PHYSICAL REVIEW B 92, 214111 (2015)

Theory of inelastic multiphonon scattering and carrier capture by defects in semiconductors:Application to capture cross sections

Georgios D. Barmparis,1,2 Yevgeniy S. Puzyrev,1,* X.-G. Zhang,3 and Sokrates T. Pantelides1,4,5

1Department of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235, USA2Crete Center for Quantum Complexity and Nanotechnology, Department of Physics, University of Crete, 71003 Heraklion, Greece

3Department of Physics and the Quantum Theory Project, University of Florida, Gainesville, Florida 32611, USA4Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

5Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, Tennessee 37235, USA(Received 20 August 2015; published 21 December 2015)

Inelastic scattering and carrier capture by defects in semiconductors are the primary causes of hot-electron-mediated degradation of power devices, which holds up their commercial development. At the same time, carriercapture is a major issue in the performance of solar cells and light-emitting diodes. A theory of nonradiative(multiphonon) inelastic scattering by defects, however, is nonexistent, while the theory for carrier capture bydefects has had a long and arduous history. Here we report the construction of a comprehensive theory ofinelastic scattering by defects, with carrier capture being a special case. We distinguish between capture underthermal equilibrium conditions and capture under nonequilibrium conditions, e.g., in the presence of an electricalcurrent or hot carriers where carriers undergo scattering by defects and are described by a mean free path.In the thermal-equilibrium case, capture is mediated by a nonadiabatic perturbation Hamiltonian, originallyidentified by Huang and Rhys and by Kubo, which is equal to linear electron-phonon coupling to first order. Inthe nonequilibrium case, we demonstrate that the primary capture mechanism is within the Born-Oppenheimerapproximation (adiabatic transitions), with coupling to the defect potential inducing Franck-Condon electronictransitions, followed by multiphonon dissipation of the transition energy, while the nonadiabatic terms are ofsecondary importance (they scale with the inverse of the mass of typical atoms in the defect complex). We reportfirst-principles density-functional-theory calculations of the capture cross section for a prototype defect usingthe projector-augmented wave, which allows us to employ all-electron wave functions. We adopt a Monte Carloscheme to sample multiphonon configurations and obtain converged results. The theory and the results representa foundation upon which to build engineering-level models for hot-electron degradation of power devices andthe performance of solar cells and light-emitting diodes.

DOI: 10.1103/PhysRevB.92.214111 PACS number(s): 72.20.Jv, 72.10.Di, 72.20.Ht

I. INTRODUCTION

Elastic scattering of electrons by phonons, impurities, andother defects limits the conductivity in metals and the carriermobility in semiconductors. The fundamental theory is wellestablished, parameter-free mobility calculations have becomepossible [1,2], and engineering-level modeling methods arewidely available. Inelastic scattering of hot electrons by defectshas long been known to cause device degradation. For example,hot electrons in Si-SiO2 structures can transfer energy andrelease hydrogen from passivated interfacial Si danglingbonds [3,4]. More recently, it was found that hot electronscause degradation of power devices based on wide-band-gapsemiconductors [5]. It has been shown that the degradationis caused by hot-electron-mediated release of hydrogen fromhydrogenated defects such as Ga vacancies or impurities [6].In other cases, carrier capture transforms benign defects tometastable configurations that cause recoverable degradation[7]. Similarly, nonradiative carrier capture by defects, whichis a special case of inelastic scattering, limits the perfor-mance of photovoltaic cells, light-emitting diodes, and otherdevices [8,9].

A theory of inelastic scattering by defects by multiphononprocesses (MPPs) does not exist while the theory of non-

*yevgeniy.s.puzyrev@vanderbilt.edu

radiative carrier capture or emission by defects by MPPshas a long and controversial history. In 1950, Huang andRhys [10] reported a theory of how the energy of latticerelaxation that accompanies the photoionization of a defectis dissipated by MPPs. The process was described withinthe Born-Oppenheimer or adiabatic approximation (BOA)and the Franck-Condon approximation (FCA). The formersays that electronic and nuclear (vibrational) wave functionsobey decoupled equations. The latter states that an electronicexcitation occurs instantaneously and relaxation processesfollow at a relatively slow pace, allowing one to write theexcitation rate (Fermi’s golden rule) as the product P = AF ,where A describes the instantaneous electronic excitation inthe initial lattice configuration and F, the so-called line-shape function, describes the MPPs that occur during latticerelaxation. In the Huang-Rhys theory, the operator that causesthe excitation is strictly the photon field and MPPs dissipateonly the energy of the ensuing lattice relaxation.

In the same paper, Huang and Rhys [10] also proposeda theory for nonradiative multiphonon transitions betweendefect levels. Such transitions are caused by the terms thatare dropped when the BOA is made, namely, derivatives of theelectronic wave functions with respect to nuclear positions(nonadiabatic terms). In 1952, Kubo [11] independentlyinvoked the same nonadiabatic terms as being responsible forthe thermal ionization of a defect. In subsequent years, Kuboand Toyozawa [12] and later Gummel and Lax [13] adopted

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BARMPARIS, PUZYREV, ZHANG, AND PANTELIDES PHYSICAL REVIEW B 92, 214111 (2015)

Kubo’s formalism to explore carrier capture and emissionusing analytical approximations. Kovarskii and Sinyavskii[14–16] published several papers expanding on Kubo’s for-malism. In 1977, in search of a practical scheme to modelelectron capture in experiments, Henry and Lang [17] adopteda Huang-Rhys analog: the electronic transition is causedinstantaneously by the perturbation potential �V generatedby atomic vibrations—the linear electron-phonon couplingpotential that is normally thought to cause elastic scatteringand is used for mobility calculations [18]. The followingyear, Ridley [19] showed that the Henry-Lang model exhibitsthe correct temperature dependence at high temperatures (thesemiclassical limit) but pointed out that the correct wayto calculate nonradiative capture cross sections is throughthe nonadiabatic perturbation terms identified by Huang andRhys [10] and by Kubo [11]. In 1981, however, Huangshowed that the nonadiabatic perturbation Hamiltonian and thelinear electron-phonon coupling perturbation Hamiltonian areequivalent to first order [20]. The issue whether such a first-order calculation is adequate remained open as, throughoutthe years of all these developments, only model calculationswere pursued, largely analytical, employing model defectwave functions. Furthermore, calculations of the line-shapefunction were typically restricted by the assumption thata single vibrational mode contributes to the MPPs. In thechemical literature, nonradiative transitions between molec-ular orbitals have been studied [21,22]. It was recognized thatthe inclusion of all vibrational modes in the MPP calculationleads to exploding computational requirements as the sizeof the molecule increases [21]. The so-called parallel-modeapproximation or simply a single vibrational mode is typicallyused [22].

The first application of modern density-functional-theory(DFT) calculations to MPPs in the case of luminescence,i.e., the classic Huang-Rhys problem, where an electronictransition is caused by the photon field and MPPs dissipatethe ensuing lattice relaxation, was reported by Alkauskaset al. [23]. Those authors studied the luminescence spectraof defects in GaN employing DFT pseudowave functions forthe electronic matrix elements and the single-phonon-modeapproximation to the Huang-Rhys line-shape function. In amore recent paper, Alkauskas et al. [24] reported calculationsof nonradiative capture of electrons by defects using thelinear electron-phonon coupling perturbation Hamiltonian,pseudowave functions, and a single-phonon mode to calculatethe MPPs that dissipate the transition energy. They pointedout that the electronic transition is a slow process becausecapture is mediated by the phonons that are localized aroundthe defect.

In this paper we first revisit the theory of carrier captureby defects. We identify two distinct regimes that are governedby different processes. One is carrier capture under thermal-equilibrium conditions; i.e., capture occurs in tandem withemission with electrons in the conduction band (or holes in thevalence band) are not being accelerated. Under these condi-tions, capture and emission are inverse processes; i.e., the roleof the initial and final states is reversed. For an electron boundat a defect, emission amounts to a transition to a band statethat is an eigenstate of the same Hamiltoninan (perfect crystalplus defect potential). Band states are occupied according to

the Fermi-Dirac distribution function. Any of these carrierscan be captured into the defect’s ground state. Under suchconditions, band carriers are effectively undergoing diffusiveBrownian motion. In this case, the Huang-Rhys-Kubo (HRK)nonadiabatic Hamiltonian perturbation is the only possiblecause of these thermal transitions.

Under nonequilibrium conditions, however, e.g., in thepresence of an electrical current, carriers are accelerated in aspecific direction and a mean free path is defined by scatteringevents. It is then standard procedure to treat the band electronsas being in eigenstates of the perfect crystal Hamiltonian andconsider scattering by the defects. In particular, one considerselastic scattering by defects as a mechanism that limits thecarrier mobility. In this case, the initial and final states areeigenstates of the perfect crystal Hamiltonian and the defectpotential acts as the perturbation that causes the transitions,i.e., the defect potential is “turned on” in order to use time-dependent perturbation theory and arrive at Fermi’s goldenrule. Clearly, hot carriers can undergo inelastic scattering aswell, dropping to a Bloch state of lower energy, with the energydissipated by MPPs. For such calculations, one must againturn on the defect potential, though the HRK nonadiabaticperturbation must also be included. Transitions caused by thedefect potential are within the BOA, whereas those caused bythe HRK perturbation Hamiltonian are nonadiabatic. Finally,under such nonequilibrium conditions, carrier capture can beviewed as a special case of inelastic scattering: if the defectpotential can cause elastic scattering and inelastic scatteringwith energy dissipation via MPPs, then it certainly should alsobe included as a cause of capture.

