Doped semiconductors in an inhomogeneous magnetic field

Solid State Communications,

Vol. 13, PP. 1713—1716, 1973.

Pergamon Press.

Printed in Great Britain

DOPED SEMICONDUCTORS IN AN INHOMOGENEOUS MAGNETIC FIELD* S.L. Cunningham and R.F. Wallis Department of Physics, University of California, Irvine, California 92664, U.S.A. (Received 9 September 1973 by A.A. Maradudin)

We show that a potential difference is generated across a semiconductor sample containing free carriers when the sample is in a magnetic field gradient in the direction of the field. A measurement of the potential difference can give direct information concerning the effective mass and effective g-factor of the free carriers.

THE PURPOSE of this note is (i) to show that a potential difference will be induced across a sample containing free charge carriers when the sample is in a magnetic field gradient and (II) to show how the magnitude of this potential is related to the properties of the sample. The mathematical formalism is general and can be applied to any system containing spin one-half charge carriers. However, for the purposes of illustration, we will specifically consider a sample of n-type InSb.

of light in vacuum, his Planck’s constant, 13 is the Bohr magneton, and B is the magnitude of the magnetic field. The constants m and g* are material dependent and are the effective mass and the effectiveg-factor, respectively. For InSb, the effective mass1 is very small; m* = 0.0 145 me where me is the free electron mass. On the other hand, the effective g-factor2 for InSb is large and negative; 50. In Fig. I we show a schematic of the energy levels for n-type InSb.

The energy,E, of a single electron in a magnetic field, which is taken to be in the z-direction, is labelled by three quantum numbers,n, the Landau level number; s, the spin quantum number = ± ~ and k~, the wave vector component in the z-direction. The energy is given by: h2k2 E(n,s,k~)= ~ (1)

For each single set of quantum numbers n, s, and k~,there are D electrons, where B D = ~!__ ~ (3) 2irch Here, L~,,is the physical dimension of the sample in the c~direction.

—

The total number of free electrons, NT, in the sample is equal to the degeneracy, D, times the number

where the cyclotron frequency is =

eB mc —---,

of occupied states, which at T = OK, can be written ,~ kp Nr=D~~1. (4)

(2)

e is the magnitude of the electron charge, c is the speed *

S=i~n’Ok~~~—kF

By examining Fig. 1, we see that the limit on the k~ sum is dependent upon the values ofn and s and is given by:

Work supported in part by the U.S. Office of Naval Research under Contract No. N000l4-69-A-02009003.

/

l2m 1~

Technical Report No. 73-61.

kF(s,n) 1713

=

[~F

g13Bs —(n

+

1714

DOPED SEMICONDUCTORS IN AN INHOMOGENEOU5 MAGNETIC FIELD

We consider the case when the magnetic field has a gradient in the z-direction, i.e., in the direction of

E

~~2”

~

// / /

__________________

field, B, the electron number density, p, and the fermi level some CF. Toz-dependence. do this and still be able to use equati~n have These are the magnetic

~

the field. (7), number we imagine ofThus hypothetical the consider sample to that perpendicular be three divided quantities into the large may z-direction. Thewe slabs areslabs sufficiently thick sotoathat a local fermi wave vector, a local electron number density, and a local fermi energy can be defined yet thin enough so that the magnetic field is very nearly a constant over the thickness of each slab. For lnSb doped with 1016 free cathers per cm3, this thickness is of the order of 0.1—1 .0 Mm.

I 1/

/

When the system of free electrons in the magnetic field gradient is in equilibrium, then the electrochemical potential is constant throughout the sample. This electrochemical potential is equal to the sum

-

w~ kz FIG. 1. Energy level diagram showing the electron energy of the lowest Landau levels as a function of k~for level numbers 0 through 4 including the spin splitting. The relative spacing is that appropriate for InSb.

of the Fermi energy and the electric potential energy. Thus we write EF(Z)

=

mt

_g*138s

[th(CF

—

—

eØ(z)

=

const.

