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2.2 Wigner-Boltzmann equation

The Wigner equation introduced in Section 2.1 describes ballistic carrier transport. A widespread adoption of a semiconductor device technology requires operation at room temperature where phonon scattering plays an important role. One of the major advantages of the Wigner formalism is that the treatment of scattering is relatively simple compared to other quantum mechanical formalisms.

The phase-space description used in the Wigner formalism invites the use of the semi-classical scattering operator used in the Boltzmann equation. The addition of this semi-classical scattering model to the WTE was first proposed in [62], as an ad hoc solution to treating scattering in RTDs. This raised questions whether the use of semi-classical scattering is justified in the Wigner formalism. A rigorous derivation within the Wigner formalism for both phonon [41, 63] and impurity scattering [64], has shown that the semi-classical scattering models can be obtained as a limiting case of full quantum models. This derivation is outlined in the following.

2.2.1 Derivation

The Wigner equation can be generalized to account for electron-phonon interactions by considering the interactions of an electron with a quantum field; a classical field interaction is not sufficient [64]. A general system comprising both the electron and phonon populations is required, but under certain approximations the effects of the phonons on the electrons can be averaged. The phonon population is described by a collection of integers {ng }, giving the number of phonons (n) in the phonon mode g. This introduces two additional variables to the Wigner function: fw(r,k,{n },{n′} ,t) g g.

A general procedure to incorporate additional physical phenomena in the Wigner formalism is to i) express the appropriate Hamiltonian for the phenomenon to be described, ii) expand the commutator bracket in the Liouville-Von Neumann equation (2.6), iii) apply the Wigner transform and iv) perform mathematical calculations and physically-motivated approximations to obtain a tractable equation. In the following, this procedure for the derivation of the scattering models for phonon and impurity scattering in the Wigner formalism is outlined.

2.2.1.1 Phonon scattering

The Hamiltonian (1.19), describing a free electron, must be augmented to account for the free phonons and their interaction with the electron:

ˆH = ˆH + ˆH + ˆH {0 2 ph e- ph } ∑ ( ) ∑ ( ) = - -ℏ--∇2 + V (r) + ℏωg a†ag + 1- + iℏ F (g ) ageig⋅r + a†e-ig⋅r . (2.13) 2m * r g g 2 g g
The second term yields the energy of all phonons: ℏωg denotes the energy of a phonon with wavevector (mode) g; the creation and annihilation operators, denoted by ag† and ag, form the number operator ag†ag. The third term describes the energy exchanged between electrons and phonons: the function F (g) describes the coupling of electrons to phonons specific to the type of phonon scattering being considered (refer to Appendix A). Each contribution to the total Hamiltonian (2.13) can be considered separately and transformed; the additional terms arising from Ĥph and Ĥe-ph augment the Wigner equation (2.11).

The contribution of the second term in (2.13) yields a contribution to the RHS of (2.11) of

{ ({ })} ( { } ) Cph = -1 ϵ({ng})- ϵ n′g fw r,k, {ng}, n′g ,t , (2.14) iℏ
where ϵ({ng }) is the total energy of the phonons, given by
∑ ϵ({ng}) = ngℏωg. (2.15) g

The contribution of the third term in (2.13) gives rise to four terms of a similar form:

∑ ( ′){ ig′⋅r∘ ------- ( g′ + { ′} ) Ce-ph = F g e ng′ + 1fw r,k- 2-,{ng}g′ , ng ,t + g′ ( ) -ig′⋅r∘ -′----- g′ + +e ng′ + 1fw r,k - 2 ,{ng} ,{ng}g′ ,t - ( ′ { } ) - e-ig′⋅r√ng-′fw r,k + g-,{ng}-g′ , n′g ,t - ( 2 ) } ig′⋅r∘ --′ g-′ - - e n g′fw r,k + 2 ,{ng },{ng}g′ ,t . (2.16)
The first (last) two terms in the curly braces correspond to phonon creation (absorption). The notation {ng }g′± signifies that the integer corresponding to mode g′ is increased/decreased by one, i.e. ng′→ ng′± 1.