In the capture case, however, there is a subtle difficulty.In order to derive a transition rate using Fermi’s goldenrule, initial and final states must be eigenstates of the sameHamiltonian. In the carrier capture case, however, the finalstate is an eigenstate of the crystal Hamiltonian plus the defectpotential, whereas the initial state is an eigenstate of theperfect-crystal Hamiltonian. The difficulty can be overcomeif we prepare a propagating state for the incoming electronthat is not aware of the bound state’s existence, with capturebeing triggered by the sudden turning-on of a suitable coupling(initial and final states must belong to the same Hamiltonian forthe concept of a transition to be meaningful). Such adiabatictransitions have not been considered so far in the context ofmultiphonon transitions at defects in semiconductors, but theyare commonly invoked in chemistry for electron transitions inmolecules [25–27].

We develop a comprehensive theory of inelastic scatteringand capture for transitions caused both by the defect po-tential (adiabatic transitions) and by the nonadiabatic HRKperturbation Hamiltonian. We show that, for carrier capture,adiabatic transitions are the zeroth-order term in an expansionin the defect-atom displacements that following capture (latticerelaxation) and are, therefore, dominant under nonequilibriumconditions. The electronic transition is caused instantaneouslyby the defect potential (it is effectively a Franck-Condontransition) and the energy is dissipated by MPPs. The nextorder in the series, which is linear in the atomic displacements,comprises two terms, only one of which has been captured byprior theories [20,24,28]. We estimate that these “linear terms”make smaller contributions to the capture rate as they scale

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with 1/m, where m is a typical nuclear mass in the defectcomplex. The adiabatic perturbation Hamiltonian that couplesthe incoming electron to the defect is constructed in terms ofHamiltonian matrices as in the Forster theory of electron andexciton transfer in molecules [25], which allows the derivationof Fermi’s golden rule for these transitions.

In addition to presenting the basic elements of the funda-mental theory, we report explicit calculations for capture crosssections as functions of energy transfer for a prototype defectusing DFT for the electronic matrix elements. We employthe projector-augmented wave (PAW) scheme [29], whichallows the use of the all-electron defect potential and wavefunctions as opposed to pseudopotentials and pseudo–wavefunctions. For calculation of the line-shape function, weintroduce a Monte Carlo scheme to sample the space of phononcombinations that contribute to the MPP energy dissipationand find that random configurations containing up to twelvedifferent phonon modes and trillions of configurations areneeded to obtain converged results.

A few more observations are in order before we describethe present theory in detail. In a perfect crystal withoutdefects, the HRK perturbation Hamiltonian is responsible forelectron-phonon scattering (only linear coupling is usuallyincluded) and for the formation of polarons, which areelectrons or holes dressed by phonons. Under strong-couplingconditions, the HRK Hamiltonian can be responsible forpolaron self-trapping. When a defect is present, the HRKHamiltonian can cause carrier capture. As Alkauskas et al.[24] pointed out, such capture is very slow. Indeed it iscaused by the derivatives of the electronic wave functionswith respect to nuclear displacements, which amounts toa “frozen electron approximation” (recall that the BOA iseffectively a “frozen nuclei approximation”). As we noted, thiskind of capture occurs under thermal equilibrium conditions,which corresponds to constant emission and capture by inverseprocesses; i.e., the band electrons are definitely “aware” of thedefects, i.e., they should not be treated as “free” carriers with amean free path, undergoing scattering by defects and phonons.In this regard, the linear coupling approximation [24] should beviewed as the zero mean-free-path limit, whereas the theoryput forward in this paper represents the limit in which themean free path is bounded only by Lcapture, the mean distancean electron travels before being captured by a defect.

The conditions under which capture cross sections aremeasured by junction capacitance methods [17] are close toequilibrium; i.e., they are slow. Similarly, in light-emittingdiodes, carriers by design have minimal acceleration throughthe pn junction. However, even in such deliberate setups,there must still be some nonequilibrium driving forces, e.g., acurrent must flow through the system, in order to carry out themeasurement or for the device to operate. The carrier mean freepath is always finite, never exactly zero. Therefore, a realisticmodel of the measured capture cross sections can be obtainedby scaling the difference between the two limits according tothe factor L/Lcapture, where L is the elastic scattering meanfree path,

σ = L

Lcaptureσadiabatic + σ nonadiabatic, (1)

where σnonadiabatic is the capture cross section due to the HRKHamiltonian and σadiabatic is the adiabatic capture cross sectioncalculated in this paper.

For scattering of a carrier into another propagating state at alower energy, the defect is left in the same charge state, whichrequires that scattering by the defect potential is elastic (noenergy can be dissipated in the FCA in such a case). We findthat inelastic scattering can still occur within the BOA by thefirst-order correction to the FCA, which are the linear termsdiscussed above.

II. FERMI GOLDEN RULE FOR ADIABATICAND NONADIABATIC TRANSITIONS

As discussed in the previous section, in order to describetransitions, it is always necessary to identify the piece of the to-tal Hamiltonian that causes the transition between eigenstatesof an approximate Hamiltonian. Let us be more specific. Inthe hydrogen atom, one usually includes only the Coulombicattraction between the proton and the electron, leaving outthe electromagnetic field at large. The calculated energylevels are only eigenstates of this approximate Hamiltonian.The electromagnetic field, treated as a perturbation, thencauses a transition from, say, a 2p state to the 1s state. InAuger transitions, one must leave out specific electron-electroninteractions, which are then introduced to cause transitions[30]. Our task here is to identify the approximate Hamiltonianwhose eigenstates are the propagating state of the incomingelectron, which is not aware of the bound state of the defectpotential, and the final state, which can be either anotherpropagating state that is not aware of the existence of a boundstate at a lower energy or the bound state itself, and determinethe perturbation Hamiltonian that causes the transition.

In the BOA, the many-electron Hamiltonian dependsparametrically on the nuclear positions and the total wavefunctions are products of many-electron wave functions andphonon wave functions. Within DFT, the many-electronwave functions are Slater determinants of Kohn-Sham wavefunctions. We start by defining the many-electron HamiltonianH 0 for the perfect crystal and the corresponding eigenvalueproblem:

H 0∣∣�0

n

⟩ = E0n

∣∣�0n

⟩. (2)

For the crystal containing a single defect, we have

H |�m〉 = Em|�m〉. (3)

One normally writes

H = H 0 + �H. (4)

The partitioning of the total Hamiltonian H according toEq. (4) is not useful for our purposes. Instead, we write

H = H 0 + H BO1 , (5)

where

H 0|�n〉 = εn|�n〉. (6)

In order to obtain an explicit description of H BO1 , which then

defines H 0 through Eq. (5), we express �H in terms of the

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complete set of functions �n:

�H =∑m

|�m〉〈�m|�H∑

n

|�n〉〈�n|

=∑mn

|�m〉�Hmn〈�n|. (7)

We then define H BO1 by

H BO1 = |�i〉�Hif 〈�f | + |�f 〉�Hf i〈�i |, (8)

where the subscripts i and f denote the eigenstates of H 0 thatare the initial and final states of our problem. This definition ofH BO

1 is analogous to the so-called Forster transition often usedin energy transfer in molecules [25]. In effect, H BO

1 eliminatesthe coupling of the incoming electron via the defect potentialto the final state, whether propagating or bound. The defectpotential �H, which can be arbitrarily strong, is still present.It is the perturbation Hamiltonian H BO

1 that is weak and cancause transitions whose rate is describable by Fermi’s goldrule, i.e., to first order in H BO

1 . Note also that the state |�i〉contains an incoming electron that “sees” the defect potentialbut does not couple to the bound state. Also, for all practicalpurposes, for carrier capture we have |�f 〉 = |�f 〉 (i.e., thebound state is not affected by the presence of an incomingelectron that does not couple to the defect).

The adiabatic transition rate is given by the usual Fermi’sgolden rule,

wBOif = 2π

�

∑f

∣∣〈Xf |〈�f |H BO1 |�i〉|Xi〉

∣∣2δ(f − i + εif ),

(9)where i,f are the total phonon energies of states |Xi,f 〉 andεif = εf − εi is the energy difference between the electronicstates |�i〉 and |�f 〉. For capture, it is usually assumed thatthere is one final electronic state with a given energy differenceεif , but there are many phonon configurations that can makeup this difference. If there are multiple electronic states at thesame energy, we need to sum Eq. (9) over all such states.

In addition to H BO1 , there are terms beyond the BOA,

usually referred to as nonadiabatic terms [10,11], which causemultiphonon transitions. These terms contain derivatives ofthe electron wave functions with respect to nuclear coordinates{Rk} and are the terms neglected when one invokes the BOA.They contribute to the total transition rate wif via the matrixelement,

−∑

k

�2

2mk

[〈Xf |〈�f |∇2Rk

(|�i〉|Xi〉)

−〈Xf |〈�f |�i〉∇2Rk

|Xi〉], (10)

where mk is the mass of atom k. This contribution is discussedin detail later.

One can define a cross section for inelastic scattering orcarrier capture by

σif = wif

vg

, (11)

where vg is the group velocity of the incident electron and

is the volume over which state |i〉 is normalized, so that vg/

represents the flux of the incoming electrons.

