« PreviousUpNext »Contents
Previous: 5.2 Two-particle Transition Rate    Top: 5 Electron-Electron Scattering    Next: 5.4 Implementation for Full-Band Structures

Investigating Hot-Carrier Effects using the Backward Monte Carlo Method

5.3 Single-particle Transition Rate

The Monte Carlo methods described above estimate the solution of the Boltzmann equation, which is the single-particle distribution function. In order to stay in this single-particle framework, the two-particle scattering rate has to be reduced to a single-particle scattering rate. For this purpose, some assumptions about the distribution of the partner-electrons have to be made. These assumptions can vary from case to case. One can assume an equilibrium distribution or a more realistic self-consistent distribution for the partner electrons.

The total transition rate for a single-particle can be obtained by summing over all initial states \( \kv _2 \) and final states \( \kv _2’ \) of the partner electrons [P5].

(5.17) \{begin}{align} P_1(\kv _1,\kv _1’) = \sum \limits _{\kv _2’}\sum \limits _{\kv _2}\,P_2(\kv _1,\kv _2,\kv _1’,\kv _2’)\, [1-f(\kv _2’)]\, 2f(\kv _2) \label {eq:sp-trans-1} \{end}{align}

Here, \( f \) is the distribution function of the partner electrons. The factor \( 2 \) in front of the distribution \( f(\kv _2) \) results from spin degeneracy. This means that, every state \( \kv _2 \) can be occupied by two electrons. One summation in (5.17) can be evaluated using the Kronecker-delta. The other summation is converted to an integral taking into account the density of states, \( \Omega /(2\pi )^3 \). Therefore, the transition rate becomes

(5.18) \{begin}{multline} P_1(\kv _1,\kv _1’) = \frac {2\pi }{\hbar } \left ( \frac {e^2}{\epsilon _s\Omega } \right )^2 \frac {\Omega }{(2\pi )^3} \\ \times \int d^3k_2 \frac {\delta \bigl [\Epsilon (\kv _1’) + \Epsilon (\kv _1 +\kv _2 - \kv _1’) - \Epsilon (\kv _1) -\Epsilon (\kv _2)\bigr ]} {\left (\abs {\kv _1 - \kv _1’} + \beta _s^2\right )^2} \\ \times \left [ 1-f(\kv _1+\kv _2-\kv _1’) \right ]\,2f(\kv _2) \label {eq:sp-trans-2} \{end}{multline}

Additionally, we define the differential transition rate \( S \).

(5.19) \{begin}{align} S_1(\kv _1,\kv _1’) = \frac {\Omega }{(2\pi )^3} P_1(\kv _1,\kv _1’) . \label {eq:sp-trans-3} \{end}{align}

We define the momentum transfer \( \qv \) as

(5.20) \{begin}{align} \qv = \kv _1’-\kv _1 = \kv _2 - \kv _2’ \label {eq:momentum-transfer} \{end}{align}

and introduce the function \( w \).

(5.21) \{begin}{align} w(\kv _2,\qv ) = f(\kv _2) [1-f(\kv _2-\qv )] \{end}{align}

Using these definitions, the differential transition rate (5.19) becomes

(5.22) \{begin}{multline} S_1(\kv _1,\kv _1+\qv ) = \\ \frac {2e^4}{(2\pi )^5\hbar \epsilon _s^2} \int \frac {\delta \bigl [\Epsilon (\kv _1 + \qv ) + \Epsilon (\kv _2 - \qv ) - \Epsilon (\kv _1) -\Epsilon (\kv _2)\bigr ]} {\left (q^2 + \beta _s^2\right )^2} \, w(\kv _2,\qv )\, d^3k_2 . \label {eq:sp-trans-4} \{end}{multline}

Note that \( S_1 \) is independent of the normalization volume \( \Omega \).

5.3.1 Spin Degeneracy and Normalization

The electron concentration \( n \) is given by:

(5.23) \{begin}{align} n =\frac {2}{(2\pi )^3}\int f(\kv ) d^3k \{end}{align}

The factor \( 2 \) accounts for spin degeneracy. We introduce the normalized distribution function \( p_0 \),

(5.24) \{begin}{align} p_0(\kv ) = \frac {f(\kv )}{\int f(\kv )\,\dint ^3 k}=\frac {2\,f(\kv )}{(2\pi )^3\,n} \{end}{align}

and define \( \mean {w} \) as

(5.25) \{begin}{align} \mean {w}(\kv _2,\qv ) = p_0(\kv )[1-f(\kv _2-\qv )] \,. \label {eq:w-mean-def} \{end}{align}

