Energy bands, carrier density and Fermi energy in Bi in a

J. Phys. C: Solid State Phys. 20 (1987) 3875-3886. Printed in the UK

Energy bands, carrier density and Fermi energy in Bi in a uniform magnetic field

M Cankurtaran, M Onder, H Celik and T Alper Department of Physics, Hacettepe University, Beytepe, Ankara, Turkey

Received 1 September 1986

Abstract. The energy eigenvalue equation for electrons in Bi was solved assuming the McClure and Choi modified non-ellipsoidal non-parabolic (MNENP) model in the presence of a uniform magnetic field H applied in thexy plane, using the first-order time-independent perturbation theory. The magnetic field dependence of the electron density and Fermi energy were investigated and the numerical results were compared with those for the Lax ellipsoidal non-parabolic and Cohen non-ellipsoidal non-parabolic model. It was found that, in the MNENP model, the electron density and Fermi energy exhibit spike-like oscillations for H along the x and z axes; the oscillations for the other orientations of H have an almost sinusoidal lineshape. These spike-like oscillations in the carrier density and Fermi energy were attributed to a drastic change in the electron Fermi surface introduced by the MNENP model.

1. Introduction

Bismuth has been the subject of a large number of experimental and theoretical inves- tigations. Several models have been developed to describe the energy band structure of Bi (Dresselhaus 1971, McClure and Choi 1977, Dorofeev and Falkovskii 1984). It has been shown that the magnetic field dependence of many physical phenomena in Bi can be explained by a two-band model. Magneto-optical results (Maltz and Dresselhaus 1970, Vecchi et af 1976), longitudinal magnetostriction (Michenaud et a1 1982) and ultrasonic quantum oscillations data (Cankurtaran et af 1985, 1986) support the Lax ellipsoidal non-parabolic (ENP) model (Lax and Mavroides 1960). However, the theoretical calculations of Cohen (1961) and the experimental results of Koch and Jensen (1969) and of Dinger and Lawson (1970,1971,1973) indicate that the energy bands in Bi conform to the Cohen non-ellipsoidal non-parabolic (NENP) model. In their work on magnetic surface resonances, Takaoka et a1 (1976) concluded that neither the Lax model nor the Cohen model is adequate and proposed a ‘hybrid’ model. McClure and Choi (1977) reviewed the band models currently in use and presented a modified model which is more general. They also showed that it is in good agreement with the data of a large number of magneto-oscillatory and resonance experiments. More recently, Chen et a1 (1984) proposed a new model which is a simplified version of the McClure and Choi model and called it the modified non-ellipsoidal non-parabolic (MNENP) model. In the same work, Chen et a1 (1984) derived the energy eigenvalues for electrons in a uniform magnetic fieldH directed along the z axis and discussed numerical results for the electron

0022-3719/87/253875 + 12 $02.50 @ 1987 IOP Publishing Ltd 3875

3876 M Cankurtaran et a1

density based on the MNENP model. The present study was motivated by the above work of Chen et a1 (1984) in which they showed that the density of electrons in the MNENP model exhibits periodic spike-like discontinuities as a function of magnetic field. We expect the discontinuities in the carrier density to be associated with a similar effect in the Fermi energy. Oscillations of the Fermi energy are required to keep the number of electrons equal to the number of holes.

In previous work (Cankurtaran et a1 1986), we calculated the variation in Fermi energy and carrier density with a magnetic field up to 3 T, in the framework of the ENP model for electrons and a parabolic band model for T holes. In the present paper, we study the energy eigenvalue equation of electrons, in the framework of NENP and MNENP models, in the presence of a uniform magnetic field H perpendicular to the z direction. We derive an equation for the energy eigenvalues of electrons and obtain an expression for the electron density for an arbitrary direction of H in the xy plane. We discuss the numerical results for the magnetic field dependence of the carrier density and electron Fermi energy in the ENP, NENP and MNENP models. We found that in the MNENP model the spike-like discontinuities in the carrier density and Fermi energy occur only in the close vicinity of the x and z axes of the electron ellipsoids; the oscillations for the other orientations of H are quite different in nature. We attribute these discontinuities to a drastic change in the electron Fermi surface introduced by the MNENP model.

