Derivation of Effective-Mass

Expressions for Electrons and

Holes in the Anisotropic

Multiband Semimetals Ar, Sb,

and Bi

ARL-TR-2152 August 2000

Frank J. Crowne 

Approved for public release; distribution unlimited.
The findings in this report are not to be construed as an

official Department of the Army position unless so

designated by other authorized documents.

Citation of manufacturer’s or trade names does not

constitute an official endorsement or approval of the use

thereof.

Destroy this report when it is no longer needed. Do not

return it to the originator.
ARL-TR-2152 August 2000

Army Research Laboratory

Adelphi, MD 20783-1197

Derivation of Effective-Mass

Expressions for Electrons and

Holes in the Anisotropic

Multiband Semimetals Ar, Sb,

and Bi

Frank J. Crowne 

Sensors and Electron Devices Directorate

Approved for public release; distribution unlimited.
Abstract

In this report, certain properties of the multicomponent plasmas present in

the group-V semimetals As, Sb, and Bi have been derived. Notable among

these properties is anisotropy of the effective masses of both electrons and

holes, which in turn leads to anisotropy in the plasma frequencies of these

materials. Because the systems of interest are particles in a host matrix

rather than bulk materials, it is likely that this anisotropy in the plasma

response will be averaged out in some way; however, this problem must be

examined in detail before such a conclusion is warranted.

ii
Contents

1 Introduction 1

2 Boltzmann-Equation Calculation of Fermi Velocities in Group-V

Semimetals 2

3 Calculations of Sums Over k-Space Minima for High-Symmetry

Band Structures 10

3.1 Simple Cubic Band Structure . . . . . . . . . . . . . . . . . . 10

3.2 (111) Cubic Pockets (Tetrahedral Band Structure) . . . . . . . 11

4 Calculations of Sums Over k-space Minima for the Band Structures of Group-V Elements 14

5 Conclusions 17

References 18

Distribution 19

Report Documentation Page 21

Figures

1 Conduction band minima (pockets) of silicon in k-space. . . . . 10

2 Conduction band minima (pockets) of germanium in k-space. . 11

3 Conduction-band and valence-band minima (pockets) of As and

Sb in k-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

iii
1. Introduction

The electrodynamic properties of the semimetallic group-V elements (arsenic, antimony, and bismuth) are complex and ill-understood in many

ways. This is primarily due to the complicated structures of their conduction and valence bands, which have many maxima and minima for the

Fermi surface to intersect, resulting in multiple “pockets” of electrons and

holes populated by carriers at all temperatures [1]. According to the Sommerfeld theory of metals, each minimum or maximum should give rise to a

different species of electron or hole, so that the electrodynamic response of

the material resembles that of a multicomponent plasma. To make matters

worse, all these materials crystallize orthorhombically, so that their charge

carriers (both electrons and holes) have anisotropic effective masses.

When nanometer-size samples of these materials (so-called quantum dots)

are prepared, the resulting systems are expected to behave in unusual ways

when probed by electromagnetic radiation. A particularly easy way to create such samples was discovered some 10 years ago, when researchers

found that under certain conditions low-temperature growth of GaAs by

molecular-beam epitaxy (MBE) resulted in material with large numbers of

ultramicroscopic inclusions of As, up to 1 percent of the host volume [2],

which arose from the aggregation of GaAs antisite defects (i.e., lattice sites at

which a gallium atom was replaced by an As atom). Although theories regarding their nature are still problematic, there are strong indications that

these inclusions, which are roughly spherical in shape, consist of metallic arsenic. Similar behavior has been observed when InAs and GaSb are

grown in this way.

There is ample reason to believe that these materials could be useful in

device engineering. However, before this usefulness can be explored, the

individual nanoparticles must be properly modeled with regard to interactions with electromagnetic waves. In this report, I derive an appropriate set

of constitutive equations that can be used with the Maxwell equations to

implement this modeling.