Thus, the differential transition rate (5.22) becomes

(5.26) \{begin}{multline} S_1(\kv _1,\kv _1+\qv ) = A \int \frac {\delta \bigl [\Epsilon (\kv _1 + \qv ) + \Epsilon (\kv _2 - \qv ) - \Epsilon (\kv _1) -\Epsilon (\kv _2)\bigr ]} {\left (q^2 + \beta _s^2\right )^2} \mean {w}(\kv _2,\qv ) d^3k_2 , \label {eq:sp-spin-trans-rate} \{end}{multline}

where the pre-factor \( A \) is of the form

(5.27) \{begin}{align} A=\frac {ne^4}{(2\pi )^2\hbar \epsilon _s^2} \,. \label {eq:pre-fac-A} \{end}{align}

The transition rate is proportional to the electron density \( n \) and does not contain the spin degeneracy factor.

5.3.2 A Model for the Partner Electrons

The scattering rate (5.22) depends on the unknown distribution function \( f(k) \). Therefore, a Boltzmann equation including this scattering rate will be nonlinear. In this work, the partner electrons are assumed to be in thermal equilibrium, described by the equilibrium distribution \( f_0 \). The equilibrium distribution can be either a Fermi-Dirac or a Maxwell-Boltzmann distribution. With this assumption, the Boltzmann equation will be linear. This assumption is valid if hot carriers in the highly doped drain region are investigated. However, by fixing the distribution of the partner electron the heating of the partner electrons due to hot carriers is neglected.

Another assumption of our model is, that the cold partner electrons are described with a parabolic and isotropic dispersion relation.

(5.28) \{begin}{align} \Epsilon (\kv ) = \frac {\hbar ^2 \kv ^2}{2m} \label {eq:energy-parabolic} \{end}{align}

Here, \( \kv \) is the wavevector relative to the valley minimum located at \( \kv _0 \)

(5.29) \{begin}{align} \kv = \kv _{FB} - \kv _0\„ \{end}{align}

where \( \kv _{FB} \) represents the wave vector in the Brillouin zone, relative to the \( \Gamma \)-point.

5.3.3 Principle of Detailed Balance

In the single-particle picture, EES is no longer an elastic process. However, it can be shown that this process still satisfies the principle of detailed balance. With the following relation of the Fermi-Dirac distribution \( f_0 \)

(5.30) \{begin}{align} \frac {1-f_0(\Epsilon )}{f_0(\Epsilon )} = \e ^{\beta (\Epsilon -E_F)} \label {eq:f0-FD} \{end}{align}

the term of (5.17) can be reformulated as

(5.31) \{begin}{align} [1-f_0(\Epsilon _2’)]\,f_0(\Epsilon _2) = [1-f_0(\Epsilon _2’)] \, [1-f_0(\Epsilon _2)] \, \e ^{-\beta (\Epsilon _2-E_F)} . \label {eq:sp-db-1} \{end}{align}

Using (5.31) and the energy balance equation

(5.32) \{begin}{align} \Epsilon _2 = \Epsilon _2’ + \Epsilon _1’ - \Epsilon _1 \{end}{align}

one can reformulate the transition rate (5.17) as

(5.33) \{begin}{align} P_1(\kv _1,\kv _1’) = \e ^{\beta (\Epsilon _1 - \Epsilon _1’)} \sum \limits _{\kv _2,\kv _2’}\,P_2(\kv _1,\kv _2,\kv _1’,\kv _2’)\, [1-f_0(\Epsilon _2’)] \, [1-f_0(\Epsilon _2)] \, \e ^{-\beta (\Epsilon _2’-E_F)} . \label {eq:trans-rate-det-bal} \{end}{align}

In (5.33) we interchange the variables \( \kv _1, \kv _2 \) and \( \kv _1’, \kv _2’ \) and employ the symmetry property (5.15).

(5.34) \{begin}{align} P_1(\kv _1’,\kv _1) &= \e ^{\beta (\Epsilon _1’ - \Epsilon _1)} \sum \limits _{\kv _2,\kv _2’}\,P_2(\kv _1’,\kv _2’,\kv _1,\kv _2)\, [1-f_0(\Epsilon _2)] \, [1-f_0(\Epsilon _2’)] \, \e ^{-\beta (\Epsilon _2-E_F)} \\ \nonumber &= \e ^{\beta (\Epsilon _1’ - \Epsilon _1)} P_1(\kv _1,\kv _1’) \{end}{align}

This equation shows that \( P_1 \) satisfies the principle of detailed balance:

