Electronic Structure

We calculate the electronic structure of the acceptor states in strained group-III germa-nium by means of the six-band Lutteinger-Kohn effective mass Hamiltonian [20] modified by the Bir-Pikus deformation theory [36]. In this scheme, the acceptor states can be expressed as

6

where |3/2,±3/2>, |3/2,±1/2> , and |1/2,±1/2> are the HH, LH, and split-off-hole (SO) band edge states, respectively. The |J^j,M^j> transform in the T_d^' group like the spherical harmonic

function YJ Mj j. The |3/2,±3/2> and |3/2,±1/2> are basis functions of the Γ representation; 8

the |1/2,±1/2> are basis functions of Γ representation. The F7 j are the envelope functions which are the solutions of the effective-mass equation,

6

where E is the energy of the acceptor states and Hij the elements of the effective-mass Hamil-tonian H. The H can be expressed as a sum of the HamilHamil-tonian without acceptors H0 (i.e., the Hamiltonian of the perfect crystal) and the remaining part caused by the presence of the ac-ceptors,

VI =H −H0, (3.3)

where V is the impurity potential, I is a 6×6 unit matrix. The general form of H⁰ can be ex-pressed as [34, 37]

14 along the crystallographic directions [100], [010], and [001], respectively; ε_ij is the symmetric strain tensor; ∆ is the spin-orbit split-off energy, γ 1, γ2, and γ3 are the Luttinger parameters; av, b, and d are the Pikus-Bir deformation potentials; m0 is the free-electron mass. For the case of a stress P along the [001] direction, only the normal strains εxx, εyy, and εzz are not vanished.

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Therefore, we need only the deformation potential b in the calculation if we neglect the P_ε wich just shifts the valence bands as a whole. The impurity potential is a sum of the Coulomb contribution VC and the central-cell correction Vcc and is expressed in a semi-empirical form:

( )

where α, β, and A are dimensionless parameters; ϵ is the dielectric constant; q is the elemen-tary charge; a^*_B=γ₁²/m e₀ ² is the effective Bohr radius. The Coulomb contribution V_C is caused by the point charge of the acceptor ion modified by the q-dependent dielectric screen-ing [25]. The V_cc includes the contributions (a) the difference in the screened potential in-duced by the positive point charge at the impurity site with the charge magnitude equal to that of the core electrons between the impurity and the host atoms, (b) the difference in the screened potentials induced by the core electrons between the impurity and the host atoms, (c) the difference in the effective repulsive potentials, which is the kinetic energy of the valence electrons in nature, localized in the central-cell region, and originates from the requirement that the wave functions of the valence electrons are orthogonal to those of the core electrons, (d) the lattice relaxation around the impurity site induced by the presence of the impurity [21-22]. The sum of the contribution (a) and the V_C is just the difference in the screened point charge potential induced by the nucleus. The sum of the potentials (a) and (b) is attractive for

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the valence electrons and localized in the central-cell region because they are induced by charges of the same magnitude but opposite sign. The effect (c) should be the primary contri-bution. Therefore, we expect that the Vcc is small for the isocoric acceptor Ga, positive (A>0) for B and Al, and negative (A<0) for In and Tl.

In order to solve the effective-mass equation (3.2), the envelope function can be ex-panded in a sum of products of radial functions and spherical harmonic functions,

( ) ( )

j jlm lm

lm

F =

∑

To save the labor in the calculation, we take into account the symmetry of the acceptor states.

In the absence of stress, the acceptor states transform like basis functions of the irreducible representations Γ , ₆ Γ , and ₇ Γ of the ₈ T_d^' group. In the presence of stress along the [001]

direction, the Γ state of the ₈ T_d^' group split into one Γ and one ₆ Γ states of the ₇ D_2d^'

group, and the Γ (₆ Γ ) state of the ₇ T_d^' group becomes the Γ (₆ Γ ) state of the ₇ D_2d^' group.

Both the Γ and the ₆ Γ states are doubly degenerate because of the time reversal symmetry. ₇ Furthermore, in spite of the lack of the inversion symmetry of the problem, all the envelope functions of an acceptor states have a common parity because the effective-mass Hamiltonian H has inversion symmetry about the impurity site. Therefore, the acceptor state can be

classi-fied to Γ⁺₆, Γ₆⁻, Γ⁺₇ ,and Γ₇⁻ states, where the even- (odd-) parity of the envelope function is denoted by the superscript + (−). For the even- (odd-) parity states, the sum in Eq. (3.7) over l runs over all nonnegative even (odd) integers; the sum over m runs over the integers

17

where n is an integer. Such a choice of the basis functions of the angular part of the envelope functions is equivalent to that in Ref. [34] although different in formulation. The radial part of the envelope function is expanded as

( )

jlm j

g r r c e_ν ^αν

ν

=

∑

where the numbers αν are chosen to form a geometric progression [26]. The value L is chosen according to the following rule

Having obtained the acceptor states, we can go on to calculate the electric-dipole transi-tions between these states. Considering the case that the temperature T=4.22 K, we suppose that all the holes are in the ground state 1Γ₈⁺ in the absence of stress, and in the 1Γ₆⁺ and the

1Γ7⁺ states, into which the 1Γ₈⁺ splits when a [001] stress is applied. Thus, the absorption coefficient of the electric-dipole transition between acceptor states can be written as

( )

( )

^{( )} ( ( ) ( ) )