1
2. Boltzmann-Equation Calculation of Fermi Velocities in Group-V

Semimetals

The most effective way to deal with metals and their interactions with

electromagnetic radiation (light, microwaves, etc) is to use the BoltzmannVlasov equation to calculate momentum (k-space) distributions for the

metallic charge carriers [3]. For semimetals, this naturally leads to a general multiband set of Boltzmann equations:

∂tfj + v(j) α (

k)∂αfj +

e

 ̄h

(∂αφ)

∂

∂kα

fj



= − 1

τj



fj − n(j)

n(j)

0

f(j)

0 (

k)



, (1)

where f(j)

(k, x, t) is the number distribution function for the jth carrier,

v

(j) α (

k) is the semiclassical (hydrodynamic) velocity of such a carrier with

momentum k, τj is its particle-number-conserving relaxation time, and

φ(x, t) is any external (electrostatic) potential applied to the system. The

hydrodynamic approximation introduces a parameterization of the solutions to these equations in terms of macroscopic variables such as density,

drift velocity, etc, which are determined self-consistently as moments of the

equations. If the particles are semiconducting with parabolic bands, we can

use Maxwell-Boltzmann functions for the perturbed carrier distributions:

f(j)

(k, x, t)=4π3n(j)(x, t)



mlm2

t

−1/2

  ̄h2

2πkBT(j)(x, t)

3/2

exp − 1

kT(j)(x, t)

G(j)

(k, x, t)

	

, (2)

where

G(j)

(k, x, t) =  ̄h2

2






k − k(j)

D (x, t) − 

k0j



α



m−1

j



αβ






k − k(j)

D (x, t) − 

k0j



β . (3)

Here n(j)

(x, t) is the hydrodynamic density of carriers in the jth carrier

pocket, k(j)

D (x, t) is the hydrodynamic drift momentum of carriers in this

pocket, k0j locates the center of the pocket in k-space, m−1

j is the effectivemass tensor of the jth carrier type with principal values ml and mt (assuming the pockets are spheroidal), kB is Boltzmann’s constant, T(j)

(x, t)

is the hydrodynamic temperature of the jth carrier type, and  ̄h is Planck’s

constant. The corresponding unperturbed distributions are

f(j)

0 (

k)=4π3n(j)

0



mlm2

t

−1/2

  ̄h2

2πkBT

3/2

exp −  ̄h2

2kT






k − k0j



α



m−1

j



αβ






k − k0j



β




, (4)

2
where n(j)

0 is the equilibrium density of carriers in the jth carrier pocket,

and T is the usual lattice temperature. For metallic particles, the correct

distributions at low temperatures are perturbed Fermi distributions:

f(j)

(k, x, t) = θ




F (x, t) − P j − G(j)(k, x, t)



, (5)

where F (x, t) is a position-dependent Fermi energy (imref) measured from

some constant reference (e.g., vacuum), and P j marks the minimum energy of the jth pocket. The corresponding equilibrium distribution is

f(j)

0 (

k) = θ



F − P j −  ̄h2

2






k − k0j



α



m−1

j



αβ






k − k0j



β



, (6)

where F is the equilibrium Fermi energy. We can relate the density n(j)

0 to

F by introducing the density-of-states mass μ = 

m2

t ml

1/3 and the quantity k2

F j = 2μ∆F j/ ̄h2, where ∆F j = F − P j (i.e., the Fermi wave vector

of pocket j), and by integrating the distribution over k-space in the usual

way. This gives

n(j)

0 = k3

F j

3π2 . (7)

We can likewise define the deviation from equilibrium in the usual way:

δf(j)

(k, x, t) = f(j)(k, x, t) − f(j)

0 (

k) . (8)

By integrating only over k-space, we obtain the perturbed and unperturbed

hydrodynamic particle densities and their difference:

n(j)

(x, t) = 2

(2π)

3



d3kf(j)(k, x, t) ,

n(j)

0 = 2

(2π)

3



d3kf(j)

0 (

k) ,

δn(j)

(x, t) = n(j)(x, t) − n(j)

0 . (9)

Note that these also enter into the number-conserving collisional relaxation

term and hence must be determined self-consistently. These equations couple to the Poisson equation in the usual way:

∇2φ = 1





j

ejδn(j)(x, t) , (10)

where the charge ej = +1 for holes, −1 for electrons.

Let us linearize the Boltzmann equations around the unperturbed

distributions:

∂tδf(j) + v(j) α (

k)∂αδf(j) +

e

 ̄h

(∂αφ)

∂

∂kα

f(j)

0 (

k)



= − 1

τj



δf(j) − δn(j)

n(j)

0

f(j)

0 (

k)



, (11)

3
where we neglect all spatial dependences to zero order. To solve these equations, we use the (wave) ansatz

δf(j)

(k, x, t) = δf(j)

q (

k)ei(q
·
x−ωt) ,

δn(j)

(x, t) = δn(j)

q ei(q
·
x−ωt) ,

φq = − 1

q2



j

ejδn(j)

q . (12)

Then the equations for the perturbed system become



−iω +

1

τj



δf(j)

q (

k) + iqαv(j) α (k)δf(j)

q (

k) = −ej

 ̄h (iqαφq)

