REFERENCEIC/72/52

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

MAGNETIC BREAKDOWN FOE BLOCK ELECTRONS

U.K. Upa&hyaya

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION 1972 MIRAMARE-TRIESTE

..•tJW.'-KU\

IC/T2/52

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

MAGNETIC BREAKDOWN FOR BLOCH ELECTRONS *

U.H. Upadhyaya **

International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

The phenomenon of magnetic •breakdown for Bloch electrons is studied

using a different approach from that followed "by earlier authors. The present

analysis is essentially "based on the fact that there is a finite probability, in

the case of the near crossing of two quantum levels, that the adiabatic path is

not followed. A rigorous formula for this probability is derived which in the

present case yields the criterion for magnetic breakdown. As the present

approach does not use any perturbation theory, the result is valid for all

strengths of magnetic field.

MIRAMARE - TRIESTE

June 1972 •

* To be submitted for publication.

** Permanent address: Department of Physics, School of Basic Sciences and

Humanities, Udaipur University, Udaipur (Rajasthan), India.

I. The phenomenon of magnetic breakdown vas first studied "by Cohen and

Falicov who postulated that when orbits of nearly the same energy in

different "bands also approach very close in k_-space, an electron may not

suffer Bragg reflection while approaching a Brilloiiin zone "boundary "but it

may instead pass through the "boundary with a finite probability, jumping on

another orbit, even though this involves tunneling through regions where no2) 3)

classically permitted energy surface exists. Later on, Blount , PippardIt)

and Chambers investigated the problem in more detail using different

approaches. Blount analysed the important special case of two bands which

are nearly degenerate and they cross each other by addition of small pertur-

bation A . In such a case the effective Hamiltonian can be reduced to a form

identical to the Hamiltonian which produces Zener breakdown in insulators.

Blount obtained the expression for the transmission amplitude following

Kane's method, which is a good approximation for the asymptotic behaviour

at low field. Pippard has considered an almost-free-electron model in which

the lattice potential is weak enough to be neglected entirely except in the

immediate neighbourhood of the Bragg reflection points. His approach

essentially consists in considering the removal of the degeneracy of Landau

levels by the weak periodic potential and he obtained Blount's criterion of

magnetic "breakdown as the condition for significant line broadening. More

recently, Chambers has generalized-Pippard's analysis to the case of electrons

in real metals. Chamber's approach involves only the local geometry of the •

k_-space orbits near the breakdown region and it is therefore more widely

applicable than the nearly-free-electron problem. However, his derivation

for th*e expression of probability of tunneling, as he himself pointed out,

makes no • pretenee • of rigour and its justification lies in the fact that the

expression reduces to the correct form for the casetjffEal^ost-free-electron

model.

In the present paper we study the problem from a different approach.

The present analysis, in essence, likens the phenomenon of tunneling of an

"electron from one semiclassical orbit to another to the violation of the

Ehrenfest principle . The Ehrenfest principle states that if the

external conditions for a system are changed slowly enough, the change of the

system can be predicted by computing quasistatically the modification of its

initial quantum state. However, the Ehrenfest principle is not a law of

nature, as the system may follow an alternative path which may be termed the

-2-

inertial or rapid path. This alternative path is favoured "by a rapid change

of the external variables. A rigorous formula for the probability that the

adiabatic path is not followed for a very general geometry of the electron

orbits near the"breakdown region is derived. This formula in the present case

is the criterion for magnetic breakdown.

II. In the semiclassical treatment of Bloch electrons in a,magnetic field,

originally due to Onsager, one treats the electron as a particle defined ~by

£(k.) and, subject to a classical equation of motion, -ak_ = F_ .where F is

the Lorentz force eV x H , 'The solutions are in the form of orbits of all

energies, from which are chosen those that satisfy the Bonr-Sommerfeld

quantization rule © P-dq = (n + r)h . Thus in k_-space an electron is

driven by. the Lorentz force round an energy contour in the plane normal to

H_ and in real space it executes an orbit similar in shape but scaled in

dimensions by l/s (s = eH/fi) and turned through — IT .

7)We take the geometry as considered by Blount in vhich two

bands that cross are coupled by a small term A , so that in the neighbourhood

of a crossing point the band energies are given by the roots of the secular

equation;

*n" k • v - E— a

T̂ - E**b

- 0 (1)

where k_ and E are measured from the crossing point. The parameters jEL

and y, give the group velocities AE/fi associated with the bands in thisneighbourhood. It is assumed that _y and v are in the plane normal to

a o

jH since the parallel components have no effect on the results. The secular

equation can be written in the form

(fik - Eg) • v a v^ . (-nk - Eg) = A2 (2)

where g is a vector in the plane of y and v such that g • v = g • v. = 1— ~a "• u ~ a r- ~D

-3-

Since this iVo-function of (k_ - Eg/tt) it is seen that the contours associated

with some given E are just obtained from the one for vhich E is zero by

a shift in the direction of £ of the appropriate size. It is therefore

only necessary to consider the zero contour for which

k • v v. • k =— ~a —D

(3)

Then for any field direction not in the direction of v - v > the orbits are

hyperbolas with asymptotes normal to v and y and the bands have equal

values of v_ In the following we shall assume that the lattice directions

have been so chosen that the variation of k carries us from one branchx

to another. The Hamiltonian corresponding to (l) can be written as

•ft k aX

k bJr

k c + -fi k bx • y

(U)

where

a = vaxv bx

b = v = v,ay by (5)

In the above the components ot^ k_ vector are treated as operators in the8)

presence of the magnetic field, following the method of Luttinger and Kohn

The introduction of the magnetic field by this method involves the replacement

of the quantity k by k -i33/3k , where 6 = eH/ficQ , e is the

magnitude of the electronic charge, H Is the magnetic field (parallel to

the z-axis), -H is Planck's constant, cQ the velocity of light,and the

vector potential is given by A = (0,H ,0) .

