12-2

978-1-4799-5288-5/14/$31.00 c⃝ 2014 IEEE 317

Electron-phonon interaction in Si nanowire devices:Low field mobility and self-consistent

EM NEGF simulationsGennady Mil’nikov1,2 and Nobuya Mori1,2

1Department of Electrical, Electronic and Information EngineeringOsaka University, Suita, Japan

2CREST, JST, Chiyoda, Tokyo 102-0075, JapanEmail: {gena |mori}@si.eei.eng.osaka-u.jp

Abstract—The paper presents a method for quantum transportsimulations in nanowire (NW) MOSFETs with inelastic scatteringprocesses incorporated. An atomistic tight-binding Hamiltonianwith realistic electron-phonon interaction is transformed into anequivalent low-dimensional transport model which can be easilyused in full-scaled NEGF simulations. The utility of the methodis demonstrated by computing IV characteristics in n-Si NWdevices.

I. Introduction

Recent technological progress has stimulated a growing in-terest to quasi-one-dimensional quantum transport in nanowire(NW) structures. Experimental studies of Ge/Si NWs haveshown excellent gate control, high drain current and reducedsensitivity to temperature [1], [2]. This opens novel oppor-tunities for the design of nanoscale devices and for exploringquantum transport in low-dimensional systems [3]. Theoreticalmodeling of such nanoscale devices are required to take intoaccount realistic band structure of mobile carries and inelasticscattering effects [4]. The non-equilibrium Green’s function(NEGF) formalism with a tight-binding type of atomisticHamiltonian provides a general framework for such studies[5], [6]. However, complexity of the electronic Hamiltonianand large number of possible electron-phonon (e-ph) scat-tering processes make the atomistic transport simulationsprohibitively difficult.

In the present work we have developed a method to over-come this difficulty. We construct a low-dimensional rep-resentation of one-electron states within an energy intervalrelevant to the transport phenomena and obtain an equivalentmodel (EM) which greatly reduces the computational costs ofatomistic transport simulations and allows inelastic effects tobe incorporated. We employ a strain-dependent sp3d5s∗ tight-binding model (TBM) [7] for the electrons and a generalizedKeating model for phonons [8], [9] in order to calculate theinelastic scattering rates and phonon-limited mobility includ-ing couplings between all possible electronic and phononicstates. We use these exact results as reference data and showthat inelastic interaction can be reproduced by a small numberof effective phonon modes with properly chosen couplingparameters. We demonstrate our approach by constructing

a computationally cheap inelastic EM in various n-Si NWdevices and computing their IV characteristics in the presenceof the inelastic e-ph scattering.

II. Electron-phonon interaction and low field mobility

We first consider the electron-phonon interaction in ananowire close to equilibrium. Under applying a constant elec-tric field E, the distribution function fnk for the electron state inthe n-th band with wave vector k is given by fnk = f 0

nk+eEηnk,where f 0

nk ≡ f 0(Enk) is the equilibrium Fermi distributionfunction. For a weak electric field the deviation ηnk can befound from the linearized Boltzmann equation

∂ f0∂Enk

Vnk =∑mν

∫dq M(nk,mk′)

[ηnk(1 − f 0

mk′ ) − ηmk′ f0nk

]

− M(mk′, nk)[ηmk′ (1 − f 0

nk) − ηnk f 0mk′], (1)

where M(nk,mk′) is the scattering rate between electron states(n, k) and (m, k′ = k + q) and Vnk = �

−1∂Enk/∂k is the groupvelocity. Once equation (1) has been solved, the mobility isgiven by

μ = e∑

nk Vnkηnk∑nk f 0

nk

. (2)

Performing such calculations in a realistic nanostructure re-quires considerable computer resources. The electron-phononinteraction is introduced by expanding a strain-dependent TBHamiltonian H = H(R0

a) + (∂H/∂Ra) δRa with respect toquantized normal modes of the atomic displacements δRa =

Ra−R0a. The contribution from a particular phonon mode (νq)

to the total rate is given by

Mνq(nk,mk′) =|〈Ψmk′ |T νq|Ψnk〉|2

2ωνqMSi

[NBδ(Emk′ − Enk − ωνq)