∂

∂kα

f(j)

0 (

k)



+

1

τj



δn(j)

q

n(j)

0

f(j)

0 (

k)



, (13)

which are easily solved:

δf(j)

q (

k) =

ej

 ̄h (qβφq)


 ∂

∂kβ f(j)

0 (

k)



+ i

τj



1

n(j)

0

f(j)

0 (

k)



δn(j)

q

ω + i

τj − qαv

(j) α (

k) . (14)

We can now substitute this expression back into the density expression

δn(j)

q = 2

(2π)

3



δf(j)

q (

k) d3k (15)

to get

δn(j)

q = ej

 ̄h (qβφq) 2

(2π)

3



∂

∂kβ f(j)

0 (

k)

ω + i

τj − qαv

(j) α (

k)

d3k +

i

τj

δn(j)

q

n(j)

0

2

(2π)

3

 f(j)

0 (

k)

ω + i

τj − qαv

(j) α (

k)

d3k , (16)

which can be written

δn(j)

q = ej

 ̄h (qβφq)β

1j +

i

τj

δn(j)

q

n(j)

0

2j . (17)

The first term in this expression contains the vector coefficient

β

1j = 2

(2π)

3



d3k

∂

∂kβ f(j)

0 (

k)

ω + i

τj − qαv

(j) α (

k)

, (18)

while the second term contains the scalar coefficient

2j = 2

(2π)

3



d3ξ f(j)

0 (ξ)

ω + i

τj − qαv

(j) α (ξ)

. (19)

4
Since each k-space pocket in the semimetal is ellipsoidal, the hydrodynamic

velocities can be defined in terms of the effective mass tensors:

v(j) α (

k)= ̄h



m−1

j



αγ






k − k0j



γ

≡  ̄h



m−1

j



αγ

ξγ , (20)

where ξγ is the deviation in k from the jth band minimum (or maximum).

Then the scalar term becomes

β

2j = 2

(2π)

3



d3ξ f(j)

0 (ξ)

ω + i

τj − qα ̄h



m−1

j



αγ

ξγ

, (21)

and the vector term becomes

β

1j = 2

(2π)

3



d3ξ

∂

∂ξβ f(j)

0 (ξ)

ω + i

τj − qα ̄h



m−1

j



αγ

ξγ

. (22)

Using the Fermi distributions

f(j)

0 (ξ) = θ



F − P j −  ̄h2

2 ξα



m−1

j



αβ ξβ



(23)

gives

∂

∂ξα

f(j)

0 (ξ) = − ̄h2



m−1

j



αβ ξβδ



F − P j −  ̄h2

2 ξα



m−1

j



αβ ξβ



. (24)

Then β

1j becomes

β

1j = − ̄h 2

(2π)3



d3ξ

 ̄h



m−1

j



βγ

γ

ω + i

τj − qα ̄h



m−1

j



αγ

ξγ

δ(F − ξj ) (25)

where

ξj = P j +  ̄h2

2 ξα



m−1

j



αβ ξβ . (26)

We can now solve the density equation



1 − i

τj

2j

n(j)

0



δn(j)

q = ej

 ̄h (qβφq)β

1j , (27)

and insert the results into Poisson’s equation to get

φq = − e2

q2 φq



j



1 − i

τj

2j

n(j)

0

−1 1

 ̄h

qββ

1j . (28)

Note that e2

j = e2 for solid-state systems. Equation (28) can only be true if

1 = − e2

q2



j



1 − i

τj

2j

n(j)

0

−1 1

 ̄h

qββ

1j , (29)

5
where

qββ

1j = − ̄h 2

(2π)

3



d3ξ

 ̄hqβ



m−1

j



βγ ξγ

ω + i

τj −  ̄hqβ



m−1

j



βγ ξγ

δ (F − ξj ) . (30)

The roots of this equation define various dispersion relations for plasma

waves ω(s)

(q).

Because of the δ-function, it is possible to evaluate expression (30) analytically. Write



m−1

j



αβ = S ̄αλDλνSνβ , (31)

where the matrix S is a rotation chosen to diagonalize m−1

j . Then

qββ

1j = − ̄h 2

(2π)

3



d3ξ  ̄hqβSβλDλνSνγξγ

ω + i

τj −  ̄hqβSβλDλνSνγξγ

δ



∆F j −  ̄h2

2 ξβSβλDλνSνγξγ



, (32)

where ∆F j = F − P j . Let xν = Sνγξγ, Qν = Sνγqγ. Then the Jacobian for

this transformation is 1, and so

qββ

1j = − ̄h 2

(2π)

3



d3x  ̄hQλDλνxν

ω + i

τj −  ̄hQλDλνxν

δ



∆F j −  ̄h2

2

xλDλνxν



.