The addition of (2.14) and (2.16) to (2.11) gives a generalized Wigner equation describing the interaction of a single electron with a many-phonon system in a quantum mechanical manner. This equation, however, is computationally completely intractable: each term in the summation for a given mode g′ involves the Wigner function of mode ng′± 1 – a recursion is formed. Since the summation is over all modes g′ (infinite), a closure relation is needed to obtain some tractable form of the equation.

The aim is to obtain with appropriate assumptions a reduced Wigner function, fw(⋅,{n },{n } ,⋅) g g, which describes only the electron subsystem. The phonon subsystem is ’traced out’ such that only the first off-diagonal terms of the phonon population are considered.

The weak-coupling limit considers the electron interacting with only a single phonon, i.e. the time between two consecutive scattering events is assumed to be long enough such that the first event is completed before the next one starts. In the weak-coupling limit F(g) is assumed to be so small that F2 (g ) becomes negligible, i.e. the diagonal terms of the phonon states are considered (n = n′) to interact only with the the first off-diagonal elements. This implies that

( { ′} ) fw r,k, {ng}, n g ,t = 0, (2.17)
if |n - n′| > 1. Correlations and many-phonon processes are thus ignored.

The phonon distribution {n } g is determined using the assumption that phonons remain in equilibrium, regardless of the intensity of the interactions with the electron. The equilibrium distribution of phonons follows Bose-Einstein statistics, such that the probability of n phonons being in mode g is given by

( ) --1--- ℏωg- P (ng) = ¯ng + 1 exp - ng kBT , (2.18)
where ng denotes the mean occupation number, given by
n¯ = ∑ n P (n ) = ----1-----. (2.19) g n g g eℏω∕kBT - 1 g
The generalized Wigner function can be expressed as the product
∏ fw (r,k,{ng} ,{ng},t) = fw(r,k,t) P (ng), (2.20) g
under the assumption that the equilibrium conditions in the phonon distribution is instantaneously recovered after an interaction.

After performing a trace operation over all the phonon states, the transport equation for the reduced Wigner function can be expressed as an integral equation:

∫ t [∫ ( ) ( ) fw (r,k, t) = fw (r,k,0 )+ dt′ dkVw r,k ′ - k fw r,k′,t′ + {0 ( ′ ) + ∑ F 2(g′) eig′⋅rf r,k- g-,t′ ′ 1 2 g ( ′ ) } ] - e-ig′⋅rf2 r,k + g-,t′ + cc , (2.21) 2
where cc denotes the complex conjugate of the first two terms in the same braces. The auxiliary functions f1,2 account for the phonon emission/absorption processes that start and end on the diagonal element f1,2(r,k± q) 2 [41]. The simplification which has been gained is obvious in comparison to (2.16), where phonon modes infinitely far away from the diagonal are considered. Indeed, a substitution of f1,2 into (2.21) yields the desired closed equation for the reduced Wigner function.

Equation (2.21), known as the Levinson equation, is quantum mechanically correct in the weak-coupling limit and accounts for collisional broadening, intra-collisional field effect (ICFE) and retardation effects.

For the case of a constant homogeneous electric field, the auxiliary equation f1,2 can be solved and the summation of the phonon-scattering terms in (2.21) reduces to

∫ ∫ t ( ( ) ( ) ( ) ( )) dk ′ dt′S k′,k,t,t′ fw rp,q,t′,k′t′,t′ - S k, k′,t,t′ fw rp,q′,t′,kt′,t′ , (2.22) 0
where
( ′) kt′ = ℏk - eE t- t , (2.23) ∫ t ky - g∕2 rp,q′,t′ = r - ℏ ′ dy--m-*--. (2.24) t
The scattering rate is defined by
( ) 2V F2 (g ){ ( ( )) ( ( ))} S k′,k,t,t′ = ------3-- n (g)cos Ω k′,k,t,t′ + (n (g)+ 1)cos Ω k,k ′,t,t′ ; (2.25) ∫(2πℏ ) ( ′ ′) t ϵ-(k-τ)--ϵ(k′τ)-+-ℏω(k′-k) Ω k,k ,t,t = t′ dτ ℏ . (2.26)