(8)

where 0(z) is the spatially dependent electric potential.

Here we have introduced EF, the Fermi energy, which is measured relative to the zero magnetic field conduction band edge. The limit on the n sum is dependent upon the value of s and is given by

nm(s)

Vol. 13, No. 10

~~c)1. (6)

We noteequation that n,,, must behave an integer, so that writing (6), we introduced theinoperator, Int, which means ‘take the largest integral part.’ The limit on the n sum is such as to make the limit on the k~sum, equation (5), always real.

Poisson’s equation relates the electric potential to the electron number density by ~Ø 4~re

2

where e

=

az

—

eo

[p(z)p

0]

(9)

0 is the low frequency dielectric constant 3 and Po is the zero magnetic density which is included due to charge neutrality. Using equation (8) we obtain

(Co 17.88 for InSb) field= electron number a2CF

—

4ne2

(p(z) — Po)

—

.

(10)

Co

The k 2 sum in equation (4) can be performed by first converting it to an integral. Using equation (5) yields — —

2ch2 eB(2m)~ C. 2i~

~ ~

[CF

g13~S— (n +

~

Thus, by combining equations (10) and (7) we have a second order differential equation for the fermi level as a function of z. f

n~O

(7) NT/(LXLYL 2). This equation implicit equation for15CF, the fermi energy, since pisisanusually known and CF generally unknown. where p is the electron number density

If the magnetic field is sufficiently large so that only one Landau level is occupied, then equation (7) takes on a particularly simple form since s = + ~ and

=

n

=

0.

T p = Bd(eF—aB) where we have defined the constants

(11)

Vol. 13, No. 10

DOPED SEMICONDUC1~ORSIN AN INHOMOGENEOUS MAGNETIC FIELD

F(z)

=

EF(z)CF

20 18

‘?

12 IC

Q

and where ~ is the value of the fermi energy when the magnetic field is uniform and everywhere equal number density and local fenni energy, respectively, to B0. The defining equation for 4 is, from equation

~

(11),

.1

14• > a,

which are the fractional changes in the electron

vs. Field Magnetic Fermi Energy

16

Po I

I

I

10

I 50

100

B (10 gauss) 12.2

i

I

‘

(16)

4

Ca) 22

1715

11.8 12.0 11.6

=

Bod(4—aBo)~.

With these definitions, equation (II) becomes Bd(4)i~[ N(z) + 1 = F(z) + 1 -~-t Po EFJ and equation (10) becomes —

where

2 = PN(z) 82F(z) az

(17)

(18)

(19)

11.4I 1.0

~4ire2P

5.0

0

10.0

~0

4

(20)

B (iO~gauss) FIG. Fermiasenergy for n-type InSb where Po = 2. 10ça) 6cm3 a function of magnetic field for 4 gauss. The high field limit is fields by given greater equation than(21). i0 (b) Same for magnetic fields between iO~gauss and 1O~gauss. The energy scale is different by a factor of 10.

d

—

2ir2ch2

(12)

and S

g13 a

=

2 —

~ 2m c

+——

2

=

me m

(13)

For this high field case, the magnetic field must be everywhere greater than ~ where ~

=

F—ps ~

(14)

(Recall that g5 is negative.) We not introduce the unitless quantities N(z)

=

—

Po and

Po

(15)

it is important to consider the size of the terms 3 and in theseaequations. choose value for BIf we assume Po = 1016 cm 0 = 5 X iO~G (which is greater than ~ 2 X l0~G), then from equation (17), = 2 X 10-14 erg, and the constant P = 1011 cm2. It is this large value ofF that is significant. Since the left hand side of equation (19) can, at most, be of

4

order I cm2, we see thatN(z) must be of order 10h1 or less. Thus, to a very good approximation,N(z) can be neglected in equation (18). This has the effect of changing the basic problem from the solution of a differential equation to the solution of an algebraic equation. Equations (16) and (18) can now be easily solved for the fermi energy to give CF

=

~zB+

~Bdj

(21)

Hence, the position dependence of the fermi energy is tied directly to the z-dependence of the magnetic field, and there is essentially no position dependence of the electron number density as can be seen by the small value of N(z). Physically, this means that is does not take very much charge relocation to create an electric field sufficiently large to counter balance the fbrce due to the spatial variations in the fermi energy.