+ (1 + NB)δ(Emk′ − Enk + ωνq)], (3)

where T νq =∑

a(∂H/∂Ra) eνa(q) is the effective interactionwithin one unit-cell of the wire, Ψnk are the unit-cell normal-ized Bloch states, and (ωνq, eνa(q)) are the phonon frequenciesand the corresponding eigenstates. Figure 1 shows an exampleof the phononic band structure in a [100] Si NW obtained fromthe generalized Keating model [8], [9]. In order to integrate

318

the delta-functions of energy conservation in Eq. (3), the gridof q-points in the first Brillouin zone must be taken denseenough. Typically, ∼ 400–600 q points are required to achieveconvergence and the total number of scattering processes maybe as large as ∼ 106–107 making the mobility calculation veryheavy.

0 1 2 3 4 5 60

20

40

60

qa0

DOS

ħω (m

eV)

(arb. units)

Fig. 1. Phononic band structure of a [100] Si NW with a rectangular crosssection 2.2 × 2.2 nm2. The DOS is shown in the right panel.

III. Calculation of electron mobility in EM representation

The first step of our method is to reduce the original TBMrepresentation. For any TB model with NTBM orbitals perunit cell, the EM method [10] offers a regular procedure toconstruct NEM � NTBM basis functions which give the samephysics within a desired energy interval. Thus, the one-electronstates can be calculated in the form

Ψnk = Φψnk (4)

where Φ is the constant NTBM × NEM real-valued basis matrixwith orthogonal columns ΦTΦ = 1 and the EM Bloch statesψnk are found from a NEM-dimensional eigenvalue problem.Figure 2 shows examples of the band structure in three n-Si NWs used in the simulations. The red points show theresults from the EM (NEM = 44 (a), 54 (b) and 57 (c))which reproduce ΔE ≈ 0.55 (a), 0.35 (b) and 0.3 eV (c) at thebottom of the conduction band. In fact, not all the states inFig. 2 are needed to compute the room temperature mobility.Similar results can be obtained by using smaller EMs withΔE ≈ 0.2 eV. The freedom in choosing EMs of different sizeand reliable energy interval provides a general tool to optimizethe EM calculations and test convergence and accuracy of theresults [10].

The matrix elements can now be evaluated in the EMrepresentation 〈Ψmk′ |T νq|Ψnk〉 = 〈ψmk′ |tνq|ψnk〉, where tνq is aconstant interaction matrix obtained by the basis transforma-tion of the nonzero blocks of T νq. The computational timeis reduced by a factor of (NEM/NTBM)2 ∼ 10−4 making the

0.0

0.2

0.4

0.6

0.0 0.0

0.2

0.2

0.4

0.3

0.3

0.10.1

0 1 2 0 1 2 0 1 2ka0ka0 ka0

E −

Ec (

eV)

(a) (b) (c)

TBMEM

Fig. 2. Conduction band in n-Si NWs with a rectangular cross section 2.2×2.2(a), 3 × 3 (b), and 4 × 4 nm2 (c) along [100] crystal direction. The red pointsrepresent the EM used in the simulations.

calculations in Eqs. (1-3) trivial. Figure 3 shows two examplesof the relaxation time τ−1

t = (∂ f0/∂Enk) Vnk/ηnk obtained bysolving the Boltzmann equation in a [100] n-Si NW withrectangular cross section 2.2 × 2.2 nm2 which only takes ∼ 1hour on an ordinary PC. The fact that τt is close to the totalscattering rate τ−1

0 =∑

mk′ W(nk,mk′) strongly suggests thatone can disregard the phonon dispersion and obtain a simplermodel with similar inelastic transport properties.

0.0 0.1 0.2 0.3E − Ec (eV) E − Ec (eV)

0.0 0.1 0.2 0.3 0.40.0

0.2

0.4

0.6

0.8

τ (1

0−13 s

)

(n = 3 × 1017 cm−3) (n = 2 × 1020

cm−3)τ0 τ0τt τt

Fig. 3. Relaxation time τt (red circles) for low and high carrier concentrationin a [100] n-Si NW with a rectangular cross section 2.2 × 2.2nm2, computedby solving the Boltzmann equation. The black circles refer to the total rateτ−1

0 =∑

mk′ W(nk,mk′).