2. Theory

2.1. The Fermi surface and band structure of Bi

The electron Fermi surface of Bi consists of three identical highly elongated ellipsoids at the L points of the reduced Brillouin zone denoted by a (principal pocket), b and c . A principal axis of each of these ellipsoids coincides with the binary axes of the crystal; the other two axes are tilted from the trigonal plane by an angle of about 6". The hole surface consists of a single ellipsoid of revolution about the trigonal axis centred at the T point of the reduced Brillouin zone. In figure 1 the projection of electron ellipsoids onto the trigonal plane is shown.

Figure 1. Schematic representation of the electron Fermi surface of Bi

Energy bands, carrier density and E F in Bi 3877

In the ENP model, any one of the three electron ellipsoids in the principal axis reference frame is described by

Here EG is the energy gap between the L-point valence and conduction bands; the energy E is relative to the bottom of the conduction band; the mi values are the effective masses at the bottom of the conduction band; p is the momentum measured from the point L; x , y and z refer to the principal axis sytem of the ellipsoid; the x axis is parallel to the binary axis, they axis is in the direction of elongation of the ellipsoid and is tilted from the bisector direction by an angle of about 6", and the z axis makes the same angle with the trigonal axis.

Taking into consideration the work of McClure and Choi (1977), Chen et a1 (1984) proposed the dispersion relation

E ( l + E / E G ) = p ; / 2 m 1 + p:/2mz + p1/2m3. (1)

E(1 + E / E G ) = P f / 2 m 1 + (p: /2m2)[1 + ( E / E G ) ( l - m 2 / m ; > 1 +p1/2m3

+ P ; / 4 m 2 m ; E G - l ( P k : / 4 m l m 2 E G +P:P:/4m2m3EG) ( 2 ) where mi is the effective mass tensor component at the top of the L-point valence band. We introduced the numerical parameter p in equation ( 2 ) to combine the NENP ( p = 0) and MNENP ( p = 1) models. The other quantities have similar meanings to those in equation (1). The experimental results of Antcliffe and Bate (1967) and Dinger and Lawson (1971) indicate that the difference between m2 and mi is quite small, and hence one can set m2 = m i , as a result of which the term in the square brackets in equation ( 2 ) can be taken to be unity. The main difference between the MNENP and NENP models is the introduction of p;p: and p ; p ; terms into the dispersion relation, to be considered as perturbations.

Because of the strong non-parabolicity of the conduction band in Bi, the effective masses are energy dependent. The effective masses at the band edge in equations (1) and ( 2 ) are related to the effective masses at the Fermi level by

mi = miF/( l + 2A) i = 1 , 2 , 3 for the ENP model, and

( 3 )

m, = mLF/(l + 2A) i = l , 3

m 2 = (n/2)2 m 2 F / [ K ( k ) 1 2 k 2 = A/(1 + 2A) ( 4 ) for the NENPmOdel (Dinger and Lawson 1971). Here A = E F / E G , where EFiS the electron Fermi energy measured from the bottom of the conduction band, K ( k ) is the complete elliptic integral of the first kind and miF are components of the effective mass tensor at the Fermi level. The experimental values of miF are usually deduced from the measured cyclotron effective masses.

2.2. Solution of the energy eigenvalue equation in a uniform magnetic field H in the

Using the time-independent first-order perturbation theory, Chen et a1 (1984) found that, for H along the 2 axis, the energy eigenvalues in the NENP and MNENP model are given by

xy plane

E ( l + E / E G ) = ( n + 4 + sy)hw, + (h2k5/2m,)[1 - p(ho , /2EG)(n + 4) ] + [ (hw, )* /8EG][3(n2 + n + b) - p(n2 + n - i)] (5 )


where n and s (= L 4 ) are the Landau and spin quantum numbers respectivly, y is the spin-splitting factor, w, = eH/m, is the cyclotron frequency, and m, = (mlmz)@ is the cyclotron mass.