An electron-phonon interaction does not occur instantaneously, but over a finite amount of time (as evidenced by the time integral in (2.21)). An electron slowly starts to ’feel’ the oscillations of phonons. This is approximately the time an electron requires to travel one wavelength of a phonon [7]. From a classical point of view, the collision duration can be defined as the amount of time an electron feels the presence of the scattering (phonon) field it interacts with. Energy is conserved by the end of the interaction. In quantum mechanics, however, the position of a particle is not precisely known and the collision duration must be defined according to the chosen uncertainty in energy (various definitions have been proposed for this [7]). The energy of the electron after a scattering event is no longer known exactly – this is termed collisional broadening. If the collision duration is longer than the mean time between scattering events, the scattering events are no longer independent. In other words, the uncertainty in position of an electron should be less than the mean free path, such that an electron is involved in only a single scattering event at a time.

During the duration of the electron-phonon interaction, external forces still act on the electron, thereby changing its wavevector. Therefore, the state of the electron changes (and thereby the scattering rate) during the collision – this is known as the ICFE. Modern semiconductor devices are subject to electric fields which are strong enough such that ICFE becomes significant and the scattering rates of instantaneous collisions do no longer apply [7].

The time scale of the phonon interaction appears in (2.26) as

′ ϵ(kτ)---ϵ(kτ)+-ℏ-ω(k′-k) (2.27) ℏ
and can be assumed to be much faster than the time scale associated with the electron dynamics, i.e. significant changes in the value of fw. This amounts to the assumption of instantaneous collisions, already mentioned for the Boltzmann collision operator in Chapter 1. In the classical limit (ℏ → 0) the time integration can be approximated and (2.25) takes the form
( ) V 2 π{ ( ( ) ) S k′,k,t,t′ = -3--- |ℏF (g)|2n(g) δ ϵ(k)- ϵ k′ - ℏωk′-k + ℏ ℏ ( ( ) )} + |ℏF (- g)|2(n(- g)+ 1) δ ϵ(k)- ϵ k′ + ℏωk′-k , (2.28)
which recovers the Boltzmann scattering term for phonon scattering and neglects the ICFE.

The above derivations illustrate well how the Wigner picture gives the opportunity to account for electron-phonon interactions at different levels, forming a hierarchy ranging from a full quantum mechanical description of scattering to the semi-classical description of scattering as used in the Boltzmann equation, which is more practical due to the much smaller computational demands. This presents a big advantage of using the Wigner formalism for scattering-aware quantum transport simulations.

2.2.1.2 Impurity scattering

An ionized impurity (dopant) exerts a force on an electron due to the Coulomb potential associated with it. To account for Coulomb interactions in microscopic simulations requires several considerations [65]. The Poisson equation appropriately models the long-range effects of a screened Coulomb potential in a continuous distribution of impurities (dopants). However, the short-range Coulomb interaction between an electron and an impurity is not captured by the Poisson equation alone. The short-range interaction is accounted for by a scattering mechanism.

The appropriate model in the Wigner formalism can be derived – as has been shown in [64] – by calculating the Wigner potential (2.10) using the Coulomb potential instead of the Hartree potential. The Coulomb potential of a discrete distribution of ionized dopants is given by

e2 ∑ exp(- β|r- rj|) Ve-ii(r) = 4πκ- ----|r--r-|----, (2.29) j j
where κ denotes the dielectric constant, β the screening factor (inverse of Debye length) and rj the position of dopant j. The corresponding evolution term emerges as
e2 ∫ ( ) 1 ∑ ( ′ ′ ) ---3-- dk′fw r,k ′ -------′2----2-× e- 2i(k- k)⋅(r-rj) - e2i(k-k)⋅(r-rj) (2.30) iℏπ κ 4(k - k) + β j
Under the assumption of a sufficiently high concentration of impurities, such that the sum can be approximated by an integral and the assumption of instantaneous collisions, this expression becomes identical to the one used in the Boltzmann equation to describe the impurity scattering (see Appendix A).

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