IV. Model of random optical phonons

In scope of the NEGF method, the full Green’s functionsand inelastic self-energies need to be computed and storiedat many energies points. Such full-scale atomistic simulationsin the original TBM picture simply cannot be done and theEM method offers a promising way to solve the problem. Forexample, in a ∼ 30-nm-length Si NW, the total number ofEM quantum states can be made as small as 2–3× 103 which

319

allows the Dyson and Keldysh equations with arbitrary self-energy terms to be solved easily without much time or storageproblems. On the contrary, constructing the self-energy termsΣe−ph themselves becomes the most consuming part of suchEM NEGF simulations. Indeed, computing the self-energyinvolves operations similar to the ones for the scattering ratesin Eq. (1) and these must be repeated at many energy pointsat each step of the self-consistent iteration.

Here we propose an inelastic EM model which generatesanalytical block-diagonal e-ph self-energy terms and thusoptimizes the NEGF simulations. The model is obtained bysubstituting the actual phonons in the system by a set of simpleequidistant dispersionless modes and adjusting the couplingparameters in order to reproduce the exact electron-phononscattering rates and phonon-limited mobility. To realize thisprogram, we assume a block-diagonal electron-phonon inter-action

Hne−ph =

∑νq

T νeiqa0n

√2MSiων

{bν(q) + b+ν (−q)}, (5)

where T ν is the EM interaction operator (NEM × NEM matrix)for the ν-th phonon mode, n numerates unit-cell and a0 is itssize in the transport direction. Equation (5) is almost the sameas the exact electron-phonon interaction term except that weneglect the phonon dispersion in T ν, ων and omit a minor partof the interaction which comes from the atomic deviationsδRa in neighbor cells. Instead of trying to estimate eachparticular term T ν we introduce equidistant fictitious phononswhich are required to “accumulate” the effect of many originalmodes with similar frequency. For example, in computing thescattering rates we meet the terms ∼ ∑ν ω

−1ν T ν

i jTνkl∗ where i, j,

k, l are the EM indices for electronic states. The contributionfrom many modes around a fictitious phonon with frequencyΩ can be approximated by∑ων≈Ω

1ων

T νi jT

νkl∗δ(ΔE ± ων) ≈ 1

ΩTΩi j T

Ωkl∗δ(ΔE ±Ω). (6)

As follows from the definition of T ν after Eq. (3), variousterms in Eq. (6) have generally very different amplitudes andphases. For a dense enough phonon spectrum, the quantityTΩTΩ in the r.h.s. is analogous to the average of a product oftwo random symmetric matrices and can be approximated by

TΩi j TΩkl∗ → AΩ

(δikδ jl + δilδ jk

). (7)

We note that Eq. (7) remains invariant under arbitrary orthog-onal transformation of the EM basis. Since the terms TΩ enterany physical results only in the form of Eq. (7) the parametersAΩ specify the electron-phonon interaction completely. Forexample, the matrix element for the scattering by the fictitiousphonon Ω is now given by (see Eq. (3))

MΩ ∼∣∣∣〈ψmk′ |TΩ|ψnk〉

∣∣∣2Ω

=AΩ

Ω

[1 +∣∣∣〈ψ∗mk′ |ψnk〉

∣∣∣2] . (8)

The inelastic Σe−ph in the NEGF formalism are calculated ina similar way. For example, the contribution from a mode Ω

in the self-consistent Born approximation is given by

i4πMSiΩ

∫dε[TrG(E − ε) + G̃T (E − ε)

]DΩ(ε) (9)

where DΩ is the usual equilibrium function for frequency Ωand G̃ is the block diagonal part of the corresponding KeldyshGreen’s function. Equation (9) gives analytical block diagonalinelastic self-energies for the system of NEGF equation whichcan be easily solved.

V. Model parameters and EM NEGF simulations

We now assume a set of phonon modes Ωn with yetunknown coupling constants An. We use equidistant modesin order to reduce the number of mutually coupled Green’sfunctions. Our strategy will be to use the exact mobilitycalculations as a reference and adjust the coupling parametersAn in order to reproduce the electron-phonon scattering rates.We thus obtain a numerically cheap inelastic transport modelwhich mimics all the properties of the original TBM andcan be employed on similar grounds for the full-scale NEGFsimulations.