To solve the energy eigenvalue equation for an arbitrary orientation of H in the x y plane, we also make use of the first-order time-independent perturbation theory. For calculational simplicity, it is useful to make a coordinate transformation that aligns the x axis alongH (figure 1 ) . Let the angle between the x axis and H (x’ axis) be a; thus, this transformation requires a rotation of the coordinate frame about the z axis by a. The vector potential in the Landau gauge then takes the simple formA = (0, 0, Hy’) . Taking the inverse of this transformation gives the equations

x = x ’ cos a - y ’ sin a

y = x’ sin a + y ‘ cos a (6a) z = 2’

p x = p x j cos a - p y c sin a

p y = p x # sin a + p y p cos a

P z = PZV.

The energy eigenvalue equation of the unperturbed system (equation ( 1 ) ) without the electron spin-splitting effect is transformed into

The eigenvalues of this equation are given by

Enk(l + E n k / E G ) = ( f i k ~ ) ’ / 2 m ~ + (n + $)hoc

where kH = kx, is the wave-vector component along H and mH is the longitudinal mass. In the present case, mH and mc are defined by

mH = m , cos2 a + m2 sin2 a

H’ =P;/4mZm;EG - !4(PZP;/4mlmZEG + P ; P ? / ~ ~ z ~ ~ E G )

m, = (mlm2m3/mH)”2 . (9)

(10)

In the NENP and MNENP models, we treat

as a perturbation term. After application of the transformation given by equations (6) to H’ , perturbation theory was applied and the energy eigenvalues were found as

Enk(l + Enk/EG) = A l k $ + A2k$ + A 3 ( 1 1 )

where

A l = (h4/4mzEG)(l + 2 & ~ o s ~ c ~ ) ~ s i n ~ a -p(h4/4mlm2EG)[4sin2 2a + (isin4a)P

+ ( 1 - 4 sin2 2a)P - (4 sin 4a)P + (a sin2 2cu)P4] (12)

Energy bands, carrier density and EF in Bi 3879

A 2 =h2/2mH + ( n + i ) h o c { & h 2 m l / m 2 m H E G ) ( 1 + 2 Q c o ~ ~ a ) ~ s i n ~ 2 a

- p ( h 2 / 4 m H E G ) [ 1 -Bsin22a- (#sin4a)P+ (#sin22a)P2]

- p ( h 2 / 4 m 2 E G ) ( l + 2Q cos2 a)' sin2 a} (13)

A 3 = ( n + f ) h w c + [3(hwc)2/8EG](ml/mH)2(n2 + n + f) x [cos4 a - p ( m 2 / 4 m 1 ) sin2 2a]

- p [ ( h o c ) 2 / 8 E , ] ( m l / m H ) ( n 2 + n - 1) cos2 a (14) with Q = ( m 2 - m1) /2mH and P = Q sin 2a. If the electron spin-splitting effect is taken into account, ( n + I)hwc in equation (14) should be replaced by ( n + h + sy)ho,.

2.3. Density of carriers in the presence of a magneticfield

The ultimate goal of this paper is to determine the magnetic field dependence of Fermi energy. To this end, we calculate the density of carriers. In Bi, there are equal numbers of electrons and holes (charge neutrality). This condition determines the Fermi energy. In the ENP model for an arbitrary orientation of H, when EF S- kT, the density of electrons per ellipsoid whose energy is below EF is expressed by

] ' I 2 (15) 2312,~

N i ( H ) = 2 m g 2 - ( n + f + sy)hw, h n,s

where i refers to any one of the electron ellipsoids. The angular dependence of N i ( H ) is introduced through mH and w,.

In the NENP and MNENP models the expressions for ",(H) depend explicitly on the orientation of H. Once the energy eigenvalues are known, it is easy to calculate the density of electrons (Kahn and Frederikse 1959). Thus, for H along the z axis, we found that the density of electrons per ellipsoid is given by

with

A , = (n + f + sy)hwc + (hwc/8EG)'[3(n2 + n + f) - p(n2 + n - I ) ] . For p = 1 ) this equation is essentially the same as that given by Chen et a1 (1984).

expressed by When H lies in the xy plane, we found that the electron density per ellipsoid is

where

D = EF(1 + E F / E G ) - A 3 .

