ON MAGNETIC SURFACE STATES INDUCED LINEAR MAGNETORESISTANCE

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ON MAGNETIC SURFACE STATES INDUCED

LINEAR MAGNETORESISTANCE

A. van Gelder

To cite this version:

(2)

JOURNAL D E PHYSIQUE _Colloque

_C6,

_{supplkment au}

_no

_{8, Tome 39, aoat 1978, page} _(3-1124

ON MAGNETIC SURFACE STATES INDUCED LINEAR MAGNETORESISTANCE

A.P. Van Gelder

Physics Laboratorg and Research I n s t i t u t e for MateriaZs University of Nijrnegen, ToernooiveZd, IVijmegen The Netherlands.

R6sumd.- Un nouveau modele quantique pour expliquer la magnltorlsistance lindaire des metaux simples est propos6. L'explication ddpend de l'existence d'etats de surface magndtiques internes au voisi- nage des dislocations ou d'autres imperfections ltendues du cr:istal.

Abstract.- A new quantum mechanical model for explaining linear magnetoresistance of simple metals is proposed, based on the existence of internal magnetic surface states of electrons in the vicinity of dislocations, or other extended crystalline imperfections.

The linear dependence of the resistivity p(B) as a function of the magnetic field B of simple metals with a closed Fermi surface has been one of the unsettled puzzles of electron transport in metals, in spite of the numerous theoretic attempts 11-41 for finding a plausible explanation of the effect. A large contribution to the linear term is usually ascribed to imperfections of the crystal

latticell-31, like dislocations or interfaces between crystallites etc. Theoretical predictions for the effects of crystalline imperfections 12-31 in- voke the concept of a volume fraction (£) of the crystal, for which the conductivity is appreciably different from that of the bulk. Although the quoted theories are basically correct and have been verified experimentally 151, it is doubtful whether

the large values for (f) required to fit the expe-

rimental data for p(B) are realistic. In particular

considering that the volume fraction of anomalously conducting metal due to dislocations or grain boun- daries is too low to explain the observed orders of magnitude of the linear term, even if shape correc-

tions are properly accounted for, the need for a different explanation becomes urgent. Besides, it is possible to demonstrate that the magnetoresistance which would follow from the classical consi-

deration of Refs. 12-31 in the limit f + 0, does

not approach a linear, but rather a logarithmic de-

pendence of B, if B -+ w, provided the two non-va-

nishing dimensions of the inclusions are of similar orders of magnitude.

The model proposed hereafter is intended for

lattice imperfections of precisely this type : with

a vanishing volume, and two dimensions equal to

R

Y and Lz, of which the latter is parallel with the

+

magnetic field B, and both are perpendicular to the

direction of the current J=Jx. Furthermore, R and

Y LZ are assumed small in comparison with theelectron mean free path for impurity-scattering, say, but

large compared with the cyclotron orbit radius R =

1

(21nE~)~ /(Be) for values of the field B in the li-

near regime. The extended crystal imperfections are assumed to be inpenetrable for the electrons, a feature which may be expressed in terms of a poten- tial barrier of the type:

V = g &(X), if

l y l

< L I2 and lzl < gZ/2. A poten- Y

tial barrier of this kind is quantummechanically capable of trapping electrons in its immediate vicinity in localized and bound states, as is well

known for the external surfaces of the crystal 16-

7/. This feature is detrimental to our explanation

of linear magnetoresistance : not $only because the

trapped electrons are strictly forbidden to parti- cipate in the conduction process, but may be consi- dered to occupy effectively open orbits, giving

rise to a ~'-de~endence for the magnetoresistance if

this 'opening' were due to .the periodic lattice. However, considering that the density of bound

states for an energy Ep (= Fermi energy) is equal

0

to 3 ~OL~\R/(~E~), were n is the electron density

the effect is quenched by a factor proportional

with R (% B-'), so that the magnetoresistance p(B)

becomes linear with B, as soon as R 6

R

.

In fact,

it is found that {p(~)-p(O)}/p(O) = h nLc& R 3 / ~ ,

Z Y if nlnc is the density of the extended crystal im-

pirfections with sizes L and kz, R = (2mEF)+/(Be),

Y

and h is a numerical factor, which we have approxi-

mated by h % 5 ~ 1 8 for this model. An extra factor

T-I may be added to this result, if the inclusions.

are randomly oriented, and L =

RZ.