We work within DFT so that the many-electron wavefunctions are Slater determinants of Kohn-Sham one-electronwave functions and the many-electron Hamiltonians are thoseof noninteracting Kohn-Sham quasiparticles in the presenceof an effective single-particle external potential. From now onwe view the Hamiltonians and wave functions in Eqs. (9) and(10) as one-electron Kohn-Sham Hamiltonians and electronwave functions without change of notation.

A. Adiabatic series

We now examine the electronic part of the transition matrixelement in the BOA by showing explicitly its dependence onthe atomic coordinates:

MBOe ({Rj }) = ∣∣〈�f ({Rj })|H BO

1 ({Rj })|�i({Rj })〉∣∣2. (12)

The BOA by itself does not separate electron and phononmatrix elements. A further approximation is needed. Weexpand

MBOe ({Rj }) = MBO

e

({R(0)

j

})+∑

k

(Rk − R(0)

k

) · ∇RkMBO

e ({Rj }) + · · · (13)

in terms of the atomic displacements Rk − R(0)k , where R(0)

k arethe atomic positions in a reference state, which is determinedlater. The transition rate is then

wBOif = 2π

�

∣∣MBOe

({R(0)

j

})∣∣2∑f

|〈Xf |Xi〉|2δ(f − i + εif )

+2π

�

∑f

∣∣∣∣∣∑

k

∇RkMBO

e

({R(0)

j

})·〈Xf |(Rk−R(0)k

)|Xi〉∣∣∣∣∣2

× δ(f − i + εif ) + · · · . (14)

Here the cross terms are dropped because the zeroth-orderand first-order terms cannot have the same final phonon wavefunctions: the number of phonons needed to ensure a nonzerooverlap matrix element are different for the two cases. The firstterm in this expansion represents a complete separation of theelectron and phonon wave functions as if they are independentof each other and corresponds to the FCA. The second termis the first-order correction to the FCA arising from the BOAperturbation Hamiltonian H BO

1 .

B. Nonadiabatic series

According to Huang [20], the nonadiabatic matrix elementdefined in Eq. (10) can be evaluated for linear phononcoupling,∑

k

⟨�f

({R(0)

j

})∣∣∇RkHe

({R(0)

j

})∣∣�i

({R(0)

j

})⟩·〈Xf |(Rk − R(0)

k

)|Xi〉, (15)

where He is the electron part of the Hamiltonian. Whenelectron-phonon coupling Hep = He({Rj }) − He({R(0)

j }) isintroduced, the electron wave functions are changed by a

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perturbation,

|δ�i({Rj })〉 =∑i ′ �=i

〈�i ′ |Hep|�i〉εi ′ − εi

∣∣�i ′({

R(0)j

})⟩(16)

(and a similar equation applies for the final states). We writeboth the initial and the final states in the form

|�i(f )({Rj })〉 = ∣∣�i(f )({

R(0)j

})⟩+ |δ�i(f )〉. (17)

Substituting this into Eq. (10) and keeping only the linearterms,

−∑

k

�2

2Mk

[〈Xf |∇2Rk

(⟨�f

({R(0)

j

})∣∣δ�i

⟩|Xi〉)− 〈Xf |⟨�f

({R(0)

j

})∣∣δ�i

⟩∇2Rk

|Xi〉]

= (i − f )〈Xf |⟨�f

({R(0)

j

})∣∣δ�i

⟩|Xi〉=∑

k

∑i ′ �=i

εif

εi ′ − εi

〈�i ′ |∇RkH |�i〉

⟨�f

({R(0)

j

})∣∣�i ′({

R(0)j

})⟩ · 〈Xf |(Rk − R(0)k

)|Xi〉

=∑

k

〈�f |∇RkHe|�i〉 · 〈Xf |(Rk − R(0)

k

)|Xi〉. (18)

Here the first equality results from the Schrodinger equationsfor the phonon wave functions, and for the second equality weused i − f = εif .

We note that the above linear-order term in the nonadiabaticseries has the same phonon matrix element as the linear-orderterm in the BOA series in the previous section. This indicatesthat the leading nonadiabatic term is a smaller contribution tothe electron capture rate compared to the zeroth-order BOAterm. The electronic matrix element in the nonadiabatic seriesis different from that in the BOA series. We show later thatboth these terms scale as 1/m, where m is the mass of a typicalatom in the defect complex.

The linear term in Eq. (15) is usually referred to as thelinear electron-phonon coupling term. A similar term has beencalculated by Alkauskas et al. [24], with the exception thatin that work the wave functions are |�i(f )〉, which are theeigenstates of the full Hamiltonian He, whereas in our casethe wave functions are �i(f ), which are the eigenstates of theHamiltonian H 0. We recover the term calculated by Alkauskaset al. if we combine the BOA and the nonadiabatic series. Wemake use of the result in Eq. (24) and get, for our final result,

wif = 2π

�

∣∣MBOe

({R(0)

j

})∣∣2∑f

|〈Xf |Xi〉|2δ(f − i + εif )

+ 2π

�

∑f

∣∣∣∣∑k

[〈�f |∇RkHe|�i〉 − ⟨�f

∣∣�0i

⟩

×〈�f |∇RkHe|�f 〉] · 〈Xf |(Rk − R(0)

k

)|Xi〉∣∣∣∣2

× δ(f − i + εif ) + · · · . (19)

Here the first term is the zeroth-order term that corresponds tothe FCA and the second term is the totality of contributionsfrom the linear terms in the two series. The first term in bracketsis precisely the term that Alkauskas et al. [24] calculated. Wenote that there exists a second term, which has the appearanceof a force term. These two terms can either add or subtract. Weshow shortly that these linear-order terms are proportional to1/m, where m is a typical atomic mass in the defect complex,

and are, therefore, significantly smaller than the zeroth-orderFranck-Condon term, which is dominant.

III. ELECTRON MATRIX ELEMENTS

We first consider the zeroth-order term in the BOA series,which yields a capture cross section that can be written in thefamiliar factorized form,

σif = Aif Fif , (20)

where Aif contains the electronic part of the matrix element,

Aif = 2π

�vg

∣∣⟨�f

({R(0)

j

})∣∣H BO1

({R(0)

j

})∣∣�i

({R(0)

j

})⟩∣∣2, (21)

and F is called the line-shape factor due to vibrations,

Fif =∑f

|〈Xf |Xi〉|2δ(f − i + εif ). (22)

Next we consider these two factors separately.Detailed derivations given in Appendixes A and B give the

final results:

MBOe = −⟨�f

∣∣�0i

⟩εif (23)

and

∇RkMBO

e + 〈�f |∇RkH |�i〉

= 〈�f |∇RkH |�i〉 − ⟨�f

∣∣�0i

⟩〈�f |∇RkH |�f 〉. (24)

For evaluation of the above matrix elements, we employthe PAW scheme, which allows us to use all-electron wavefunctions instead of pseudo–wave functions. Details are givenin Appendix C.

IV. PHONON MATRIX ELEMENTS

First, we consider the effect of displacements for a classicalHamiltonian. We derive this Hamiltonian for the ion motion,from which the phonon wave functions and matrix elementscan be calculated. For this purpose we start with a supercellcontaining a number of atoms na , with the defect site at

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its center. This supercell is repeated N times using theBorn–von Karman periodic boundary condition. For the initialstate, the equilibrium positions of the atoms are Rk , wherethe subscript k runs through both the atomic index withinthe supercell and the Cartesian components. Each atomoscillates around its equilibrium position with displacementukl , where the subscript l labels different copies of the supercellunder the Born–von Karman periodicity. Using the harmonicapproximation for the potential energy, under which only termsthat are second order in displacements make a contribution, andintroducing force constants �kl,k′l′ , we can write [25]

H ′i = 1

N

∑kl

[1

2mk

(dukl

dt

)2

+ 1

2N

∑k′l′

ukl�kl,k′l′uk′l′

],

(25)

for the initial state, where the atomic mass mk also carries thesubscript k for convenience even though it depends only onthe atomic index, and not the coordinate component index.

