Electron-longitudinal acoustic phonon interaction – Wikipedia

The electron-LA phonon interaction is an interaction that can take place between an electron and a longitudinal acoustic (LA) phonon in a material such as a semiconductor.

Displacement operator of the LA phonon[edit]

The equations of motion of the atoms of mass M which locates in the periodic lattice is

${displaystyle M{frac {d^{2}}{dt^{2}}}u_{n}=-k_{0}(u_{n-1}+u_{n+1}-2u_{n})}$

,

where

${displaystyle u_{n}}$

is the displacement of the nth atom from their equilibrium positions.

Defining the displacement

${displaystyle u_{ell }}$

of the

${displaystyle ell }$

th atom by

${displaystyle u_{ell }=x_{ell }-ell a}$

, where

${displaystyle x_{ell }}$

is the coordinates of the

${displaystyle ell }$

th atom and

${displaystyle a}$

is the lattice constant,

the displacement is given by

${displaystyle u_{l}=Ae^{i(qell a-omega t)}}$

Then using Fourier transform:

${displaystyle Q_{q}={frac {1}{sqrt {N}}}sum _{ell }u_{ell }e^{-iqaell }}$

and

${displaystyle u_{ell }={frac {1}{sqrt {N}}}sum _{q}Q_{q}e^{iqaell }}$

.

Since

${displaystyle u_{ell }}$

is a Hermite operator,

${displaystyle u_{ell }={frac {1}{2{sqrt {N}}}}sum _{q}(Q_{q}e^{iqaell }+Q_{q}^{dagger }e^{-iqaell })}$

From the definition of the creation and annihilation operator

${displaystyle a_{q}^{dagger }={frac {q}{sqrt {2Mhbar omega _{q}}}}(Momega _{q}Q_{-q}-iP_{q}),;a_{q}={frac {q}{sqrt {2Mhbar omega _{q}}}}(Momega _{q}Q_{-q}+iP_{q})}$

${displaystyle Q_{q}}$

is written as

${displaystyle Q_{q}={sqrt {frac {hbar }{2Momega _{q}}}}(a_{-q}^{dagger }+a_{q})}$

Then

${displaystyle u_{ell }}$

expressed as

${displaystyle u_{ell }=sum _{q}{sqrt {frac {hbar }{2MNomega _{q}}}}(a_{q}e^{iqaell }+a_{q}^{dagger }e^{-iqaell })}$

Hence, using the continuum model, the displacement operator for the 3-dimensional case is

${displaystyle u(r)=sum _{q}{sqrt {frac {hbar }{2MNomega _{q}}}}e_{q}[a_{q}e^{iqcdot r}+a_{q}^{dagger }e^{-iqcdot r}]}$

,

where

${displaystyle e_{q}}$

is the unit vector along the displacement direction.

Interaction Hamiltonian[edit]

The electron-longitudinal acoustic phonon interaction Hamiltonian is defined as

${displaystyle H_{text{el}}}$

${displaystyle H_{text{el}}=D_{text{ac}}{frac {delta V}{V}}=D_{text{ac}},div ,u(r)}$

,

where

${displaystyle D_{text{ac}}}$

is the deformation potential for electron scattering by acoustic phonons.^[1]

Inserting the displacement vector to the Hamiltonian results to

${displaystyle H_{text{el}}=D_{text{ac}}sum _{q}{sqrt {frac {hbar }{2MNomega _{q}}}}(ie_{q}cdot q)[a_{q}e^{iqcdot r}-a_{q}^{dagger }e^{-iqcdot r}]}$

Scattering probability[edit]

The scattering probability for electrons from

${displaystyle |krangle }$

to

${displaystyle |k’rangle }$

states is

${displaystyle P(k,k’)={frac {2pi }{hbar }}mid langle k’,q’|H_{text{el}}| k,qrangle mid ^{2}delta [varepsilon (k’)-varepsilon (k)mp hbar omega _{q}]}$

${displaystyle ={frac {2pi }{hbar }}left|D_{text{ac}}sum _{q}{sqrt {frac {hbar }{2MNomega _{q}}}}(ie_{q}cdot q){sqrt {n_{q}+{frac {1}{2}}mp {frac {1}{2}}}},{frac {1}{L^{3}}}int d^{3}r,u_{k’}^{ast }(r)u_{k}(r)e^{i(k-k’pm q)cdot r}right|^{2}delta [varepsilon (k’)-varepsilon (k)mp hbar omega _{q}]}$

Replace the integral over the whole space with a summation of unit cell integrations

${displaystyle P(k,k’)={frac {2pi }{hbar }}left(D_{text{ac}}sum _{q}{sqrt {frac {hbar }{2MNomega _{q}}}}|q|{sqrt {n_{q}+{frac {1}{2}}mp {frac {1}{2}}}},I(k,k’)delta _{k’,kpm q}right)^{2}delta [varepsilon (k’)-varepsilon (k)mp hbar omega _{q}],}$

where

${displaystyle I(k,k’)=Omega int _{Omega }d^{3}r,u_{k’}^{ast }(r)u_{k}(r)}$

,

${displaystyle Omega }$

is the volume of a unit cell.

${displaystyle P(k,k’)={begin{cases}{frac {2pi }{hbar }}D_{text{ac}}^{2}{frac {hbar }{2MNomega _{q}}}|q|^{2}n_{q}&(k’=k+q;{text{absorption}}),\{frac {2pi }{hbar }}D_{text{ac}}^{2}{frac {hbar }{2MNomega _{q}}}|q|^{2}(n_{q}+1)&(k’=k-q;{text{emission}}).end{cases}}}$

See also[edit]

References[edit]