Y

(3)

Although our (qualitative) arguments in fa-

vour of a linear contribution to p(B) bear some re-

semblance with the work of Hsu and Falicov 141 for intrinsic mechanisms, our quoted result for the linear magnetoresistance was obtained by solving a Boltzmam-like transport equation for the electrons in the presence of a static magnetic field. The dif- ference between our approach and the usual one con- cerns the fact, that our equation describes the gain- and loss contributions of the distribution function due to drift and collisions between the stationary states of the electrons, as opposed to

+ + -

the conventional (r-v) variables as 'states'. In -P -P

spite of the fact, that neither the (r-v) variables,

+-

nor even the (v)-ones, are reminiscent of the good

quantumnumbers (perhaps unless B = O), our calcula-

ted magnetoresistance does not differ from the semi-

classical result : p (B) = p(0) if the inclusions

are absent. However, if the inclusions are taken into consideration, via the existence of the mentioned localized bound states, the linear term des- cribed above for the magnetoresistance is found.

In order to see how the linear term is obtained by solving the transport equation, let us start from the following expression for the conduc-

tivity :

B = (ln0e)2(2)1

6

{E(~)-E~} &P(ilit)

k

(kx

-

):k ', which is valid for free electrons

+

with momentum variables (quantum numbers) k, energy

+- 0

E(k), Fermi energy EF, density n

,

charge e, where

-+ -+

Q is the volume of the system. Furthermore, ~(klk')

is the transition-rate duetoimpurity-scattering

+- -+

between the two states k and k', and the factor

(kx

-

2):k is the transport efficiency, usually de-

noted as (1

-

cos6). The given expression is in

principle applicable only in absence of a magnetic field, where pure momentum states are stationary,

buC may also be used if B f 0, provided the quantum

+-

numbers k are understood to refer to the stationary

Landau-states of electrons within the system ; kx

in particular may be chosen (gauge) to represent

the momentum vector along the X (= current) direc-

tion. Using these stationary Landau-states, it is

rigourously possible to show, that p(B) = ~ ( 0 ) for

electrons with a quadratic dispersion relation. For

the proof, it is necessary to realize, that the -h +-

wave functions <rlk> are of the type : e ikxx sin

(kZz) X a harmonic oscillator function along the

Y-direction, which is effectively nonvanishing only

if Iy-Fikx/(~e)

I

< R, if the shape of the system is

rectangular, and if y is the coordinate along the

'Hall-direction', perpendicular to the current and +

the applied magnetic field B.

In order to understand the role of the inclusions, or the localized bound states, in our explanation of the linear magnetoresistance, it is necessary to realize the importance of the localization along the Y-direction of-the Landau-states, in con- nection with the transport efficiency factor (kx-k~?

The order of magnitude of this factor is

g,

because

the respective wave functions must overlap each

other,

if

the inclusions are absent

(G

is the wave

number corresponding with E F ) . However, if the in-

clusions are present, the wave functions of and

g'

need no longer to be overlapping, because it is possible to scatter indirectly (via the bound states)

from

C

to

C'

,

provided ~Ic~-kiI < Bee h, if R is

Y Y

the size of the inclusion along the Y-direction. The efficiency factor (kx-kk) for these anomalous

processes is obviously proportional to B ~ , if

R <<

R

As mentioned earlier, this quadratic B- Y'

dependence is quenched to a linear one due to the fact that thenumber of bound states adds another

-

1

factor R or B

,

entering through the magnitude of

+, +-

the extra contribution to ~(klk') from the trapped electronic states. This extra contribution to the transition rate was found by setting up coupled transport equations for both the current-carrying Landau-states and the trapped ones, the coupling being due to scattering. The applied electric field along the X-direction is:sensed only by the Landau- states, because the trapped states cannot carry a current and give rise to a polarization only. It is not difficult to understand from the foregoing how

the powers

R

R 3 enter into the linear term, which

Z Y

gives rise to a linear contribution of 10 % for a

field fo 10 kG for randomly oriented inclusions with an average size <R> s0.2 mm, and an average

spacing of 1 mm. The onset of the linear regime is

expected for fields close to 500 gauss in this case. References

/ I / Herring, C., J. Appl. Phys.

2

(1960) 1939.

/ 2 / Sampsell, J.B. and Garland, J.C., Phys. Rev. B13 (1967) 583.

/3/ Stroud, D. and Pan, F.P., Phys. Rev.

B13

(1976) 1434.

141 Hsu, W.Y. and Falicov, L.M., Phys. Stat. Sol. B67 (1975) 325.

-

/S/ Van Kempen, H. To be published.

/6/ Prange, R.E. and Tsu-Wei Nee, Phys. Rev.

168

(1968) 779.