(5.35) \{begin}{align} P_1(\kv _1’,\kv _1)\,\e ^{-\beta \,\Epsilon _1’} = P_1(\kv _1,\kv _1’)\,\e ^{-\beta \,\Epsilon _1}. \{end}{align}

5.3.4 Angular Integration of the Transition Rate

The energy transfer of both involved particles is defined by

(5.36–5.37) \{begin}{align} \Delta _1(\kv _1,\qv ) &= \Epsilon (\kv _1+\qv ) - \Epsilon (\kv _1)\„\\ \Delta _2(\kv _2,\qv ) &= \Epsilon (\kv _2-\qv ) - \Epsilon (\kv _2)\,. \{end}{align}

Thus, energy conservation of one scattering event can be formulated as

(5.38) \{begin}{align} \Delta _1(\kv _1,\qv ) + \Delta _2(\kv _2,\qv ) = 0 \{end}{align}

Note, that the equilibrium distribution \( f_0(\Epsilon ) \) is a function of energy, and so is function \( \mean {w} \), defined by (5.25).

(5.39) \{begin}{align} \mean {w}_2(\Epsilon _2,\Delta _1) = p_0(\Epsilon _2) \left [ (1-f_0(\Epsilon _2-\Delta _1) \right ] \{end}{align}

The transition rate (5.26) can be expressed in terms of \( \Delta _1 \) and \( \Delta _2 \) as follows:

(5.40) \{begin}{align} \label {eq:EES-angular-TR} S_1(\kv _1,\kv _1+\qv ) = A \int \frac {\delta \bigl [\Delta _1(\kv _1,\qv ) + \Delta _2(\kv _2,\qv )\bigr ]} {\left (q^2 + \beta _s^2\right )^2} \,\mean {w}_0(\Epsilon _2,\Delta _1)\, d^3k_2 . \{end}{align}

The \( \kv _2 \)-integration

(5.41) \{begin}{align} I(\kv _1,\qv ) = \int \delta \bigl [\Delta _1(\kv _1,\qv ) + \Delta _2(\kv _2,\qv )\bigr ] \mean {w}_0(\Epsilon _2,\Delta _1)\, d^3k_2 \label {eq:EES-k2-angular-int} \{end}{align}

can be evaluated in spherical polar coordinates, where \( \qv \) is the polar axis. With the parabolic band approximation (5.28) the energy difference for the partner electron can be expressed as

(5.42) \{begin}{align} \Delta _2(\kv ,\qv )=\Epsilon (\kv _2 - \qv ) - \Epsilon (\kv _2) & = \frac {\hbar ^2}{2m} \left (q^2 - 2\, k_2\, q\, \cos \vartheta \right )\,. \label {eq:tr-energy-diff} \{end}{align}

Substitution of (5.42) in (5.41) gives

(5.43) \{begin}{align} I(\kv _1,\qv ) &= 2\pi \int _0^\pi \d \vartheta \,\sin \vartheta \int _0^\infty \d k_2 \,k_2^2\, \delta \left [\Delta _1 + \frac {\hbar ^2}{2m} \biggl (q^2 - 2\, k_2\, q\, \cos \vartheta _2\biggr )\right ]\, \mean {w}_0(\Epsilon _2,\Delta _1) \{end}{align}

In the next step we substitute \( \chi = \cos \vartheta \) and define the wave number \( \kappa \) as

(5.44) \{begin}{align} \kappa = \frac {m}{\hbar ^2} \frac {\Delta _1}{q} + \frac {q}{2}\„ \label {eq:kappa} \{end}{align}

to obtain

(5.45) \{begin}{align} I(\kv _1,\qv ) &=2\pi \int \limits _0^\infty \,\dint k_2\,k_2^2\,\mean {w}_0(\Epsilon _2,\Delta _1)\,\int _{-1}^1 \delta \left (\frac {\hbar ^2\,k_2\,q}{m} \left (\frac {\kappa }{k_2}-\chi \right ) \right )\, \d \chi \label {eq:angular-int-1} \{end}{align}

The \( \chi \)-integration can be carried out using the \( \delta \)-function.