(33)

Now, since D is diagonal, we can easily define D = D1/2D1/2 as a scaling

transformation. If yν =



D1/2



νγ

xγ, Λν =  ̄h



D1/2



νγ

Qγ, then

qββ

1j = − ̄h



m2

t ml

1/2 2

(2π)

3



d3y

Λνyν

ω + i

τj − Λνyν

δ



∆F j −  ̄h2

2 y2



, (34)

where the factor in front is the Jacobian of the scaling transformation. In

spherical coordinates, this expression becomes

qββ

1j = −  ̄h

2π2



m2

t ml

1/2  ∞

0

y2 dy δ ∆F j −  ̄h2

2 y2

  1

−1

d(cos θ) Λy cos θ

ω + i

τj − Λy cos θ , (35)

where

Λ = 

 ̄h2



q2

⊥

mt

+ q2

z

ml

1/2

=



 ̄h2qα




m−1

j



αβ qβ

1/2

. (36)

Here, q⊥ and qz are components of q perpendicular and parallel to the c-axis

of the particle material in the principal-axis system of cylindrical coordinates. I have used the fact that as a scalar |Λ| must be rotationally invariant.

Introducing the density-of-states mass μ = 

m2

t ml

1/3 and the Fermi wave

vector k2

F j = 2μ∆F j/ ̄h2 for pocket j as we did above, we obtain

qββ

1j = −μkF j

2π2 ̄h

 1

−1

dξ ξ

Γ − ξ , (37)

6
where

Γ =

ω + 1

τj



2∆F j qα




m−1

j



αβ qβ

1/2 (38)

contains the tensor nature of the effective mass both explicitly and implicitly through the Fermi energy. The integral is trivial:

qββ

1j = μkF j

2π2 ̄h



2 − Γ ln Γ+1

Γ − 1

	 ≡ μkF j

π2 ̄h f1(Γ) . (39)

The quantity 2j can also be evaluated explicitly. Analysis similar to that

given above leads to the double integral

2j = 1

2π2



m2

t ml

1/2  ∞

0

y2 dyθ ∆F j −  ̄h2

2 y2

  1

−1

d(cos θ) 1

ω + i

τj − Λy cos θ . (40)

Introducing the notation yF =

2∆F

 ̄h2 and  = cos θ as before, and using

the properties of the Fermi function, we can rewrite the above as

2j = 1

2π2



m2

t ml

1/2  yF

0

y2 dy  1

−1

dξ 1

ω + i

τj − Λyξ . (41)

Some algebra gives the following expression for this integral:

2j = 3n(j)

0

2




ω + i

τj





Γ2 +




1 − Γ2

 Γ

2

ln Γ+1

Γ − 1

	 ≡

n(j)

0

ω + i

τj

f2(Γ) , (42)

where Γ is the quantity introduced above.

Let us solve the plasma dispersion relation to lowest order in q, i.e., the

long-wavelength limit. To do so, we take the small-q limit of our explicit

expressions in equations (30) and (42), which corresponds to Γ → ∞. Some

algebra shows that the function f2(Γ) defined in equation (42) goes to 1 in

this limit, so to lowest order

2j = 1

ω + i

τj



d3ξf(j)

0 (ξ) = n(j)

0

ω + i

τj

. (43)

The same limit for qββ

1j gives

qββ

1j ≈ μkF j

π2 ̄h



− 1

3Γ2

	

= −2μkF j

3π2 ̄h







∆F j q2

⊥

mt + q2

z

ml



(ω + i/τj )

2







. (44)

7
But

−2μ∆F jkF j

3π2 ̄h







q2

⊥

mt + q2

z

ml

(ω + i/τj )

2





 = − ̄h2 2μ∆F j

 ̄h2

kF J

3π2







q2

⊥

mt + (q2

z)

ml

(ω + i/τj )

2







= − ̄h

k3

F j

3π2







qα



m−1

j



αβqβ

(ω + i/τj )

2







= −  ̄hn(j)

0







qα



m−1

j



αβqβ

(ω + i/τj )

2





 , (45)

so that equation (29) becomes

1 =

e2

q2



j



1 − i

τj

1

ω + i

τj





−1

n(j)

0




ω + i

τj

2 qα



m−1

j



αβ qβ . (46)

Assuming that the relaxation times are the same for all the pockets, we

obtain

1 − e2

q2

1

ω




ω + i

τ

qα









j

n(j)

0



m−1

j









αβ

qβ (47)

or, introducing the unit vector qˆ = q/ |q|,

1 − e2



1

ω




ω + i

τ

 

j



n(j)

0 qˆα



m−1

j



αβ qˆβ

	

= 0 . (48)

This expression, which is clearly in the form of a requirement that a

“Drude”-type conductivity for the system should vanish, defines the

“plasma frequency” of the multicomponent system as the sum in the numerator. In general this sum will depend on the direction of qˆ.

