Boltzmann Kinematic Equation

The strategy for obtaining the non-equilibrium populations is as follows.

First we neglect the low-lying HH2P± and HH1S temporarily and solve the kinetics of the subsystem containing HH1 and LH1S in order to obtain the relation between f₂ and f_k, with considerations of phonon scattering within HH1 and the resonant transition between the continuous HH1 and the local-ized LH1S. This is justified because the resonant scattering is much faster than the decay through spontaneous emission from LH1S to HH2P±.[10]

Afterwards the occupation probability f₁ of HH2P± is determined by the its balance with non-equilibrium subband distribution fk through impact ionization, thermal recombination, and their inverse processes Auger recom-bination and thermal excitation. Detailed calculations are given below.

For a given number of holes in the subsystem containing HH1 and LH1S, the non-equilibrium distribution f_k in HH1 and occupation of LH1S f₂ are studied by solving the Boltzmann kinetic equation numerically for various electric fields and acceptor densities. In the subsystem the holes in HH1 acquire kinetic energy from the constant electric field F applied along the x axis. For moderate electric field and low temperature, it is adequate to adopt the concept of streaming motion[22] in which the only significant scattering is due to optical phonon (energy ¯hω₀). This is implemented by introducing a particle drain in momentum space such that once a specific hole drifts with velocity eF/¯h through the energy surface ε = ¯hω0 (denoted by Π) in the momentum space, the hole will experience a optical phonon scattering and simultaneously reemerge as a hole of energy less than ²₀.[10, 11] Hence fk = 0 for ε(k) ≥ ¯hω0. The energy ²0 is determined by the requirement

that in the presence of constant electric field F the probability for a hole being able to drift beyond the constant energy surface ε = ¯hω0+ ²0 without emitting one optical phonon is negligibly small. The quantity ²₀ is equal to the product of external force eF , carrier velocity √

2m^∗¯hω₀/¯h and inverse of the average optical phonon emitting rate νA. m^∗ stands for the effective mass. Note that the energy-independent optical phonon emitting rate is due to the constant density of states in two dimension. Therefore the excess energy can be expressed as

²₀ = eF νA

s2¯hω₀

m^∗ . (3.25)

The reemerging holes can be modeled as a particle source[10, 11]

S(k, t) =

e

¯

h[^R_Πf_k(t)F · dS]

[^R Θ(²₀− ε(k⁰))d²k⁰]Θ(²₀− ε(k)) , (3.26) where Θ is the step function. The meaning of the above expression is that the holes reemerging rate is uniform for energy within ²₀, and the total reemer-gence rate must match the collection of the outward carrier flux ^eF_¯h fkpassing through the surface Π in the momentum space.

In order to properly account for the temperature effects, we include the acoustic phonon scattering. The acoustic phonon scattering rate W_k,k^acu0 is of the form[23]

W_k,k^acu0 = 2πΞ²q²

%ω_qW A(n_q+1 2 ∓1

2)δ [ε(k⁰) − ε(k) ∓ ¯hωq] , (3.27) where % is the mass density of solid lattice and Ξ is the lattice deformation potential. The acoustic phonon involved in the transition has wave number q = k⁰−k and its dispersion is given by ωq = cq where c is the sound velocity in the solid. Emission and absorption of phonon in the processes correspond to + and − respectively. The product W A represents the QW volume.

We assume homogeneity in the x and y directions so that the distribution are function of variables k_x and k_y only. The set of kinetic equations can be written as resonant scattering. They are functionals of the the distribution functions.

The explicit expression for the collision terms are

C₁[f_k, f₂] = n_aA {W_k^res(f₂− f_k)} +^X The kinetic equations Eq. (3.28) and Eq. (3.29) are solved numerically by starting with the equilibrium distribution and then integrating forward in time until a steady state is reached. Note that the sum of densities n_af₂+

1 A

P

kfk is a conserved quantity in the time evolution, guaranteed by cance-lation of collision terms and the boundary conditions at surface Π. In this way not only the steady state but also the transient of the system can be modeled. The occupations of LH1S f2 and the HH1 fk are obtained up to an arbitrary total number of holes in the subsystem. In particular the relation between f₂and f_kat steady state can be readily seen by setting the left hand side of Eq. (3.29) equal to zero

f₂ = Now we consider the special case with no electric field. The subsystem is in thermal equilibrium. The occupations of HH1 and LH1S obey the Boltz-mann statistics guaranteed by the presence of delta function in the expression for resonant scattering as well as the fact that the scattering between HH1 states k and k⁰ due to acoustic phonon emission and absorption satisfies the relations

W_k^acu0_k W_kk^acu0

= 1 + nq

n_q = exp {−β [ε(k⁰) − ε(k)]} . (3.33) ε(k⁰) > ε(k) is assumed without loss of generality and q is the wavevector of the phonon involved in the process. Therefore in equilibrium f₂ is given by

f₂ = N/A

1 A

P

ke^−βε(k)+ n_ae^−βEr e^−βEr , (3.34) where N represents the total number of holes in the subsystem.

In order to describe the effect of the electric field on the distribution, we define a dimensionless parameter λ(F, T ) by

λ(F, T ) ≡

0 5 10 15 20 25 30 35 40 0

0.5 1 1.5 2 2.5

Incident kinetic energy (meV) Impact ionization rate (1012 nm2 sec−1)

HH2P±→HH1

HH1S→HH1

Figure 3.3: Impact ionization rates w^ip as functions of kinetic energy ε of incident subband hole for HH1S to HH1 and HH2P± to HH1 are respectively shown.

λ(F, T ) is the fraction of holes in HH1 for the subsystem. For low temper-ature at equilibrium virtually all holes stay near the HH1 minimum so λ is close to unity. In the presence of the electric field the population of LH1S increases as a consequence of Eq. (3.32), since holes in HH1 acquire kinetic energy from field so the non-equilibrium distribution fk has larger value at ε(k) = E_r. Therefore for given n_a, λ(F, T ) is expected to decrease as electric field increases. Increase of acceptor density n_a also raise f₂ because the dis-tribution in HH1 becomes more concentrated on ε(k) ≤ Er. This is because the stronger resonance scattering inhibits the holes to acquire energy higher than the resonance energy E_r.