(5.46) \{begin}{multline} \int _{-1}^1 \delta \left (\frac {\hbar ^2\,k_2\,q}{m} \left (\frac {\kappa }{k_2}-\chi \right ) \right )\, \d \chi = \frac {m}{\hbar ^2 q k_2} \int _{-1}^1 \delta \left (\frac {\kappa }{k_2} - \chi \right )\, \d \chi \\ = \frac {m}{\hbar ^2 q k_2} \left (\Theta (\kappa + k_2) - \Theta (\kappa - k_2)\right ) = \frac {m}{\hbar ^2 q k_2}\; \Theta (k_2 - \abs {\kappa }) \{end}{multline}

Here, \( \Theta \) is the unit step function. The integral (5.45) now simplifies to

(5.47) \{begin}{align} I(\kv _1,\qv ) = 2\pi \frac {m}{\hbar ^2 q} \int \limits _{\abs {\kappa }}^\infty \d k_2 \;k_2\; \mean {w}_0(\Epsilon _2,\Delta _1)\,. \label {eq:angular-int-2} \{end}{align}

The arguments \( \kv _1 \) and \( \qv \) enter the expression via the lower integration limit \( \abs {\kappa (\kv _1,\qv )} \) and the argument \( \Delta _1 \). With (5.47), the transition rate (5.40) becomes:

(5.48) \{begin}{align} S_1(\kv _1,\kv _1+\qv ) = B\frac {\beta _s^2}{q\left (q^2 + \beta _s^2\right )^2} \int \limits _{\abs {\kappa }}^\infty \mean {w}_0(\Epsilon _2,\Delta _1)\,k_2 \,d^3k_2 , \label {eq:partial-trans-rate} \{end}{align}

with the pre-factor B defined as

(5.49) \{begin}{align} B=2\pi \,A\frac {m}{\hbar ^2\,\beta _s^2}=\frac {n\,e^4\, m}{2\pi \,\hbar ^3\,\epsilon _s^2\,\beta _s^2} \,. \label {eq:pre-fac-B} \{end}{align}

5.3.5 Transition Rate for Boltzmann Statistics

To simplify notation we introduce the reduced Fermi energy \( \eta \) and the thermal wave number \( \tau \).

(5.50–5.51) \{begin}{align} \eta &= \beta E_F = \frac {E_F}{k_BT} \label {eq:scale-eta}\\ \tau ^2 &= \frac {2 m k_BT}{\hbar ^2}\label {eq:scale-tau} \{end}{align}

To obtain the normalized distribution \( p_0 \), one has to calculate the the normalization factor \( C_\mathrm {MB} \) for the Maxwell-Boltzmann distribution, as shown in Appendix A.4.

(5.52) \{begin}{align} \label {eq:CMB} p_0(\kv )&= \frac {f_0(\kv )}{C_\mathrm {MB}}=\frac {\e ^{-k^2/\tau ^2}}{\pi ^{3/2}\tau ^3} \{end}{align}

Note, that the normalized Maxwell-Boltzmann distribution is independent of the Fermi level \( \eta \). The integral in the partial evaluated transition rate (5.48) can be evaluated as shown in Appendix A.4:

(5.53) \{begin}{align} \int \limits _{\abs {\kappa }}^\infty \mean {w}_0(\Epsilon _2,\Delta _1)\,k \,dk = \frac {\e ^{-\kappa ^2/\tau ^2}}{2\pi ^{3/2}\,\tau } \label {eq:int-sp-trans-MB} \{end}{align}

Using (5.53) and denoting the final state as \( \kv _1’ = \kv _1 + \qv \), the equation of the transition rate (5.48) becomes:

(5.54) \{begin}{align} S_1(\kv _1, \kv _1’) = \frac {C}{2\pi }\; \frac {\beta _s^2}{q \left (q^2 + \beta _s^2\right )^2} \, \e ^{-\kappa ^2/\tau ^2} \„\label {eq:S1} \{end}{align}

with

(5.55–5.56) \{begin}{align} q&= \abs {\kv _1’ - \kv _1} \qquad \mathrm {and}\\ C&=\frac {\hbar }{\sqrt {2\pi \,m\,k_BT}}B\,. \label {eq:pf-C} \{end}{align}

From the definitions (5.44) and (5.51) we obtain

(5.57) \{begin}{align} \frac {\kappa ^2}{\tau ^2} &= \frac {\beta (E_q + \Delta _1)^2}{4\,E_q}\label {eq:k2t2} \{end}{align}

with

\[ E_q = \frac {\hbar ^2\,q^2}{2m}\,. \]

Consequently, the transition rate (5.54) can be reformulated to:

(5.58) \{begin}{align} S_1(\kv _1, \kv _1’) = \frac {C}{2\pi }\; \frac {\beta _s^2}{q \left (q^2 + \beta _s^2\right )^2} \times \exp \left (-\frac {(\Epsilon (\kv _1’) - \Epsilon (\kv _1) + E_q)^2}{4 E_q k_B T}\right )\,. \label {eq:S1-reformulated} \{end}{align}

5.3.6 Transition Rate for Fermi Dirac Statistics

For Fermi Dirac statistics, the normalized distribution \( p_0 \) can be obtained by calculating the normalization factor \( C_\mathrm {FD} \), see Appendix A.5