When an electron is absorbed or emitted from the lattice,the equilibrium positions of the atoms change. The newequilibrium positions are Rk + �k . The new Hamiltonian hasthe same form after the initial displacement vectors ukl arereplaced by u′

kl = ukl − �k . The final-state Hamiltonian isthen written as

H ′f = 1

N

∑kl

{1

2mk

[d(ukl − �k)

dt

]2

+ 1

2N

∑k′l′

(ukl − �k)�kl,k′l′ (uk′l′ − �k′)

}, (26)

where we make the assumption that force constants do notchange due to the electron capture or absorption. Sincedisplacements �k do not depend on time, the kinetic energyterm remains unchanged. Expanding the potential energy tofirst order in displacements reproduces the same term in theoriginal Hamiltonian plus a term that includes uk′l′�k:

H ′f = H ′

i − 1

N

∑kl,k′l′

�kl,k′l′�kuk′l′ . (27)

Transforming to the normal-mode representation in terms ofthe generalized coordinates,

qj = 1√N

∑kl

√mkuklwj,kl, (28)

where wj,kl is the klth element of the eigenvector for modej . Note that in this definition of the generalized coordinateqj , it has absorbed the mass factor

√mk . The Hamiltonian is

expressed as

H ′f = 1

2

∑j

q2j + 1

2

∑j

ω2j q

2j

− 1√N

∑j

qj

∑kk′

Dkk′(kj )wjk′√

mk�k, (29)

where ωj are the eigenfrequencies. A phase factor of the formexp(ikj · rl′ ), where kj is the wave vector of mode j , from

wj,k′l′ is absorbed into the force constant matrix �, yielding thedynamical matrix D and reducing wj,k′l′ to wjk′ (independentof l′). Since we assume that force constants remain the sameafter electron capture,

∑k′

Dkk′(kj )wjk′ = ω2jwjk. (30)

The linear term causes a general coordinate displacement,

δqj = − 1√N

∑k

√mk�kwjk. (31)

We can express the normal coordinates of the lattice for thefinal (f ) state, qj,f , in terms of those for the initial (i) state, qj ,

qf,j = qj + δqj , (32)

so that the final Hamiltonian is

H ′f = 1

2

∑j

q2f,j + 1

2

∑j

ω2j q

2f,j . (33)

A. Zeroth-order phonon matrix elements

We have derived the expression for the generalized co-ordinates resulting from the lattice displacements. Thesegeneralized displacements enter the phonon wave functions|Xni

j(qj )〉 and |X

nf

j(qj + δqj )〉, respectively, in the quantized

versions of the harmonic oscillator Hamiltonians H ′i and H ′

f .Now we turn to the evaluation of phonon matrix elements〈X

nf

j(qj + δqj )|Xni

j(qj )〉. When the displacements δqj are

small, we can show that the dominant contribution comesfrom single-phonon emission or absorption for each normalmode. Suppose the initial state of mode j has n phonons andits final state has n + p phonons (we dropped the index forthe mode, since it is present in the notation of generalizedcoordinate). Using the integrals provided in Appendix D, thematrix elements for the phonon part are

〈Xn+1(qj + δqj )|Xn(qj )〉 = −√

(n + 1)ωj

2�δqj , (34)

〈Xn−1(qj + δqj )|Xn(qj )〉 =√

nωj

2�δqj . (35)

The integrals for phonon modes that maintain the sameoccupation numbers are calculated to second order in qj :

〈Xn(qj + δqj )|Xn(qj )〉 = 1 − (2n + 1)ωj

4�δq2

j . (36)

Now we consider how to evaluate Eq. (22). The total numberof phonon modes in the supercell is M = 3(na − 1) excludingthe translational motion, and the total number of phonon modesin the entire system is MN , since the supercell is repeated N

times. We assume that there is a one-to-one correspondencebetween phonon bands before and those after capture. The

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THEORY OF INELASTIC MULTIPHONON SCATTERING . . . PHYSICAL REVIEW B 92, 214111 (2015)

wave function of the initial phonon state is

|Xi〉 =MN∏j=1

∣∣Xnij

⟩, (37)

and that of any one of the final phonon states is

|Xf 〉 =MN∏j=1

∣∣Xn

f

j

⟩, (38)

where nij and n

f

j are the occupation numbers of phonon modej before and after the capture and are also used to label thewave functions. The total phonon energies for initial and finalconfigurations are

i = 1

N

MN∑j=1

nijhωi

j (39)

and

f = 1

N

MN∑j=1

nf

j hωf

j , (40)

respectively, where ωij and ω

f

j are the phonon frequency ofmode j in the initial and final configurations of the defect,respectively. With the overlap matrix for each individual modeexpressed as Eq. (D1) and using Eqs. (37)–(40), Eq. (22) now

takes the form

Fif =∑{nf

j }

⎧⎨⎩

MN∏j=1

∣∣∣∣∫

Xn

f

j(qj + δqj )Xni

j(qj )dqj

∣∣∣∣2⎫⎬⎭

× δ

⎛⎝ 1

N

MN∑j=1

(n

f

j �ωf

j − nij�ωi

j

)+ εif

⎞⎠, (41)

where nf

j = nij − 1,ni

j ,nij + 1. We see below that as the limit

of N → ∞ is taken, the discrete modes in N will becomecontinuous spectra in k over the Brillouin zone of the reciprocalspace.

Now we are ready to put all the phonon matrix elementstogether and perform the configurational sum. To do this wefollow the steps of Huang and Rhys [10] but generalize themfor a system with multiple phonon frequencies. For multiplephonon bands, we assume that the frequency variation withineach band is much smaller than the frequency differencebetween the bands. This is the flat band approximation,which is complemented with the requirement of finite spacingbetween the bands. We finally find

Fj = exp

[pj�ωj

2kT− Sj coth

(�ωj

2kT

)]Ipj

[Sj

sinh(�ωj/2kT )

](42)

and

F = 1

k

∑{pj }

⎧⎨⎩⎛⎝ M∏

j=1

Fj

⎞⎠ M∑

j=1

⎧⎨⎩pj + Sj

sinh(�ωj/2kT )

Ipj +1

[Sj

sinh(�ωj /2kT )

]Ipj

[Sj

sinh(�ωj /2kT )

]⎫⎬⎭D(ωj )

⎫⎬⎭∣∣∣∣∣∣∑M

j=1 pj �ωj +εif =0

, (43)

where D(ωj ) is the phonon density of states

Sj = ωj

2�Nδq2

j (44)

and Ip is the modified Bessel function of order p.

B. Linear phonon matrix elements

To evaluate the phonon matrix elements for the linear term,we rewrite it in terms of the normal mode coordinates qj ,

∑f

∣∣∣∣∣∣∑

j

Mj 〈Xf |qj |Xi〉∣∣∣∣∣∣2

=∑f

∑j

|Mj 〈Xf |qj |Xi〉|2

= 1

2

∑f

∂2

∂λ2

∣∣∣∣∣∣〈Xf |∏j

[1 + λMjqj exp(iφj )]|Xi〉∣∣∣∣∣∣2∣∣∣∣∣∣∣λ=0

,

(45)

where φj is a random phase introduced to cancel out the crossterms, and

Mj = 〈�f |∂qjHe|�i〉 − 〈�f |�i〉〈�f |∂qj

He|�f 〉. (46)

The rest of the steps are exactly the same as for the zeroth-ordermatrix elements. Using the integrals provided in Appendix D,the matrix elements for the phonon part are

〈Xn+1(qj + δqj )|[1 + λMjqj exp(iφj )]|Xn(qj )〉

= −√

(n + 1)ωj

2�

[δqj − λ�Mj

ωj

exp(iφj )

], (47)

〈Xn−1(qj + δqj )|[1 + λMjqj exp(iφj )]|Xn(qj )〉

=√

nωj

2�

[δqj + λ�Mj

ωj

exp(iφj )

], (48)

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BARMPARIS, PUZYREV, ZHANG, AND PANTELIDES PHYSICAL REVIEW B 92, 214111 (2015)

and

〈Xn(qj + δqj )|[1 + λMjqj exp(iφj )]|Xn(qj )〉

= 1 − (2n + 1)ωj

4�δq2

j − 1

2λMjδqj exp(iφj )

= 1 − Sj

2N− 1

2λMjδqj exp(iφj ). (49)

Define

S±(λ) =(

n + 1n

)ωj

2�N

∣∣∣∣δqj ∓ λ�Mj

ωj

exp(iφj )

∣∣∣∣2

≈(

n + 1n

)ωj

2�Nδq2

j

∣∣∣∣exp

[∓2

λ�Mj

ωjδqj

exp(iφj )

]∣∣∣∣.(50)

The approximation in the second step is accurate to λ2, withthe consideration that terms such as λ2 sin 2φj and λ2 cos 2φj

drop out after the configurational average. Then√S+S− ≈

√n(n + 1)Sj , (51)

S+(λ)

S−(λ)≈ n + 1

n

∣∣∣∣exp

[−4

λ�Mj

ωjδqj

exp(iφj )

]∣∣∣∣. (52)

The λ-dependent line-shape factor for a single phonon band is

Fj (λ)=exp

[pj�ωj

2kT−Sj coth

(�ωj

2kT

)−λNδqj |Mj exp(iφj )|

]

× Ipj

[Sj

sinh(�ωj/2kT )

]∣∣∣∣exp

[−2λpj

�Mj

ωjδqj

exp(iφj )

]∣∣∣∣.(53)

Let us now compare the two λ factors by evaluating theratio

Nωjδq2j

2�= mωjδR

2

�. (54)

For a hydrogenated vacancy defect our calculation shows thatδR ≈ 0.2 A for the nearest Si atom. Using m ≈ 4.66 × 10−26

kg for the Si atom and ωj ≈ 1012 s−1, we have

Nωjδq2j

2�≈ 0.09. (55)

Thus the first λ factor has a much smaller contribution than thesecond one. The final linear phonon squared matrix element is

F1 = 1

2k

∑{pj }

⎧⎨⎩ ∂2

∂λ2

⎡⎣ M∏

j=1

Fj (λ)

⎤⎦∣∣∣∣∣∣λ=0

M∑j=1

⎧⎨⎩pj + Sj

sinh(�ωj/2kT )

Ipj +1

[Sj

sinh(�ωj /2kT )

]Ipj

[Sj

sinh(�ωj /2kT )

]⎫⎬⎭D(ωj )

⎫⎬⎭∣∣∣∣∣∣∑M

j=1 pj �ωj +εif =0

. (56)