Because of the large energy gap between the valence and the nearest conduction band at the T point of the reduced Brillouin zone, a parabolic dispersion relation for the T holes may be assumed. Thus, for the hole ellipsoid, the density of carriers whose energy is above the Fermi level is given by

2312,~ N,(H) = 7 M g 2 E [ (E, , - E F ) - ( N + 4 + SY,)~WC]'/~ (18)

N.S


where E,, is the overlap energy between the valence and conduction bands, and all other quantities have similar meanings to those in equation (15).

The charge neutrality condition is

This equation determines the Fermi energy. The sum in equations (15)-(18) is over those values of n and s such that the radicand is positive.

3. Numerical results and discussion

In the rest of this paper, we present and discuss the numerical results for the magnetic field dependence of carrier density and Fermi energy in the framework of the ENP, NENP and MNENP models for H directed along the z and y axes. To obtain the carrier density and Fermi energy as a function of the magnetic field, we impose the charge neutrality condition. In the calculations the electron and hole densities were estimated from equations (15)-(18) using the band parameters given in table 1. The value of the Fermi energy is varied at a certain magnetic field and the value which gives the best charge neutrality is selected. The calculations are repeated for fields up to 5 Tin steps of 0.002 T. Using the procedure, we simultaneously obtained the magnetic field values at which the Landau levels of the carriers in each ellipsoid cross the Fermi level.

Table 1. Band parameters for carriers in Bi used in the calculations: mu is the rest mass of the electron.

Experimental value Reference

27.2 Heremans and Hansen (1979) 13.6 Vecchi and Dresselhaus (1974) 38.5 Smith et a1 (1964)

0.005 97 mo 1.33 mo 0.0144 m, 0.064 mo Smith et al(1964) 0.064 mo 0.69 m,, 1.063,0.37 0.082, 1.94

Dinger and Lawson (1973)

The band-edge effective masses used in the calculation are obtained, through equations (3) and (4), using the effective masses at the Fermi level which were deduced from the cyclotron masses measured by Dinger and Lawson (1973). The hole effective masses and the spin-splitting factors for both electrons and holes are those determined by Smith et a1 (1964). It should be noted that, in the MNENP model, it is impossible to relate the band-edge masses to those at the Fermi level using the procedure described by Dinger and Lawson (1971). Thus, in both the NENP and the MNENP models, we used


the band-edge masses obtained from equations (4). An exception was made in the MNENP model for H along they axis, where we used the effective masses at the Fermi level. For H parallel to they axis, we set the angle a equal to 90", 210" and -30" for ellipsoids a , b and c respectively.

t HkENP

5 . 5

4.5

3.5

I I I I I I I I I 1 1 2 3 4

H ( T I

Figure 2. Magnetic field dependence of the total carrier density in Bi for H parallel to the z axis.

In figures 2 and 3, we show the magnetic field dependence of the electron density in the ENP, NENP and MNENP models for H along the z and y axes, respectively. It is seen that in these three band models the carrier density oscillates as a function of magnetic field. The behaviours of the electron density in the ENP and NENP models are quite similar; the amplitudes, the numbers and the periods of oscillations are almost the same. However, in the MNENP model, for H along the z axis, the electron density exhibits periodic spike-like discontinuities as a function of the magnetic field. Each of these


1 L H ( T I

Figure 3. Magnetic field dependence of the carrier density in each electron pocket in Bi for H parallel to the y axis. The quantum limit fields H& and Hi;' are indicated by arrows on the corresponding curves.

spikes corresponds to a value of H at which the denominator of the square root in equation (16) becomes zero. The period of these spikes is eh/2m,EG and is completely different from those given in the ENP and NENP models.

For H parallel to the y axis, the electron density in each pocket in the ENP and NENP models oscillates at low fields and varies almost linearly at high fields. Both N : and N:%c oscillate up to fields If& and Hi;', respectively, which are known as quantum limit fields and are marked with arrows on the corresponding curves. The estimated values of the quantum limit fields in the ENP and NENP models are within 5%. The results obtained here are in reasonable agreement with those reported by Takano and Kawamura (1970) and Michenaud et a1 (1981). The monotonic increase in the electron density in each pocket at fields higher than the quantum limit has been explained elsewhere (see, e.g., Edelman 1975, Cankurtaran eta1 1986). The values of HbU and the other characteristics of oscillations of the carrier density in the ENP and NENP models are similar. This fact


indicates that the additional p$/4m2m;EG term in the NENP model is really a perturbation, although it introduces some changes in the carrier density for H along the z axis. However, in the MNENP model the oscillations of the electron density in the y direction are quite different from those obtained for H parallel to the z axis. The oscillations in the present case do not show a spike-like character and have a much smaller amplitude. It is seen from figure 3 that, in the MNENP model, the density of electrons in each ellipsoid becomes zero at a certain value of the magnetic field. Above this magnetic field value the argument of the outermost square root in equation (17) is negative. This implies that the MNENP model cannot explain the high-field magneto-oscillatory phenomena.