Thus the time-independent Schrodinger equation in the presence of

magnetic field in k_ - space can be written as

•fik a +-35k bx y - ignb

3k

•nk c +-fik bx y 3k

0 , ( 6 )

where following the discussion after Eq.(2), E is taken to be zero. In the

component form, Eq.. (6) reads

-U-

V + V " (7)

k c + k "fa - igb ~x y 3k * (8)

Elimination of ik from this system yields

dkiSb.{(a+c)k + 2bk } r|- - kfac - k k b(a+c)

x y

~ - kV + i-fi

(9)

obeys a similar equation with c and a interchanged. It is convenient

to define a nev function <j> :

1>J\) = * (kx) exp (10)

Substituting (10) in (9) one obtains the following equation for

A f.(a-c)dk 2 "x (11)

This is the equation for a parabolic cylinder function, a special type of9)confluent hypergeometric function . Its asymptotics is well understood.

In terms of Whittaker functions, Eq.(ll) has two solutions and,correspondingly,

the solutions for Eq.(lO) can be written as

(k ) -v expax *

-1/2

W • A * -

3b(a-c)3b x (12)

-5-

{(a+c) 7T + 2bk2 x*1: yJ -1/2

- iA , - iA [ 2 ^ x

2B pb(a-e)

The asymptotic expressions for the, solutions (13) and (lk) are, with a

modification of the factor,

- -— {ak2 + 2bk k } - k (111)

20b2bk k

X 31iA

(15)

The solutior (lU ) corresponds to a situation where the probability is finite that

the Etate ^ was occupied (large k ) in the remote past. In solution (15)

this same probability is zero. One can easily verify from the equation for

•y, (see the remark after Eq,(lO )) that the corresponding probabilities for

this state are inverted, being finite for the second solution and zero for,the

first. If ve assume that we start out with the system in \ft »we must adopt.(1)(k ) and derive, from analysisthe solution i|r '{k ) and its companion

the value of ij) (k ) in the remote future.

One can check that the asymptotic expansion (lM of the function,(12)

is valid in the first, fourth and third quadrant of the complex plane, including

the limiting lines. From our choice of the geometry of electron orbits one can

go from one branch to another by introducing

-k- = ex (16)

Inserting this into (lU) we get for the reduction in amplitude along the

inertial path

,2 •

expTTA

(a-e)ffbB. _• (17)

Its square measures the probability that the Ehrenfest principle is violated, whioh is

-6-

the criterion for the magnetic breakdown in the present situation. Thus the

tunneling probability P is

exp

The above expression can be written in Blount's notation as

(18)

P = expTTA2C,

v v .-fieH1 x y1(19)

This result is in agreement with the one obtained by Blount in WKB

approximation. It may be interesting to compare our results with those

obtained by Reitz . Reitz calculated the transition probability for an

electron on one orbit in real space to jump to another orbit having a different

centre using the time-dependent perturbation theory. Thus Reitz's derivation

has the advantage of being . quantum mechanical and the results are there-

fore formally applicable to arbitrary magnetic field. Unlike the present

result, Reitz obtains an oscillatory component in the expression for transition

probability. However, in averaging over the different orbits on the Fermi

surface which contribute in a particular experiment, it is found that the

oscillatory behaviour tends to wash out and in the semiclassical limit Reitz "s

criterion for breakdown is of the present form.

Apart from its pedagogic value, as the present approach does not use

perturbation theory, it shows that the result (19) is valid for all strengths

of magnetic field.

• ACKNOWLEDGMENTS

I take this opportunity to thank Prof. G.H. Wannier for bringing my

attention to this problem and Prof. Abdus Salam, the International Atomic

Energy Agency and UNESCO for hospitality at the International Centre for

Theoretical Physics, Trieste, where part of the work was done.

-T-

REFERENCES

1) M.H. Cohen and L.M. Falicov, Phys. Rev. Letters 8_, 231 (l96*l).

2) E.I. Blount, Pbys. Rev. 126., 1636 U962).

3) A.B. Pippard, Proc. Roy. Soc. (London) A270, 1 (1962).

k) R.G. Chambers, Proc. Phys. Soc. (London) 88_, 701 (1966).

5) E.O. Kane, J. Phys. Chem. Solids 12_, l8l (i960).

6) G.H. Wannier, Physics 1_, 251 (1965). :

7) E. Brovn, Solid State Physics 22_, 313 (1968) (edited ty Seitz,

Turnbull and Ehrenreich).

8) J.M. Luttinger and ¥. Kohn, Phys. Rev. £1. 8^9 (1955).

9) E.T. Whittaker and G.K. Watson, A Course of Modern Analysis, Chapter XVI,

(Cambridge University Press, 1962).

10) J.R. Reitz, J. Phys. Chem. Solids 25_, 53

-8-

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