C. Ratio of zeroth-order and linear terms

From the different expressions for the zeroth-order andthe linear phonon matrix elements, we can estimate the ratiobetween the linear term and the zeroth-order term in thetransition rate. This is of the order of

2

∣∣∣∣ Mj�pj

MBOe ωj δqj

∣∣∣∣2

. (57)

To estimate Mj/MBOe , we note that the leading term in Mj is

[see Eq. (B4)]

Mj ≈ −εif

⟨∂�f

∂qj

∣∣∣∣�i

⟩. (58)

To estimate ∂�f /∂qj , we assume rigid atomic orbitals, wherethe atomic wave functions move rigidly in space with eachatom. The derivative of such a wave function with respect toatomic displacements simply reflects the change in the relativespatial phase, which is dictated by the phonon wave vector,

∂�f

∂qj

≈ i

√N

m

2π

λj

�f exp(iφ), (59)

where λj is the acoustic wavelength for mode j , m is the massof an atom, and φ is the phase factor due to the movement of theatoms, which is different in each Born–von Karman supercell.Integrating over all N Born–von Karman supercells, the sum

of the exp(iφ) factors scales as 1/N for large N . Thus,

Mj ≈ i2π√Nmλj

MBOe . (60)

Finally, pj is mostly 0, occasionally taking the values ±1,and δqj ≈ √

(m/N )δR, where δR is the largest atomicdisplacement and m is the mass of the corresponding atom. Theratio between the linear and the zeroth-order terms simplifiesto

2

(�

cmδR

)2

, (61)

where c is the sound velocity in the material. For a hy-drogenated vacancy defect our calculation shows that δR ≈0.2 A for the nearest Si atom. Using this number and c ≈8 × 103 m/s for bulk silicon and m ≈ 4.66 × 10−26 kg for theSi atom, we find

2

(�

cmδR

)2

≈ 3.6 × 10−4. (62)

Thus the linear phonon term (nonadiabatic term) is severalorders of magnitude smaller than the leading BOA term.

D. Monte Carlo method for the configurational sum

The summation over all configurations {pj } involves alarge number of terms when P =∑j |pj | is greater thana few. We use a Monte Carlo approach to calculate thissum. For a given number of phonon modes, P , and a given

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THEORY OF INELASTIC MULTIPHONON SCATTERING . . . PHYSICAL REVIEW B 92, 214111 (2015)

number of bands, B, we use Monte Carlo to construct a fixednumber of configurations, K . We rewrite the sum over theconfigurations as a sum over the number of phonons P ofa configuration, a sum over the number of bands B used to

construct a configuration with P phonons, and a sum over theconfigurations sampled (Monte Carlo steps). In each MonteCarlo step, we randomly pick B bands and then we constructall the possible configurations with P phonons constructed bythese bands.

In order to generate and count the configurations correctly, we first rewrite Eq. (43) as

F = 1

k

∑P

P∑B=1

wB

K∑{pj }′

⎧⎨⎩

M∏j=1

Fj

M∑j=1

⎧⎨⎩pj + Sj

sinh(�ωj/2kT )

Ipj +1

[Sj

sinh(�ωj /2kT )

]Ipj

[Sj

sinh(�ωj /2kT )

]⎫⎬⎭D(ωj )

⎫⎬⎭∣∣∣∣∣∣∑M

j=1 pj �ωj +εif =0

. (63)

Then we normalize the sum so that the total weight, wB , ineach subgroup of configurations (configurations with the samenumber of bands) is equal to the total number of possibleconfigurations for this number of bands:

wB = 1

K

M!

B!(M − B)!. (64)

All configurations with up to four phonon modes are con-structed and calculated explicitly. For configurations with morethan four phonons, all the configurations constructed with upto three bands are calculated explicitly and the above equationsare used to calculate the line-shape function for configurationswith more than three bands.

The last step in the Monte Carlo scheme is to collect theline-shape function into different energy bins for a distribution.To do this, we note that with an incomplete sampling of thephase space via Monte Carlo, we may not be able to resolve

the energy distribution to arbitrary accuracy. Specifically,when we sample one configuration and weigh it accordingto Eq. (64), we are effectively using it to approximateseveral configurations with different energies. Thus, the energyresolution must be consistent with the number of configurationsamples: fewer configurations should correspond to a coarserenergy resolution. For this reason, we define the energy binwidth separately for each value of P based on the requirementthat there is at least one configuration inside each energy bin.To ensure the correct normalization, we rewrite the phonondensity of states for band j as

D(ωj ) = 1

�E

∫D(E)dE = k

�E, (65)

where �E is the energy bin width and we assume that thephonon band is sufficiently flat so that it falls entirely withinone energy bin. Then Eq. (63) becomes

F = 1

�E

∑P

P∑B=1

wB

K∑{pj }′

⎧⎨⎩⎛⎝ M∏

j=1

Fj

⎞⎠ M∑

j=1

⎧⎨⎩pj + Sj

sinh(�ωj/2kT )

Ipj +1

[Sj

sinh(�ωj /2kT )

]Ipj

[Sj

sinh(�ωj /2kT )

]⎫⎬⎭⎫⎬⎭∣∣∣∣∣∣∑M

j=1 pj �ωj +εif =0

. (66)

The evaluation of the linear phonon terms is similar.

V. APPLICATION TO A DEFECT IN SILICON

In this paper, we present only one application of the theoryand computer codes for the capture cross section of a prototypedefect in Si, namely, a triply hydrogenated vacancy with abare dangling bond. Our purpose here is to demonstrate thefeasibility of calculations, especially the first-ever calculationof the line-shape function that is converged with respectto the number of phonon modes that are used to constructrandom configurations whose energy is equal to the amount ofenergy that needs to be dissipated following the instantaneouselectronic transition. We defer calculations for defects forwhich experimental data are available to a future paper wherewe anticipate using hybrid functionals in the DFT calculationsof the electronic matrix elements. Such calculations arecomputationally demanding but would provide more accuratetransition energies and electronic matrix elements. In addition,we plan to code the additional contributions from the linearterms which we estimated to be significantly smaller because

they scale with the inverse of the mass of a typical atom inthe defect cluster. It will be interesting to see how the twoterms in brackets in Eq. (19) add or subtract for differentdefects.

In Fig. 1, we show the values of calculated electronicmatrix elements as a function of the energy. At each energyvalue, there are a number of k points that contribute. Theircontributions are indicated by (red) squares. The size of theenergy bin is determined by the number of k points. For theexample shown in Fig. 1 the average matrix element as afunction of energy is shown by the (blue) line. The size of theenergy bin fixes the resolution. A smooth curve can only beobtained with very small energy bins, which requires a verylarge number of k points. It is clear from the figure that thecapture electronic matrix element is relatively constant as afunction of the energy, whereby it seems best at this point totake it to be a constant, either an average value or the value atthe threshold for capture, which introduces an error bar of afactor of ∼1.7 (clearly, to validate the theory against accurateexperimental data, we need a very accurate calculation in thenear-threshold region).

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BARMPARIS, PUZYREV, ZHANG, AND PANTELIDES PHYSICAL REVIEW B 92, 214111 (2015)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

|ΔH

if|2 x

10-5

(Har

tree2 )

Electron energy (eV)

|ΔHif (E,k)|2

|ΔHif (E)|2

FIG. 1. (Color online) Calculated electronic matrix elements as afunction of the initial-state electron energy for a triply hydrogenatedvacancy in Si with a bare dangling bond: (red) squares, matrix elementvalues at each energy for different k points; (blue) curve, averagedmatrix element over all k points for each energy.

In Fig. 2, we show the calculated capture cross section usinga constant matrix element to show clearly the convergenceof the line-shape function as we increase the number ofphonon modes that are used to construct configurations (theelectronic matrix element is just a multiplier that sets theabsolute value). The dominant contribution to the line-shapefunction comes from the balance between the modes withthe largest general coordinate displacement and the growthof the number of allowed combinations with smaller generalcoordinate displacement. Note that the curves are smoothbecause we employ millions of configurations at each energyand therefore we have very tiny energy bins. It is clear thata single-phonon-mode approximation would be very poor

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0.30 0.40 0.50 0.60 0.70

σ(E)

x 1

0-17 (c

m2 )

Electron energy (eV)

9 phonon modes10 phonon modes11 phonon modes12 phonon modes

FIG. 2. (Color online) Calculated electron capture cross sectionusing a constant electron matrix element and different numbers ofphonon modes.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

4 5 6 7 8 9 10 11 12

σ(E)

x 1

0-17 (c

m2 )

Number of phonon modes

FIG. 3. (Color online) Convergence of the calculated electroncapture cross section at the threshold as a function of the numberof phonon modes.

indeed. In Fig. 3 we show the convergence of the capturecross section at threshold (for electrons at the bottom of theconduction band), which is what is usually measured. Oncemore, it is clear that the single-phonon-mode approximationwould be inadequate.

For a calculation of the cross section using electronic matrixelements that depend on the energy, the resolution is limitedby the energy bin size. We show the result in Fig. 4. Clearly,the size of the energy bin is important. For capture crosssections, one is often interested only in the threshold value. Thecalculations presented here are a prelude to calculations of hot-electron inelastic multiphonon scattering, for which the energydependence is important. The energy dependence is alsoimportant in luminescence curves, i.e., the classic Huang-Rhysproblem that was treated in the single-phonon approximation

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.3 0.4 0.5 0.6 0.7

σ(E)

x 1

0-17 (c

m2 )

Electron energy (eV)

40 meV energy bin50 meV energy bin75 meV energy bin

FIG. 4. (Color online) Calculated full capture cross section usingthe electron matrix element from Fig. 1 and 12 phonon modes in theline-shape function.