In practice, we find the coupling parameters which minimizethe quantity

S =∑

i

[∑nk

F(Enk − Ei)Δτ0(nk)]2, (10)

where Δτ0 is the deviation of the scattering rate from theexact result and F is a zero centered weight function, whichis introduced in order to avoid too strong influence of anyparticular point (nk) and smoothen the contribution from manystates around the reference energy Ei. We use F = f0(1 − f0),

E − Ec (eV)

n (cm−3)

E − Ec (eV)

n (cm−3)

μ (c

m2

/ Vs)

τ 0 (1

0−13

s)

0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3 0.4

0.4

0.0

0.2

0.6

0.8

180

200

220

240

235

230

2251018 1019 1020 1018 1019 1020

Ω (meV) Ω (meV)20 40 60 20 40 60

ΔΩ = 20 meV ΔΩ = 5 meV

A / Ω

A / Ω

ΔΩ = 20 meV

ΔΩ = 5 meV

Fig. 4. Constructing the EM with equidistant dispersionless phonons forΔΩ = 20 and 5 meV. Upper panels: optimized scattering time τ0 (red circles)versus the exact data from Fig. 3 (black circles). The insets show the effectivestrength A/Ω of the electron-phonon interaction. Lower panels: the roomtemperature mobility as a function of the carrier concentration. The redtriangles refer to the equidistant phonon model and the black lines to thefull spectrum in Fig. 1.

320

n (cm−3)1018 1019 1020

200

400

600

800μ

(cm

2 /Vs)

(a)

(b)

(c)

Fig. 5. The room temperature mobility as a function of the carrierconcentration in [100] Si NWs in Fig. 2. The black lines show the exactresult for the full phononic spectrum. The red triangles refer to the optimizedmodel of 13 equidistant optical modes with ΔΩ = 5 meV.

where f0 is the Fermi function for T = 100 K with 10–20reference Fermi energies. Figure 4 shows τ0(nk) as a functionof energy E(nk) in two optimized models with ΔΩ = 20 meV(3 modes) and ΔΩ = 5 meV (13 modes). The correspondingroom temperature mobility is shown in the lower panels.The model with ΔΩ = 5 meV is found to be in quantitativeagreement with the exact results and can be used in the NEGFsimulations. Similar accuracy has been found in the optimizedmodel for n-Si NWs of larger cross section (see Fig. 5).

Figure 6 presents the calculated IV characteristics in a[100] n-Si NW MOSFET at room temperature with dopantconcentration of 1020 cm−3 at applied bias VSD = 0.1 V.

Figure 7 also shows an example of the mobile chargedistribution at two contacts and illustrates convergence ofthe self-consistent Born iterations. The reduction of the drain

10−3

10−2

10−1

10−4 0

3

6

9

12

−0.3 −0.2 −0.1 0.0 0.1 0.2 0.3VG (V)

I D (μ

A)

BallisticScattering

100

101

Fig. 6. Transport characteristic Id–VG at VSD = 0.1 V of a n-Si NW FETwith rectangular cross section 2.2 × 2.2 nm2. The black lines with trianglesrefer to the ballistic current and the red lines with triangles to the current withelectron-phonon scattering.

Niter2 4 6 8 10 2 4 6 8 10

I D

Ioff (10−4 μA) Ion (μA)2

3

1

E (eV)

3

6

9

1.41.3 1.5 1.4 1.61.5

dQ /

dE (a

.u.)

Drain DrainSource Source

Off state On stateBallisticScattering

E (eV)

Niter

1.6 1.3

Fig. 7. Mobile charge spectrum (upper panels) and self-consistent Borniterations for the drain current (lower panels) in a [100] n-Si NW with arectangular cross section 2.2 × 2.2 nm2. The ballistic charge spectrum in thesame potential is shown for comparison.

current and the phonon subband structure are found to be ina qualitative agreement with the results of the perturbationtheory which further justifies the applicability of the inelasticEM with optimized equilibrium parameters. We finally notethat the EM approach makes it possible to perform the full-scale NEGF simulations with all the phonon modes includedand thus obtain a numerical estimate for the accuracy of thepresent method away from equilibrium. Such test calculationsare currently in progress.

VI. Summary

The method is formulated to construct a low-dimensionalmodel Hamiltonian which reproduces both ballistic and non-ballistic transport in NWs with atomistic resolution. The modelmakes it possible to utilize the full power of the NEGFformalism and study inelastic scattering processes in realisticnanostructures.

References

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