DOPED SEMICONDUCTORS IN AN INHOMOGENEOUS MAGNETIC FIELD Vol. 13, No. 10

1716

The neglect of the term N(z), which is the equivalent of replacing p(z) by Po in equation (11), is also a valid approximation for smaller magnetic fields. Thus, we can

write equation(7) as Po

=

Bd ~

~

[CF —

a(n, s)B] ~

(22)

2p~ aB

0~ =

±

a

—~j~

-~—

.

(25)

±

Hence, for a field gradient of io~G/cm, we see that a~-±2X~ cm2.Thus,onlyaverysmall fraction of the charges in the bulk accumulate on the surface.

where a(n,s)

=

he g5fls + (n + ~)—--. m C

(23)

In Fig. 2 we show the solution of equation (22) where we plot the fermi energy as a function of magnetic field for the case of InSb referred to above, Since the local fermi level is a function only of the local magnetic field, and since the local fermi level is directly related to the local potential through equation (8), the potential difference across a sample is dependent only upon the magnitude of the magnetic field at each surface. For example, if our InSb sample has a field of 5 X I0~G at one surface and a field of 6 X ~ G at the other surface, then there will be an electric potential difference across the sample of 2 X l0~V [from Fig. 2 or equation (21)]. result is independent of the length of the sample. This potential difference is related to the migration of charge from one surface to the other. The amount of charge involved is related to the gradient of the potential (and thus to the gradient of the fermi energy) through Gauss’ law. In this way we fmd aEF(Z) az

=

~

~ 4ire 2 ~ Co

(24)

In conclusion, we have shown that a magnetic field gradient can induce a potential difference across a sample containing free carriers. If the number of

carriers is of the order of 1016 cm3. thenthe potential differences are of the order of i0~V for intermediate size magnetic fields (‘- 5 X I O~G) and magnetic field gradients (“- 5 X iO~gauss/cm). The exact magnitude of the potential difference depends upon the effective mass and the effective g-factor, and in the high field limit, this dependence is simple. We note that this potential difference cannot be directly measured with a standard voltmeter because the electrochemical potential is constant throughout the sample, however, methods can be devised to measure the effect of the surface charge. One method is to (i) establish the surface charge by having a magnetic field gradient superimposed on a constant magnetic field, (ii) electrically isolate the two surfaces from each other, (iii) turn off the magnetic field gradient, (iv) drain the charge from the semiconductor to a capacitor, and (v) repeat the process until sufficient charge has accumulated to permit measurement of the potential across the capacitor. The measurement of this potential as a function of the magnetic field represents new method for obtaining values for the effective massa and effective g-factor.

where a is the surface electron number density and the + or — sign used to indicate the surface located at the + or —z coordinate. In the high field limit we obtain, using equations (24) and (21),

Acknowledgments We wish to thank Dr. William Parker and Mr. Charles Falco for helpful discussions. —

REFERENCES

1.

PALIK E.D., P1CUS G.S., TEITLER S. and WALLIS R.F.,Phys. Rev. 122,475 (1961).

2. 3.

ROTH L.M., LAX B. and ZWERDLING S.,Phys. Rev. 114,90(1959). HASS M. and HENVIS B.W.,J. P&vs. Chem. Solids 23, 1099 (1962).

Nous montrons qu’ une difference de potentiel apparait aux bornes d’un semi conducteur contenant des porteurs libres quand l’echantillon est soumis a un gradient de champ magnétique dans la direction du champ. Une mesure de cétte difference de potentiel peut douver une information directe sur Ia masse effective et le facteur de Landé de ces porteurs libres.