H, 0). In our case, the density of states p(s) takes the form:

p(~)=~I ~ v(~)+~ [~ Nm with: ~“na(E)

=

0

E[-~-+ 1hw

—

E(Nm, 0)12 +

F2]’

71,J

312V hm 2V’2ir2h3

(9)

‘~/[~—

w (n + y)hw]

(10)

the density of states of the nth Landau band of the perfect crystal. The second term of(9) gives the contribution to p(s) from n~ impurities. Writing p(s) po(E~)+ pj(~),we have ~2= ~1o + ~i and

=

n~ji+n1(~i—2pBH)

+~flfpBHR2(~,

(14)

with ji = p(H, 0)’~p + e. 3. MAGNETIC SUSCEPTIBILITY AND SPECIFIC HEAT

Nm1

a

0).

&7 1(p,H,0)

=

ir5(x—x

In this approximation, equation (13) becomes:

—

0

0(x)

=

On having recourse to the thermodynamical relation: 1 (a2~z(pHo)~ (IS) 2 I~.T ~(H,0) ~ ~H we obtain the magnetic susceptibility of the system. In the present case, it is of the form: ~(H, 0) ~o(H, 0) + xi(H, 0), (16) and xo(H,0)

~JN,.L~\Rs, 2 ~V)io)

=

2(x)_2xR3i2(x)+x2Ri,2(x) 3 [R312(x)]’’

(17)

,5/2

(~)

~

3~(~—2n~.

0(p,H, 0) = —~Np0\Ilo / R512(x). (11) Above, Po (3N/8irV)213(h2/2m) is the chemical potential of the electron gas atH= T= 0;moreover,x /1(H, O)/PBH

and

R~(x) = px~’+ 2p ~ (x

—

2n)~0(x — 2n).

(12)

n=1

For the properties of the function R~(x),we refer to Rumer and Ryvkin [6]. Similarly, we obtain for ~2i(p,H, 0)

x1(H,0)

=

2c~~ n=1 ~ (2n) p(H,0)

\PB11

)

(18)

Equation (18) is valid for ‘~p,~‘ F. The expression (17), because of the function R 1, 2(x) in its numerator, exhibits (infinite) discontinuities in the points x = 2n, i.e. at p = nhw; whereas the expression (18) has peaks in the points where ~ = 2n, i.e. where p = nhw — e. Hence, with varying magnetic field strength, as the quasi-local levels cross the Fermi level, the susceptibility ~1(H, 0) becomes extremal whereas the total suceptibility of the system for a given nth Landau band, crossing the Fermi

Vol. 24, No. 9

IMPURITIES IN SEMI-METAL IN EXTERNAL MAGNETIC FIELD

level, has two close-lying peaks (which are infinite at T= 0): the one is related with an anomaly in density of states of the perfect crystal, and the other with the quasi-local levels, On insertion of equation (14) into (7), we obtain: 1 2nEPBH—nEPBH ~ —

c21(p,H, T)

=

n~p

n1

x 1(_~~_2n~+ 2kT ln 2

L

—

—

/

\I.LBH

ch

—

PBH

2ni1~H~ 2kT

(19)

j

xi(H T)

— —

kT

n~1

2 1 + ch[(p (2n) + —nhw)/kTJ’

(20)

The effect of finite temperature resides in making the heights of the peaks finite and their widths of the order of kT/2hw. We shall now calculate the contribution of the quasi-local levels to the electron specific heat of the system. Applying the relation 2c2

C 1~H,T) = —T

a

(21)

1(p,H, 2 T) aT

and with regard to (19), we obtain:

<~

I we have

1(H, T) tends to zero atH-~-0. By (23), the quasi-local levels contribute linearly in T to the specific heat and their contribution oscillates as the magnetic field varies. If therather calculations are the performed withone theobtains Lorentz function than with 5-function an expression for C 1 in which, under the sum in the second term of (23), there occurs a factor exp (2irr’F/hw) for (kT, F). Obviously, in order that the oscillations of the specific heat can be observed, the condition F kThas moreover to be fulfilled. ~>

=

2kn1 ~

(22)

I (p+e—nhw)/2kT ~ch[(p + —nhw)/2kTJ

In order to separate the oscillating part of C1, we transform the sum in (22) by having recourse to Poisson’s formula, thus obtaining at ji/kT>> I: tkT~ C 1(H,T) = \hw J ( ~+\ + 2 ~ f(vX) cost 27w___J]~ (23) \ where:

hw)

2~ B~+, 22t~~’1_1 (2n)!

6 ~ (— 1)~x

6 cthx shx

—

2x—1 3x 2cth ; shx

4. CONCLUSIONS From equation (23), we note that the amplitude of the oscillating part of C 1 is twice larger (at x <~1) than the non-oscillating part [given by the first term of (23)]. On considering the oscillating parts C0 and C1 we note that they are comparable if the ratio of impurity concentration and electron concentration fulfills the following equation: 312 (25) (hw/p) 4~Jir which, for semi.metals of the Bi type and degenerate semiconductors, requires that Cl/Ce I.

Cf/Ce

=

‘~

~ C1(H, T)

=

Above, B,~are Bernoulli numbers. For x f(x)_~ I, and C

<~

whence:

f(x)

649

x

— —

2ir2kT .

(24)

The oscillation period of C 0 is the same as the= of effects of the de Haas—van Aiphen type (~(1/H) dh/mcF);whereas the “period” of oscillations* C1 lies close to i~(l/H);consequently, the specific heat of the system measured vs the magnetic field strength has to exhibit a beat effect, similarly to the magnetic susceptibility [2, 5]. The author wishes to thank Prof. J. Morkowski and Dr. A.R. Ferchmin most sincerely for their interest and valuable discussions.

ACknowledgements

—

The argument of cosine in (23) does depend on the * magnetic field through e = e(H) and p = p(H). Therefore, the oscillations of C 1 are not strictly periodic. However, and pperiodicity does depend slightly ontoH, that deviationspfrom are only not expected beso large. “~

REFERENCES 1. 2. 3.

BYCHKOVYu.A.,Zh. Eksp. i Teor. Fiz. 39,689,1401 (1960);BRAILSFORD A.D.,Phys. Rev. 149,456 (1966). YERMOLAYEV A.M. & KAGANOV M.I.,Zh. Eksp. I Teor. Fiz. Pis’ma 6,984(1967); YERMOLAYEV A.M., Zh. E7csp. i Teor. Fiz. 54, 1259 (1968). KUKUSHKIN L.S.,Zh. Eksp. I Teor. Fiz. 54, 1213 (1968).

650

IMPURITIES IN SEMI-METAL IN EXTERNAL MAGNETIC FIELD

Vol. 24, No. 9

4.

KOCHKIN A.P., Teor. iMat. Fiz. 13,251 (1972).

5.

KAGANOV M.I. & KLAMA S. (to be published).

6.

RUMER Yu.B. & RYVKIN M.S., Thermodynamics, Statistical Physics and Kinetics (in Russian) p. 247. Nauka, Moscow (1972).