To derive an analogue of the linearized Thomas-Fermi equation and a corresponding screening length, we need to take these expansions to higher

order in q. Saving next-order terms in f1(Γ) and f2(Γ) gives

f1(Γ) ≈ −

1

3Γ2 − 15Γ4 , f2(Γ) ≈ 1 +

1

5Γ2 . (49)

Let us rewrite equation (29) as

1 = − e2

q2



j



1 − i

τj

f2 (Γ)

ω + i

τj





μkF j

π2 ̄h f1 (Γ) (50)

and note that

μkF j

π2 ̄h = 32

n(j)

0

∆F j

. (51)

8
Then equation (50) becomes

1 = − 1

q2



j

λ2

j



1 − i

τj

f2(Γ)

ω + i

τj





−1

f1(Γ) , (52)

where λ2

j is the Thomas-Fermi screening length for carriers in the jth pocket:

λ2

j = 3

2

n(j)

0 e2

∆F j

. (53)

With these results, we can easily derive the expression

1 = 

j

n(j)

0 e2

q2

qα



m−1

j



αβ qβ

ω




ω + i

τj







1 + 6

5

∆F j




ω + i

τj

2



1 +

2

3

i

ωτj



qα



m−1

j



αβ qβ





 . (54)

The peculiar fraction (6/5) that occurs in this expression derives from

our linearization around a spatially independent zero-order distribution.

A more careful linearization is needed to recover the true hydrodynamic

value of this coefficient and, indeed, to recover any of the expressions encountered in classical fluid mechanics. This will be the subject of a subsequent report.

9
3. Calculations of Sums Over k-Space Minima for High-Symmetry

Band Structures

3.1 Simple Cubic Band Structure

A simple cubic band structure is appropriate for elemental silicon (see fig.

1). In order to calculate the effective masses for this method, we assume

that the densities in the carrier pockets are the same, and that the principalaxis projections are xˆxˆ, yˆyˆ, and zˆzˆ. Since there are six pockets, each holds

1/6 of the electrons.

Let us write the effective masses as follows:



m(1)−1=



m(−1)−1= 1

ml

xˆxˆ +

1

mt

(ˆyyˆ + ˆzzˆ) ,



m(2)−1=



m(−2)−1= 1

ml

yˆyˆ +

1

mt

(ˆxxˆ + ˆzzˆ) , and



m(3)−1=



m(−3)−1= 1

ml

zˆzˆ +

1

mt

(ˆyyˆ + ˆxxˆ) . (55)

Then the sum is simple:



j

n(j)

0



m−1

j



= 1

6

n0

1

ml

+

2

mt



2[ˆxxˆ + ˆyyˆ + ˆzzˆ] = 1

3

n0

1

ml

+

2

mt

 ↔

1 , (56)

where ↔

1 is the identity matrix. This shows that the mass required is just the

usual isotropic optical mass.

Figure 1. Conduction

band minima (pockets)

of silicon in k-space.

10
3.2 (111) Cubic Pockets (Tetrahedral Band Structure)

A tetrahedral band structure is appropriate for elemental germanium (see

fig. 2). Here we have four pockets, at the vertices of a tetrahedron. The

effective-mass tensors are



m(1)−1=



m(−1)−1= 1

ml

eˆ1eˆ1 +

1

mt




sˆ1sˆ1 + t

ˆ1t

ˆ1



,



m(2)−1=



m(−2)−1= 1

ml

eˆ2eˆ2 +

1

mt




sˆ2sˆ2 + t

ˆ2t

ˆ2



,



m(3)−1=



m(−3)−1= 1

ml

eˆ3eˆ3 +

1

mt




sˆ3sˆ3 + t

ˆ3t

ˆ3



, and



m(4)−1=



m(−4)−1= 1

ml

eˆ4eˆ4 +

1

mt




sˆ4sˆ4 + t

ˆ4t

ˆ4



, (57)