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THEORY OF INELASTIC MULTIPHONON SCATTERING . . . PHYSICAL REVIEW B 92, 214111 (2015)

0.0

0.1

0.2

0.3

0.4

0.5

0 2 4 6 8 10

Ato

mic

dis

plac

emen

t (Å

)

Distance from the vacancy site (Å)

3x3x3 k−pointsSingle k point

FIG. 5. (Color online) Atomic displacements of the triply hydro-genated Si vacancy as a function of the distance from the vacancy sitefor a 64-atom supercell.

in Ref. [23] (in the case of luminescence, MPPs dissipateonly the relaxation energy of the defect, when one expectsthe phonon mode corresponding to the actual relaxation todominate; nevertheless, a fully convergent calculation wouldbe needed to establish the degree of accuracy one obtainswith the single-mode approximation). The accuracy of thecalculation of the line-shape function is controlled by theaccuracy of the calculation of the generalized displacements.The latter depends on the accuracy of the calculation of theatomic displacements. We found that the accuracy is enhancedsignificantly if we allow the entire supercell to relax, whichallows the defect’s neighbors to relax more freely. At thesame time, a dense k-point mesh is necessary. In Fig. 5, wepresent the atomic displacements of the triply hydrogenated Sivacancy as a function of the distance from the vacancy site fora 64-atom supercell. Using only one k point and not allowingthe supercell to relax we get only the Si atom near the defectto move significantly, while the rest of the crystal remainsessentially frozen (blue circles). This kind of relaxation leadsto only a few phonon modes being significant, and thus thesystem is artificially able to dissipate energy efficiently atcertain frequencies. On the other hand, the well-relaxed crystalof the (3 × 3 × 3) k-point grid (red circles) has more atomscontributing to the generalized displacements, and thus almostall the phonon modes contribute in the dissipation to theenergy of the incoming electron. The use of supercells withmore than 64 atoms would be prohibitively expensive for theline-shape-function calculation.

The above MPP calculations leave out the effect of thephonons that are emitted during the process, i. e. only equilib-rium phonons are assumed to be available for absorption. Thiseffect will be explored in future work.

VI. SUMMARY

We have presented a comprehensive theory of inelastic mul-tiphonon carrier capture and scattering processes. We showedthat, under nonequilibrium conditions, i.e., in the presence

of currents or hot electrons, the defect potential is primarilyresponsible for capture through a zeroth-order term in anexpansion in terms of the atomic displacements (relaxation)that accompanies capture. These terms have not been includedin any prior theory. Instead, the focus has always been onthe linear terms, which we showed here to be much smallerbecause they depend on the inverse of the mass of typicalatoms in the defect complex. The linear terms are dominantonly in the limit of thermal equilibrium. For the first time, weused accurate all-electron wave functions obtained by the PAWmethod for the electronic matrix elements and an accurateMonte Carlo scheme to sample random configurations of upto 12 distinct phonon modes for the line-shape functions toachieve convergence (a single-phonon-mode approximationhas been standard in prior calculations). We have presentedresults for a prototype defect. More accurate hybrid exchange-correlation functionals are needed to produce results that areaccurate enough for comparison with experimental data. Inaddition, a reliable comparison with data can only be madewith experimental measurements of capture cross sectionssimultaneously with determination of the elastic mean freepath and the capture mean free path, as they appear in Eq. (1).

ACKNOWLEDGMENTS

We would like to thank Chris Van de Walle and AudriusAlkauskas for valuable discussions. This work was supportedin part by the Samsung Advanced Institute of Technology(SAIT)’s Global Research Outreach (GRO) Program, by theAFOSR and AFRL through the Hi-REV program, and by NSFGrant No. ECCS-1508898. A portion of this research wasconducted at the Center for Nanophase Materials Sciences,which is sponsored at Oak Ridge National Laboratory bythe Division of Scientific User Facilities. The computationwas done using the utilities of the National Energy ResearchScientific Computing Center (NERSC) and resources of theOak Ridge Leadership Computing Facility at the Oak RidgeNational Laboratory, which is supported by the Office ofScience of the U.S. Department of Energy under ContractNo. DE-AC05-00OR22725. G. D. Barmparis acknowledgessupport from EU/FP7-REGPOT-2012-2013-1 under GrantAgreement No. 316165. The work was also supported bythe McMinn Endowment at Vanderbilt University. This paperwas authored by UT-Battelle, LLC, under Contract No.DE-AC05-00OR22725 with the U.S. Department of Energy.The United States Government retains and the publisher, byaccepting the article for publication, acknowledges that theUnited States Government retains a nonexclusive, paid-up,irrevocable, worldwide license to publish or reproduce thepublished form of the manuscript, or allow others to do so,for United States Government purposes. The Department ofEnergy will provide public access to these results of federallysponsored research in accordance with the DOE Public AccessPlan (http://energy.gov/downloads/doe-public-access-plan).

APPENDIX A: EVALUATION OF THE ELECTRONICMATRIX ELEMENT FOR THE BOA TRANSITION

In the basis of |�n〉, the unperturbed Hamiltonian H 0 isdiagonal with eigenenergies εn. The total electron Hamiltonian

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BARMPARIS, PUZYREV, ZHANG, AND PANTELIDES PHYSICAL REVIEW B 92, 214111 (2015)

H = H 0 + H BO1 has coupling terms only between state |�i〉

and state |�f 〉. We can, therefore, construct solutions of(H 0 + H BO

1

)|�〉 = E|�〉 (A1)

in the form |�〉 = a|�i〉 + b|�f 〉, so that(εi �Hif

�Hf i εf

)(a

b

)= E

(a

b

). (A2)

There are two sets of solutions,

Ei(f ) = 1

2[εi + εf ±

√(εi − εf )2 + 4|�Hif |2], (A3)

where state i takes the positive sign and state f takes the minussign, since Ei > Ef . The coefficients satisfy

εiai + �Hif bi = Eiai (A4)

and

|ai |2 + |bi |2 = 1. (A5)

There is an arbitrary phase factor within ai . We can define aset of solutions as

ai = b∗f =

√√√√1

2+√

1

4−∣∣∣∣ �Hif

Ei − Ef

∣∣∣∣2

(A6)

and

bi = −a∗f = �Hif

Ei − Ef

1√12 +

√14 −

∣∣∣ �Hif

Ei−Ef

∣∣∣2. (A7)

If we can compute the overlap integral 〈�f |�i〉 = af , thenwe can solve for |�Hif |2 from |af |2 and find

|�Hif |2 = |〈�f |�i〉|2 − |〈�f |�i〉|4(1 − 2|〈�f |�i〉|2)2

ε2if . (A8)

To be consistent with the phase of Eq. (A7), we have

MBOe = 〈�f |H BO

1 |�i〉 = �Hif

= −√

1 − |〈�f |�i〉|21 − 2|〈�f |�i〉|2 〈�f |�i〉εif . (A9)

The wave function |�i〉 is related to that of a perfect crystal|�(0)

i 〉 through a perturbation expansion,

∣∣�(0)i

⟩ = |�i〉 −∑

i ′ �=i,f

〈�i ′ |�H |�i〉εi ′ − εi

|�i ′ 〉. (A10)

Because H1 has only nonzero elements between state |�i〉 andstate |�f 〉, for j �= i,f , the wave functions |�j 〉 = |�j 〉 sothat 〈�f |�j 〉 = 0. Thus, to first order in the defect potential,

〈�f |�i〉 = 〈�f |�0i 〉, (A11)

and assuming that |〈�f |�0i 〉| � 1, we arrive at Eq. (23), which

simplifies the evaluation of the overlap integral.

APPENDIX B: EVALUATION OF THE GRADIENT TERMS

Using the result in the previous section for the matrixelement MBO

e , we now calculate the gradient terms inEq. (19), ∇Rk

MBOe + 〈�f |∇Rk

He|�i〉. Neglecting higher order|〈�f |�i〉|2 terms, the first gradient term is

∇RkMBO

e =−(〈∇Rk�f |�i〉+〈�f |∇Rk

�i〉)εif −〈�f|�i〉∇Rk

εif

= −(〈∇Rk�f |�i〉 + 〈�f |∇Rk

�i〉)εif

−〈�f |�i〉〈�f |∇RkH0|�f 〉, (B1)

where in the last step we have used the fact that ∇Rkεi = 0

(the initial state is at equilibrium) and the Helmann-Feynmantheorem for ∇Rk

εf . From Eq. (16) we have

∣∣∇Rk�i

⟩ =∑i ′ �=i

〈�i ′ |∇RkHe|�i〉

εi ′ − εi

|�i ′ 〉, (B2)

where we have used ∇RkHel = ∇Rk

He. Because |�i ′ 〉 = |�i ′ 〉for i ′ �= i,f and 〈�f |�f 〉 = 1 + O(|〈�f |�i〉|2), we have

⟨�f

∣∣∇Rk�i

⟩ = 〈�f |∇RkHe|�i〉

εif

〈�f |�f 〉=〈�f |∇RkHe|�i〉

εif

.