(5.59) \{begin}{align} p_0(\kv )&=\frac {f_0(\kv )}{C_\mathrm {FD}}=\frac {1}{\pi ^{3/2}\,\tau ^3\,\mathcal {F}_{1/2}(\eta ) \left ( \e ^{k^2/\tau ^2-\eta }\,+1 \right )} \label {eq:p0-FD} \{end}{align}

where \( \mathcal {F}_{1/2} \) denotes the Fermi integral of order \( 1/2 \). The integral in the partial evaluated transition rate (5.48) can be evaluated by substituting \( \mean {w}=w/C_\mathrm {FD} \) as shown in Appendix A.5:

(5.60) \{begin}{align} \int \limits _{\abs {\kappa }}^\infty \mean {w}_0(\Epsilon _2,\Delta _1)\,k \,d^3k &= \left (\frac {\tau ^2\e ^\eta }{2C_\mathrm {FD}} \right ) \frac {\e ^{-\eta }}{1-\e ^{\beta \Delta _1}}\ln \left ( \frac {1+\e ^{\eta -\kappa ^2/\tau ^2}}{1+\e ^{\eta +\beta \Delta _1-\kappa ^2/\tau ^2}}\right ) \label {eq:partial-trans-rate-2} \{end}{align}

With (5.60), the scattering rate (5.48) becomes

(5.61) \{begin}{align} S_1(\kv _1, \kv _1 + \qv ) = \frac {C}{2\pi }\frac {\e ^\eta }{\mathcal {F}_{1/2}(\eta )} \frac {\beta _s^2}{q \left (q^2 + \beta _s^2\right )^2} \frac {\e ^{-\eta }}{1-\e ^{\beta \Delta _1}}\ln \left ( \frac {1+\e ^{\eta -\kappa ^2/\tau ^2}}{1+\e ^{\eta +\beta \Delta _1-\kappa ^2/\tau ^2}}\right )\,. \label {eq:S1-FD} \{end}{align}

Finally, with (5.57) and the relation

(5.62) \{begin}{align} \frac {\kappa ^2}{\tau ^2} -\beta \Delta _1 &= \frac {\beta (E_q - \Delta _1)^2}{4\,E_q}\label {eq:k2t2-bD} \„ \{end}{align}

the scattering rate (5.61) can be reformulated to:

(5.63) \{begin}{align} S_1(\kv _1, \kv _1 + \qv ) = \frac {C}{2\pi }\frac {\e ^\eta }{\mathcal {F}_{1/2}(\eta )} \frac {\beta _s^2}{q \left (q^2 + \beta _s^2\right )^2} \frac {\e ^{-\eta }}{1-\e ^{\beta \Delta _1}}\ln \left ( \frac {1+\e ^{\eta -\frac {\beta (E_q + \Delta _1)^2}{4\,E_q}}}{1+\e ^{\eta -\frac {\beta (E_q - \Delta _1)^2}{4\,E_q}}}\right ) \label {eq:S1-FD-2} \{end}{align}

Boltzmann Limit

For small carrier concentration the Boltzmann limit must be recovered. We consider the limit \( \eta \rightarrow -\infty \). In this regime the Fermi-integral behaves as \( \e ^\eta \). It holds

(5.64) \{begin}{align} \lim \limits _{\eta \rightarrow -\infty } \frac {\e ^\eta }{\mathcal {F}_{1/2}(\eta )} = 1 \,. \{end}{align}

The limit of the following expression is calculated using the rule of l’Hôpital.

(5.65–5.66) \{begin}{align} &\lim \limits _{\eta \rightarrow -\infty } \frac {\ln \left ( 1+\e ^{\eta -\kappa ^2/\tau ^2} \right ) - \ln \left ( 1+\e ^{\eta +\beta \Delta _1-\kappa ^2/\tau ^2} \right )}{\e ^\eta } =\\ &\lim \limits _{x \rightarrow 0} \frac {\ln \left ( 1+x\,\e ^{-\kappa ^2/\tau ^2} \right ) - \ln \left ( 1+x\,\e ^{\beta \Delta _1-\kappa ^2/\tau ^2} \right )}{x} = \e ^{-\kappa ^2/\tau ^2}\left ( 1-\e ^{\beta \Delta _1} \right ) \{end}{align}

Taking these limits, the scattering rate (5.61) simplifies to the Boltzmann result (5.54).

« PreviousUpNext »Contents
Previous: 5.2 Two-particle Transition Rate    Top: 5 Electron-Electron Scattering    Next: 5.4 Implementation for Full-Band Structures