where the vectors eˆi, sˆi,t

ˆi ≡ eˆi ⊗ sˆi mark the principal axes of the pocket

ellipsoids:

eˆ1 = √

1

3 (ˆx + ˆy + ˆz) , sˆ1 = √

1

2 (−xˆ + ˆy) , t

ˆ1 = √

1

6 (−xˆ − yˆ +2ˆz) ;

eˆ2 = √

1

3 (−xˆ + ˆy − zˆ) , sˆ2 = √

1

2 (−xˆ − yˆ) , t

ˆ2 = √

1

6 (−xˆ + ˆy +2ˆz) ;

eˆ3 = √

1

3 (−xˆ − yˆ + ˆz) , sˆ3 = √

1

2 (ˆx − yˆ) , t

ˆ3 = √

1

6 (ˆx + ˆy +2ˆz) ; and

eˆ4 = √

1

3 (ˆx − yˆ − zˆ) , sˆ4 = √

1

2 (ˆx + ˆy) , t

ˆ4 = √

1

6 (ˆx − yˆ +2ˆz) .

(58)

Figure 2. Conduction

band minima (pockets)

of germanium in

k-space.

11
Then the dyadic products can be written in matrix form as follows:

eˆ1eˆ1 = 1

3





111

111

111



 , sˆ1sˆ1 = 1

2





1 −1 0

−110

0 00



 , t

ˆ1t

ˆ1 = 1

6





1 1 −2

1 1 −2

−2 −2 4



 ;

eˆ2eˆ2 = 1

3





1 −1 1

−1 1 −1

1 −1 1



 , sˆ2sˆ2 = 1

2





110

110

000



 , t

ˆ2t

ˆ2 = 1

6





1 −1 −2

−11 2

−22 4



 ;

eˆ3eˆ3 = 1

3





1 1 −1

1 1 −1

−1 −1 1



 , sˆ3sˆ3 = 1

2





1 −1 0

−110

0 00



 , t

ˆ3t

ˆ3 = 1

6





112

112

224



 ;

eˆ4eˆ4 = 1

3





1 −1 −1

−11 1

−11 1



 , sˆ4sˆ4 = 1

2





110

110

000



 , t

ˆ4t

ˆ4 = 1

6





1 −1 2

−1 1 −2

2 −2 4



 .

(59)

Performing the summation once more, we obtain



m(1)−1= 1

3ml





111

111

111



 +

1

mt







1

2





1 −1 0

−110

0 00



 +

1

6





1 1 −2

1 1 −2

−2 −2 4











= 1

ml





1

3

1

3

1

3

1

3

1

3

1

3

1

3

1

3

1

3





+

1

mt





2

3 −13 −13

−1

3

2

3 −13

−1

3 −13

2

3



 ,



m(2)−1= 1

3ml





1 −1 1

−1 1 −1

1 −1 1



 +

1

mt







1

2





110

110

000



 +

1

6





1 −1 −2

−11 2

−22 4











= 1

ml





1

3 −13

1

3

−1

3

1

3 −13

1

3 −13

1

3





+

1

mt





2

3

1

3 −13

1

3

2

3

1

3

−1

3

1

3

2

3



 ,



m(3)−1= 1

3ml





1 1 −1

1 1 −1

−1 −1 1



 +

1

mt







1

2





1 −1 0

−110

0 00



 +

1

6





112

112

224











= 1

ml





1

3

1

3 −13

1

3

1

3 −13

−1

3 −13

1

3





+

1

mt





2

3 −13

1

3

−1

3

2

3

1

3

1

3

1

3

2

3



 , (60)

12
and so



m(4)−1= 1

3ml





1 −1 −1

−11 1

−11 1



 +

1

mt







1

2





110

110

000



 +

1

6





1 −1 2

−1 1 −2

2 −2 4











= 1

ml





1

3 −13 −13

−1

3

1

3

1

3

−1

3

1

3

1

3





+

1

mt





2

3

1

3

1

3

1

3

2

3 −13

1

3 −13

2

3



 . (61)

Each pocket holds 1/4 of the electrons. Then the total mass sum equals



j

n(j)

0



m−1

j



= 1

4

n0

 1

ml

	 



4

3 0 0

0 4

3 0

0 0 4

3



 +

1

mt





8

3 0 0

0 8

3 0

0 0 8

3



 = 1

3

n0

1

ml

+

2

mt

 ↔

1 , (62)

which is again the isotropic optical mass.