(B3)Similarly,

〈∇Rk�f |�i〉 = −〈�f |∇Rk

He|�i〉εif

〈�i |�i〉

= −〈�f |∇RkHe|�i〉

εif

. (B4)

Combining these results and noting that H BO1 does not have

diagonal components, we arrive at

∇RkMBO

e + 〈�f |∇RkH |�i〉

= 〈�f |∇RkH |�i〉 − 〈�f |�i〉〈�f |∇Rk

H |�f 〉. (B5)

We can use Eq. (A11) and approximate |�f 〉 ≈ |�f 〉 to getEq. (24).

APPENDIX C: EVALUATION OF THE OVERLAPINTEGRAL WITHIN THE PAW

Consider the problem of evaluating the overlap integral〈�|�〉 between two wave functions from two different solids(e.g., one is a perfect crystal and the other contains a defect).Using the PAW expansion of the full wave functions,

|�〉 = |�〉 + |�AE〉a − |�PS〉a, (C1)

where |�〉 is the pseudo–wave function and |�AE〉a and |�PS〉aare the atomic wave functions inside the augmentation sphereof each atom a, and similarly,

|�〉 = |�〉 + |�AE〉b − |�PS〉b. (C2)

Now 〈�|�〉 is given as

〈�|�〉 = (〈�| + a〈�AE| − a〈�PS|)(|�〉 + |�AE〉b − |�PS〉b)

= 〈�|�〉 + 〈�|�AE〉b − 〈�|�PS〉b + a〈�AE| �〉− a〈�PS| �〉+ ( a〈�AE| − a〈�PS|)(|�AE〉b−|�PS〉b).

(C3)

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THEORY OF INELASTIC MULTIPHONON SCATTERING . . . PHYSICAL REVIEW B 92, 214111 (2015)

The first term, 〈�|�〉, is the overlap of the pseudo–wavefunctions and can be easily calculated since the pseudo–wavefunctions are expanded in the same base set of plane waves.

In order to evaluate the terms 〈�|�AE〉b − 〈�|�PS〉b anda〈�AE| �〉 − a〈�PS| �〉, we make use of the unitary operatorsconstructed by the projectors |p〉 and the pseudo–atomic wavefunctions |φ〉, ∑

b,ib

∣∣pbib

⟩⟨φb

ib

∣∣ = 1 (C4)

and ∑a,ia

∣∣φaia

⟩⟨pa

ia

∣∣ = 1, (C5)

inside the augmentation sphere of each atom b of theperfect crystal and each atom a of the solid with the defect,respectively. Thus,

〈�|�AE〉b − 〈�|�PS〉b=∑b,ib

(⟨�∣∣pb

ib

⟩⟨φb

ib

∣∣�AE⟩b− ⟨�∣∣pb

ib

⟩⟨φb

ib

∣∣�PS⟩b

)(C6)

and

a〈�AE| �〉 − a〈�PS| �〉=∑a,ia

(a

⟨�AE

∣∣ φaia

⟩⟨pa

ia

∣∣�⟩−a

⟨�PS

∣∣ φaia

⟩⟨pa

ia

∣∣�⟩). (C7)

Equations (C6) and (C7) ensure that in the case where thetwo solids are identical, i.e., |�〉 and |�〉 are eigenstates of thesame Hamiltonian and the augmentation spheres are identical,the one center expansion

∑i |φ〉〈p|�〉 of the pseudo–wave

function is identical to the pseudo–wave function |�〉 insidethe augmentation sphere and

〈�f |�AE〉 − 〈�|�PS〉 = 〈�PS|�AE〉 − 〈�PS|�PS〉. (C8)

To evaluate Eqs. (C6) and (C7), we need the projections of thepseudo–wave functions of the first solid to the projectors of theatomic wave functions of the second solid, 〈�|pb

ib〉, and vice

versa for the projections 〈paia|�〉. This can be easily calculated

since both the pseudo–wave functions and the projectors areexpanded in the same base set of plane waves.

The difficulty in evaluating the last term in Eq. (C3)( a〈�AE| − a〈�PS|)(|�AE〉b − |�PS〉b) is that the cutoff spheresfor the two wave functions are usually not identical. Wecan bypass this difficulty by evaluating the integral with theassistance of a complete set of plane waves |k〉:

( a〈�AE| − a〈�PS|)(|�AE⟩b− |�PS

i

⟩b

)=∑

k

( a〈�AE| − a〈�PS|)|k〉〈k|(|�AE〉b − |�PS〉b)

=∑

k

( a〈�AE| k〉 − a〈�PS| k〉)(〈k|�AE〉b − 〈k|�PS〉b).

The plane waves can be expanded in either sphere as

eik·r = 4π∑lm

iljl(kr)Y ∗lm(k)Ylm(r), (C9)

and using

|k〉 = 1√V

eik·r, (C10)

the all-electron and the pseudoatomic wave functions arewritten as

|�AE〉b =∑b,ib

RAEb,ib

Ylb,mb

⟨pb,ib

∣∣�i

⟩, (C11)

|�PS〉b =∑b,ib

RPSb,ib

Ylb,mb

⟨pb,ib

∣∣�i

⟩, (C12)

a〈�AE| k〉 − a〈�PS| k〉 = 4π√V

∑a,ia

⟨�∣∣pa,ia

⟩eik·Ra ila Y ∗

la ,ma(k)

×∫ ra

0jla (kr)

(RAE

a,ia− RPS

a,ia

)r2dr,

(C13)

and

〈k|�AE〉b−〈k|�PS〉b = 4π√V

∑b,ib

⟨pb,ib

∣∣�⟩e−ik·Rb (−i)lbYlb,mb(k)

×∫ rb

0jlb (kr)

(RAE

b,ib− RPS

b,ib

)r2dr.

(C14)

APPENDIX D: PHONON INTEGRALS

The overlap matrix between the initial and the final statesfor mode j is⟨X

nf

j(qj + δqj )

∣∣Xnij(qj )⟩ = ∫ Xnj +pj

(qj + δqj )Xnj(qj )dqj ,

(D1)

where nij = nj and n

f

j = nj + pj . For convenience, we dropthe subscript j for nj and pj . Expanding Xn(qj + δqj ) interms of δqj ,

Xn(qj + δqj ) =∑

l

1

l!

dlXn(qj )

dqlj

δqlj . (D2)

Defining the raising and lowering operators

a± = ∓√

�

2ωj

d

dqj

+√

ωj

2�qj , (D3)

we have

a+Xn(qj ) = √n + 1Xn+1(qj ) (D4)

and

a−Xn(qj ) = √nXn−1(qj ). (D5)

Subtracting the two, we find

d

dqj

Xn(qj ) =√

ωj

2�(a− − a+)Xn(qj )

=√

nωj

2�Xn−1(qj ) −

√(n + 1)ωj

2�Xn+1(qj ).

(D6)

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Using this recursive relation, we find that the lowest order term for∫

Xn(qj + δqj )Xn+k(qj )dqj is δq|k|j . Therefore, for small

δqj , only k = ±1 terms dominate. This means that each mode would emit or absorb at most a single phonon.The results for the integrals are ∫

dXn(q)

dqXn+1(q)dq = −

√(n + 1)ωj

2�(D7)

(note that this was incorrectly given as −(√

�mω/2)√

n + 1 in Ref. [10]) and∫dXn(q)

dqXn−1(q)dq =

√nωj

2�. (D8)

For linear phonon matrix elements, we have

qjXn(qj ) =√

�

2ωj

(a− + a+)Xn(qj )

=√

n�

2ωj

Xn−1(qj ) +√

(n + 1)�

2ωj

Xn+1(qj ). (D9)

The integrals needed are ∫Xn(q)Xn+1(q)qdq =

√(n + 1)�

2ωj

, (D10)

∫Xn(q)Xn−1(q)qdq =

√n�

2ωj

, (D11)

and ∫dXn(q)

dqXn(q)qdq = −1

2. (D12)

APPENDIX E: LINE-SHAPE FUNCTION

We first consider a single phonon band; i.e., all of the phonon modes ωj = ω(kj ) form a single continuous band described bywave vectors kj . Because of the Born–von Karman periodic boundary condition, the phonon band is discretized into N modes.Suppose that s modes go down by one quantum and s + p modes go up by one quantum. Then the line-shape factor, Eq. (41)with M = 1, contains contributions formed from the products⎧⎨⎩

N∏j=1

tj

⎫⎬⎭{∏

k∈s

fk,−

}⎧⎨⎩∏

l∈s+p

fl+

⎫⎬⎭δ

⎛⎝∑

l∈s+p

[ni

l�(ω

f

l − ωil

)+�ωf

l

]+∑k∈s

[ni

k�(ω

f

k − ωik

)− �ωf

k

]+∑

m�{s,s+p}ni

m�(ωf

m − ωim

)+εif

⎞⎠,

×⎧⎨⎩

N∏j=1

tj

⎫⎬⎭{∏

k∈s

fk,−

}⎧⎨⎩∏

l∈s+p

fl,+

⎫⎬⎭δ

⎛⎝∑

l∈s+p

�ωf

l −∑k∈s

�ωf

k +N∑

j=1

nij�(ω

f

j − ωij

)+ εif

⎞⎠, (E1)

where∑N

j=1 nij�(ωf

j − ωij ) is the energy difference because of the different phonon frequencies of the initial and final

configurations of the defect, and tj , f−, and f+ are defined as

tj =∣∣∣∣∫

Xnj(qj )Xnj

(qj + δqj )dqj

∣∣∣∣2

, fk,− =∣∣∫ Xnk

(qk)Xnk−1(qk + δqk)dqk

∣∣2∣∣∫ Xnk(qk)Xnk

(qk + δqk)dqk

∣∣2 ,

fl,+ =∣∣∫ Xnl

(ql)Xnl+1(ql + δql)dql

∣∣2∣∣∫ Xnl(ql)Xnl

(ql + δql)dql

∣∣2 . (E2)

A naive way to sum over all possible configurations is to neglect the difference in the frequencies and apply the same countingmethod as Huang and Rhys [10] to write the configurational sum for all such combinations of phonons as

1

s!(s + p)!