“ t l 21 0 L h

1 2 3 L H IT1

Figure 4. Magnetic field dependence of the electron Fermi energy in Bi for H along the z axis.

Plots of the variation in Fermi energy with magnetic field are shown in figures 4 and 5 for H along the z and y axes, respectively. For H parallel to the z axis the variation in EF in the ENP and NENP models is within 5% of the zero-field Fermi energy EF(0) , and it is significant only above 0.5 T. The oscillations of EF in these models have a sinusoidal lineshape. However, in the MNENP model the function EF(H) exhibits deep spike- like discontinuities, which correspond to the singularities in the electron density. The variation in E, in the last case is almost about 50% of EF(0). Such a large oscillatory variation in E,(H) has never been presented in any experimental or theoretical work. The period of the spike-like oscillations in EF is equal to that of the carrier density. Thus the origin of these discontinuities in both EF(H) and N,(H) is the same and can be explained by considering the shape of the Fermi surface suggested by the MNENP model.

C25


H ( T I Figure 5. Magnetic field dependence of the electron Fermi energy in Bi for H along the y axis.

From the symmetry considerations of the energy equation in the MNENP model, similar discontinuities for the electron density (at least for the principal pocket) and Fermi energy are also expected for H parallel to the x axis.

In the MNENP model the oscillations of EF as a function of the magnetic field along they axis are also different from those in the ENP and NENP models. However, now they do not show a spike-like character and are much smaller in amplitude. In the ENP and NENP models, at fields higher than the quantum limit, the increase in the electron density in the last Landau levels causes the Fermi energy to lower.

It is well known that the ENP and NENP models suggest pseudo-ellipsoidal constant- energy surfaces. So the path of an electron inp space is always closed for any orientation of H. However, in order for the Fermi surface suggested by the MNENP model to be closed, the following inequality has necessarily to be satisfied:

- 2 < E,/EG < 1. (20) In most of the published work on Bi the sign of E, was taken to be positive, and the

ratio EF/EG was found to be about 2. Thus the conditions in equation (20) are not fulfilled. For H along they axis the cross section of the Fermi surface determined by the MNENP model with a plane perpendicular to H is an ellipse and its extrema1 area is given by

A,,, = 2 ~ r ( m , m ~ ) ~ / ~ E ~ ( l + E F / E G ) . (21) However, for H along the z or x axis, the MNENP model does not always suggest closed orbitals, provided that the cyclotron frequency is not defined. We attribute the spike-


like discontinuities in the electron density and Fermi energy to the open Fermi surface proposed by the MNENP model (figure 6). It should be interesting to study the energy eigenvalues and carrier density in the presence of a magnetic field, on the basis of the original McClure and Choi model using the two sets of data reported in the same paper (McClure and Choi 1977).

Figure 6. Principal cross sections of the electron Fermi surface suggested by the v h t \ P

model p r o = [2m,EF(1 + &/&)I’ ’. p , ~ = (2m2EG)’ ’ and p ( , = [2m?EF(1 + &/EG)]’ The three cross sections are on the same scale

The spike-like discontinuities in the electron density occur only forH along the z and x axes. We do not expect this type of oscillations for the other orientations of H . Therefore the large-amplitude spike-like oscillations in the magnetoresistance and Hall effect in Bi calculated by Wu and Lin (1984,1985) should disappear for the orientations of H away from the x and z axes of the electron ellipsoid.

Acknowledgments

This work was supported by the Scientific and Technical Research Council of Turkey (Project TBAG-738) and Hacettepe University Research Fund (Project 85-01-010- 26).

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Energy bands, carrier density and Fermi energy in Bi in a

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