13
4. Calculations of Sums Over k-space Minima for the Band Structures

of Group-V Elements

The band structures of the group-V elements are described in detail by Lin

and Falicov [4] and Shapira and Williamson [5]. For the conduction band,

the pattern of ellipsoidal pockets shown in figure 3 is obtained; the angle

with z-axis θC is material-dependent. I assume here that the valence band

has a similar structure, although the actual Fermi surface of As is a single

multiconnected surface rather than a collection of pockets.

Let

σ = cos θC , τ = sin θC . (63)

Then the hexagonal symmetry in the basal plane motivates us to introduce

the quantities

α = 1/2 , β = √

3/2 . (64)

This allows us to write the longitudinal-mass principal axis vectors as

follows:

eˆ1 = τxˆ + σz , ˆ

eˆ2 = ατxˆ + βτyˆ − σz , ˆ

eˆ3 = −ατxˆ + βτyˆ + σz , ˆ

eˆ4 = −τxˆ − σz , ˆ

eˆ5 = −ατxˆ − βτyˆ + σz , ˆ and

eˆ6 = ατxˆ − βτyˆ − σz . ˆ (65)

Figure 3.

Conduction-band and

valence-band minima

(pockets) of As and Sb

in k-space. Upper

pockets are filled with

electrons, lower ones

with holes.

(a) Three-dimensional

view, and (b) view

down c-axis (z-axis in

(a)).

θC

14
Then the other principal axes are

sˆ1 = ˆy , t

ˆ1 = −σxˆ + τzˆ ;

sˆ2 = −βxˆ + αy , ˆ t

ˆ2 = ασxˆ + βσyˆ + τzˆ ;

sˆ3 = −βxˆ − αy , ˆ t

ˆ3 = −ασxˆ + βσyˆ − τzˆ ;

sˆ4 = −y , ˆ t

ˆ4 = −σxˆ + τzˆ ;

sˆ5 = βxˆ − αy , ˆ t

ˆ5 = ασxˆ + βσyˆ + τzˆ ; and

sˆ6 = βxˆ + αy , ˆ t

ˆ2 = ασxˆ − βσyˆ + τz . ˆ

(66)

As before, we find the projection operators

eˆ1eˆ1 =





τ 2 0 στ

000

στ 0 σ2



 , sˆ1sˆ1 =





000

010

000



 , t

ˆ1t

ˆ1 =





σ2 0 −στ

000

−στ 0 τ 2



 ;

eˆ2eˆ2 =





α2τ 2 αβτ 2 −ατσ

αβτ 2 β2τ 2 −βτσ

−ατσ −βτσ σ2



 , sˆ2sˆ2 =





β2 −αβ 0

−αβ α2 0

0 00



 , t

ˆ2t

ˆ2 =





α2σ2 αβσ2 ατσ

αβσ2 β2σ2 βτσ

ατσ βτσ τ 2



 ;

eˆ3eˆ3 =





α2τ 2 −αβτ 2 −ατσ

−αβτ 2 β2τ 2 βτσ

−ατσ βτσ σ2



 , sˆ3sˆ3 =





β2 αβ 0

αβ α2 0

0 00



 , t

ˆ3t

ˆ3 =





α2σ2 −αβσ2 ατσ

−αβσ2 β2σ2 −βτσ

ατσ −βτσ τ 2



 ;

eˆ4eˆ4 =





τ 2 0 στ

000

στ 0 σ2



 , sˆ4sˆ4 =





000

010

000



 , t

ˆ4t

ˆ4 =





σ2 0 −στ

000

−στ 0 τ 2



 ;

eˆ5eˆ5 =





α2τ 2 αβτ 2 −ατσ

αβτ 2 β2τ 2 −βτσ

−ατσ −βτσ σ2



 , sˆ5sˆ5 =





β2 −αβ 0

−αβ α2 0

0 00



 , t

ˆ5t

ˆ5 =





α2σ2 αβσ2 ατσ

αβσ2 β2σ2 βτσ

ατσ βτσ τ 2



 ;

eˆ6eˆ6 =





α2τ 2 −αβτ 2 −ατσ

−αβτ 2 β2τ 2 βτσ

−ατσ βτσ σ2



 , sˆ6sˆ6 =





β2 αβ 0

αβ α2 0

0 00



 , t

ˆ6t

ˆ6 =





α2σ2 −αβσ2 ατσ

−αβσ2 β2σ2 −βτσ

ατσ −βτσ τ 2



 .