⎧⎨⎩

N∏j=1

tj

⎫⎬⎭{

N∑k=1

fk,−

}s{ N∑l=1

fl,+

}s+p

δ

⎛⎝∑

l∈s+p

�ωf

l −∑k∈s

�ωf

k +N∑

j=1

nij�(ω

f

j − ωij

)+ εif

⎞⎠. (E3)

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THEORY OF INELASTIC MULTIPHONON SCATTERING . . . PHYSICAL REVIEW B 92, 214111 (2015)

This would not be correct if the frequencies are different for each mode. Furthermore, the summation over configurations forlarge N is needed to integrate out the δ function. Therefore the δ function cannot be left outside the summations. Let us considerone term in the δ function at a time. Consider one the plus terms �ω

fm and insert the δ function into one of the summations:

1

s!(s + p)!

⎧⎨⎩

N∏j=1

t

⎫⎬⎭{

N∑k=1

fk,−

}s{ N∑l=1

fl,+

}s+p−1 N∑m=1

⎧⎨⎩fm,+δ

⎛⎝�ωf

m +∑

l∈s+p−1

�ωf

l −∑k∈s

�ωf

k +N∑

j=1

nij�(ω

f

j − ωij

)+ εif

⎞⎠⎫⎬⎭.

(E4)

For large N , each of the summations inside the curly braces can be converted into integrals and evaluated,

S± =N∑

k=1

fk,± = N

k

∫fk,±dk =

(n + 1

n

)ω

2�Nδq2, (E5)

where k is the volume of the reciprocal-space Brillouin zone. In the last step we have assumed that the frequency anddisplacement do not change with k.

In order to evaluate the last factor, which includes the δ function, we note that each term in the summation over m has adifferent ω

fm, which spans the entire phonon band when m scans from 1 to N . Thus as we convert the sum over m to the integral

over k, the argument ωfm is also converted to ωk,

N∑m=1

⎧⎨⎩fm,+δ

⎛⎝�ωf

m +∑

l∈s+p−1

�ωf

l −∑k∈s

�ωf

k +N∑

j=1

nij�(ω

f

j − ωij

)+ εif

⎞⎠⎫⎬⎭

≈ N

k

∫fk,+δ

⎛⎝�ω

f

k +∑

l∈s+p−1

�ωf

l −∑k∈s

�ωf

k +N∑

j=1

nij�(ω

f

j − ωij

)+ εif

⎞⎠dk

= S+D(ω)

k

∣∣∣∣�ωf +∑l∈s+p−1 �ω

f

l −∑k∈s �ωf

k +∑Nj=1 ni

j �(ωf

j −ωij )+εif =0

, (E6)

where D(ω) is the phonon density of states. Combining the above equations and then setting all frequencies to ω, Eq. (E4)becomes

1

s!(s + p)!

⎧⎨⎩

N∏j=1

t

⎫⎬⎭Ss

−Ss+p+

D(ω)

k

∣∣∣∣p�ω+∑N

j=1 nij �(ωf

j −ωij )+εif =0

. (E7)

But there is one such contribution for each ωk or ωl in the δ function, regardless of the sign of the frequency. For s modessubtracting a phonon and s + p modes adding a phonon there is a total of 2s + p such contributions. We thus sum over all theterms and obtain

2s + p

s!(s + p)!

⎧⎨⎩

N∏j=1

tj

⎫⎬⎭Ss

−Ss+p+

D(ω)

k

∣∣∣∣p�ω+∑N

j=1 nij �(ωf

j −ωij )+εif =0

. (E8)

Finally, the factor∏N

j=1 tj is

N∏j=1

∣∣∣∣∫

Xnj(qj )Xnj

(qj + δqj )dqj

∣∣∣∣2

=[

1 − (2n + 1)ω

4�δq2

]2N

= exp [−(S+ + S−)]. (E9)

The line-shape factor for a single phonon band is

D(ω)

k

∣∣∣∣p�ω+εif =0

exp [−(S+ + S−)]∞∑

s=0

2s + p

s!(s + p)!S

s+p+ Ss

− = D(ω)

k

∣∣∣∣p�ω+εif =0

exp [−(S+ + S−)]

(S+S−

)p/2

×[pIp(2√

S+S−) + 2√

S+S−Ip+1(2√

S+S−)]. (E10)

To generalize the above expression to multiple phonon bands, the normalization factor must be evaluated with a summationover both the band index and the k points within each band. If we use Fj to denote the factor for a band that adds net pj phonons,i.e.,

Fj =∞∑

sj =0

1

sj !(sj + pj )!

{N∏

m=1

tjm

}{N∑

k=1

fjk,−

}sj{

N∑l=1

fjl,+

}sj +pj

, (E11)

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BARMPARIS, PUZYREV, ZHANG, AND PANTELIDES PHYSICAL REVIEW B 92, 214111 (2015)

then, in a similar manner as for the case of a single phonon band, Fj is evaluated to be

Fj =(

nj + 1

nj

)pj /2

exp[−Sj (2nj + 1)]Ipj[2Sj

√nj (nj + 1)]. (E12)

Now we insert the δ function into the product of Fj in the same manner as in the case of a single band to form the fullline-shape factor, one phonon band at a time. For now let us consider the case where all pj ’s are positive. We have∏M

j=1 Fj

Fj ′′

∞∑sj ′′=0

2sj ′′ + pj ′′

sj ′′!(sj ′′ + pj ′′ )!

{N∏

m=1

tj ′′m

}{N∑

k=1

fj ′′k,−

}sj ′′{ N∑l=1

fj ′′l,+

}sj ′′ +pj ′′ −1

×N∑

m=1

fj ′′m,+δ

⎛⎝�ω

f

j ′′m +∑

l∈sj ′′ +pj ′′ −1

�ωf

j ′′l −∑k∈sj ′′

�ωf

j ′′k +N∑

l′=1

nij ′′l′�

(ω

f

j ′′l′ − ωij ′′l′)

+∑

j ′ �=j ′′,l∈sj ′ +pj ′

�ωf

j ′l −∑

j ′ �=j ′′,l∈sj ′

�ωf

j ′l +N∑

j ′ �=j ′′,k′=1

nij ′k′�

(ω

f

j ′k′ − ωij ′k′)+ εif

⎞⎠

=⎛⎝ M∏

j=1

Fj

⎞⎠D(ωj ′′ )

k

∣∣∣∣∑j ′ pj ′ �ωj ′+∑Md

j=1

∑Nl=1 ni

jl�(ωf

jl−ωijl )+εif =0

×{

pj ′′ + 2Sj ′′√

nj ′′ (nj ′′ + 1)Ipj ′′ +1[2Sj ′′

√nj ′′ (nj ′′ + 1)]

Ipj ′′ [2Sj ′′√

nj ′′ (nj ′′ + 1)]

}, (E13)

where j ′′ is one of the phonon bands and we have used Eqs. (E5) and (E6). Summing over all possible j ′′ terms and with anadditional summation over all configurations {pj }, we find

F = 1

k

∑{pj }

⎧⎨⎩⎛⎝ M∏

j=1

Fj

⎞⎠ M∑

j=1

{pj + 2Sj

√nj (nj + 1)

Ipj +1[2Sj

√nj (nj + 1)]

Ipj[2Sj

√nj (nj + 1)]

}D(ωj )

⎫⎬⎭∣∣∣∣∣∣∑M

j=1 pj �ωj +εif =0

. (E14)

If some of the pj ’s are negative, we need to switch the roles of S+ and S− following Ref. [10]. Redefining sj + pj → sj andsj → sj − pj in Eq. (E13), the factor corresponding to pj becomes

−pj + 2Sj

√nj (nj + 1)

I−pj +1[2Sj

√nj (nj + 1)]

I−pj[2Sj

√nj (nj + 1)]

= pj + 2Sj

√nj (nj + 1)

Ipj +1[2Sj

√nj (nj + 1)]

Ipj[2Sj

√nj (nj + 1)]

, (E15)

using the recurrence relation for the Bessel functions. Therefore Eq. (E14) is valid for both positive and negative pj ’s. Applyingthe thermodynamic average to the occupation numbers, nj is replaced by the Bose-Einstein distribution function,

nj → 1

exp(�ωj/kT ) − 1, (E16)

nj + 1

nj

→ exp

(�ω

kT

), (E17)

2nj + 1 → coth

(�ω

2kT

), (E18)

and

2√

nj (nj + 1) → 1

sinh(�ω/2kT ); (E19)

we obtain Eqs. (42) and (43).

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THEORY OF INELASTIC MULTIPHONON SCATTERING . . . PHYSICAL REVIEW B 92, 214111 (2015)

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