(67)

The dyadic sums are then



6

i=1

eˆieˆi =





4α2τ 2+2τ 2 0 2(1−2α)τσ

0 4β2τ 2 0

2(1−2α)τσ 0 6σ2



 ,



6

i=1

t

ˆitˆi =





4α2σ2+2σ2 0 −2(1−2α)τσ

0 4β2σ2 0

−2(1−2α)τσ 0 6τ 2



 , and



6

i=1

sˆisˆi =





4β2 0 0

0 4α2+2 0

0 00



 . (68)

Hence,



6

i=1

sˆisˆi + t

ˆit

ˆi =





4β2+4α2σ2+2σ2 0 −2(1−2α)τσ

0 4α2+4β2σ2+2 0

−2(1−2α)τσ 0 6τ 2



 . (69)

Note that because α = 1/2, this tensor is diagonal! Since β2 = 3/4, the above

15
becomes



6

i=1

sˆisˆi + t

ˆit

ˆi =





3(1+σ2) 0 0

0 3(1+σ2) 0

0 0 6τ 2



 . (70)

Likewise, we have



6

i=1

eˆieˆi =





3τ 2 0 0

0 3τ 2 0

0 06σ2



 (71)

so that the averaged optical effective-mass tensor is



j

n(j)

0



m−1

j



= 1

6

n0







1

ml





3τ 2 0 0

0 3τ 2 0

0 06σ2



 +

1

mt





3

*

1 + σ2+ 0 0

0 3 *

1 + σ2+ 0

0 06τ 2











= 1

2

n0





1

ml

τ 2 + 1

mt

*

1 + σ2+ 0 0

0 1

ml

τ 2 + 1

mt

*

1 + σ2+ 0

0 0 1

ml

σ2 + 1

mt

τ 2



 . (72)

In keeping with our expectations, the effective mass tensor is somewhat

anisotropic. However, there is an angle at which it becomes isotropic; this

is

cos 2θC = ml

mt − ml

. (73)

It is easy to show that the reported angles for the group-V materials do not

satisfy this criterion, so that anisotropy will be a real issue in any study of

their plasma response.

16
5. Conclusions

In this report, certain properties of the multicomponent plasmas present in

the group-V semimetals As, Sb, and Bi have been derived. Notable among

these properties is anisotropy of the effective masses of both electrons and

holes, which in turn leads to anisotropy in the plasma frequencies of these

materials. Because the systems of interest are particles in a host matrix

rather than bulk materials, it is likely that this anisotropy in the plasma

response will be averaged out in some way; however, this problem must be

examined in detail before such a conclusion is warranted.

17
References

1. J. M. Ziman, Principles of the Theory of Solids, ch. 4, Cambridge University Press, Cambridge (1964).

2. Z. Liliental-Weber, A. Claverie, J. Washburn, F. Smith, and R. Calawa,

“Microstructure of Annealed Low-Temperature-Grown GaAs Layers,”

Appl. Phys. A 53 (1991), 141.

3. K. Bløtekjær, “Transport Equations for Electrons in Two-Valley Semiconductors,” IEEE Trans. Electron Devices ED-17 (1970), 38.

4. P. J. Lin and L. M. Falicov, “Fermi Surface of Arsenic,” Phys. Rev. 142

(1966), 441.

5. Y. Shapira and S. J. Williamson, “Quantum Oscillations in the Ultrasonic Attenuation and Magnetic Susceptibility of Arsenic,” Phys. Lett.

14 (1965), 73.

18
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Derivation of Effective-Mass Expressions for Electrons

and Holes in the Anisotropic Multiband Semimetals Ar, Sb, and Bi

August 2000 Interim, 1/99–9/99

In this report, certain properties of the multicomponent plasmas present in the group-V

semimetals As, Sb, and Bi have been derived. Notable among these properties is anisotropy of

the effective masses of both electrons and holes, which in turn leads to anisotropy in the plasma

frequencies of these materials. Because the systems of interest are particulates in a host matrix

rather than bulk materials, it is likely that this anisotropy in the plasma response will be averaged

out in some way; however, this problem must be examined in detail before such a conclusion is

warranted.

Semimetals, effective mass, plasma frequency

Unclassified

ARL-TR-2152

ONE6K1

622705.H94

AH94

62120A

ARL PR:

AMS code:

DA PR:

PE:

Approved for public release;

distribution unlimited.

26

Unclassified Unclassified UL

2800 Powder Mill Road

Adelphi, MD 20783-1197

U.S. Army Research Laboratory

Attn: AMSRL-SE-RE fcrown@arl.mil

U.S. Army Research Laboratory

21

Frank J. Crowne

email:

2800 Powder Mill Road

Adelphi, MD 20783-1197