Multiscale Thermal Modeling in Nanoelectronics

International Journal for Multiscale Computational Engineering, 3(1)107-133(2005)

Review: Multiscale Thermal Modeling in Nanoelectronics

S. Sinha∗ & K. E. Goodson Thermosciences Division, Mechanical Engineering Department Stanford University, CA 94305-3030

ABSTRACT

Subcontinuum phonon conduction phenomena impede the cooling of field-effect transistors with gate lengths less than 100 nm, which degrades their performance and reliability. Thermal modeling of these nanodevices requires attention to a broad range of length scales and physical phenomena, ranging from continuum heat diffusion to atomic-scale interactions and phonon confinement. This review describes the state of the art in subcontinuum thermal modeling. Although the focus is on the silicon field-effect transistor, the models are general enough to apply to other semiconductor devices as well. Special attention is given to the recent advances in applying statistical and atomistic simulation methods to thermal transport.

*Corresponding author: [email protected]

1. INTRODUCTION tors. Compared to a bulk device, novel devices such as the strained silicon FET and the vari- Ever since Dennard et al. [1] demonstrated that ous silicon-on-insulator (SOI) [4,5] devices offer faster semiconductor devices could be fabri- better electrical performance, though at the cost cated by reducing device dimensions such that of a higher thermal resistance. This is more so the electric field remained constant, device scal- in the case of more revolutionary designs such ing has been the driver in VLSI technology. In as the three-dimensional FINFET [6]. keeping with Gordon Moore’s 1965 forecast [2], A digital circuit is typically designed such semiconductor devices have continuously di- that the peak transistor temperature does not minished in size over the past four decades exceed 80–100◦C [7]. In current bulk silicon yielding larger scale integration and faster com- devices, the temperature rise is assumed to be puting. However, traditional device scaling largely dictated by heat transfer outside the faces new challenges in the near future as the transistor. The thermal resistances due to the minimum device feature length decreases be- silicon substrate, the thermal interface material low 100 nm. Besides the complexity of de- (TIM), the external heat sink, and convection to signing devices at these length scales, the twin the ambient add up to about 0.5 K/W [8]. How- issues of power dissipation and heat removal ever, for nanotransistors with dimensions as emerge as dominant concerns. The fundamen- small as several tens of nanometers, thermal re- tal unit of an integrated circuit, the metal-oxide sistances arising from subcontinuum phenom- semiconductor field-effect transistor (MOSFET) ena inside the device become equally impor- produces nearly 0.5 mW of Joule heat per unit tant. By using an undoped silicon film as the micrometer width in the current 90 nm gate transistor channel, SOI devices such as the dual length technology. Heat generation occurs gate transistor [4], the ultrathin-body transis- within the inversion layer at the interface of The tor [5], and the FINFET [6] offer better cur- f and g processes are shown in a (110)-plane rent control due to improved electrostatics and to identify the wave vectors of the involved reduced short-channel effects. However, the phonons. The f-process phonon has the wave buried oxide layer increases the thermal resis- vector qf and the g-process phonon has the tance of the device, resulting in a relatively wave vector qg purely from geometry consid- higher temperature rise compared to a bulk de- erations.the channel region and the drain ter- vice for the same power dissipated. Further, minal, over a length of around 20 nm, with the the spatial extent of the heat source can be as peak power density approaching 5 W/µm3 [3]. small as 4 nm (full width at half maximum) in At this length scale, the assumption of contin- a nanotransistor, as shown in Fig. 2, and the uum heat conduction breaks down. Instead, peak power density may approach 60 W/µm3. conduction is described in terms of the energy Thermal measurements near such highly local- quanta of lattice vibrations called phonons. Un- ized heat sources [9] indicate that heat conduc- derstanding thermal transport at the device tion is impeded in the vicinity of the source due level is becoming increasingly important as fu- to ballistic phonon transport. The heat source ture technology evolves toward devices with in a nanotransistor exhibits similar behavior by larger internal thermal resistances and higher forming a localized phonon hotspot with a high heat generation per unit volume. Figure 1 thermal resistance. Thus, innovations on the shows a schematic of the traditional bulk sil- electrical side invariably lead to a poor thermal icon MOSFET as well as various exploratory design, augmenting the thermal resistance in- candidates for future commercial nanotransis- ternal to the device. Performance optimization

International Journal for Multiscale Computational Engineering REVIEW: MULTISCALE THERMAL MODELING IN NANOELECTRONICS 109

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FIGURE 1. The schematics show various FET device structures starting from the bulk silicon MOSFET on the top left. The locations of the source (S), drain (D) and gate (G) terminals are indicated as is location of the buried oxide (BOX) layer. The stained silicon FET uses a strained silicon layer as the channel for electron flow, which enhances the mobility of electrons. The use of dual gates and an ultrathin film silicon channel enables better current control in short-channel devices. The FINFET is a three-dimensional structure with the gate surrounding a vertically oriented fin that serves as the channel

GATE SOURCE DRAIN 120

BURIED OXIDE 100

BULK SILICON 60 y [nm]

0 0 50 100 x [nm]

FIGURE 2. The location of the heat source is shown in an ultrathin body SOI transistor with a gate length of 18 nm [7]. The contours correspond to the spatial distribution of the heat generated inside the device and are equispaced at 0.5 W/µm3 with a peak value of 60 W/µm3 at the center

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requires electrothermal models that incorpo- of plane waves, with the frequency and wave rate microscopic electron-phonon and phonon- vector obeying a nonlinear dispersion relation- phonon interactions. ship. These so-called normal modes of the crys- In this paper, we review heat conduction tal can have multiple polarizations, depend- modeling in semiconductor devices at multi- ing on whether the displacement is perpendicu- ple length scales, from the lattice spacing at lar (transverse) or parallel (longitudinal) to the a few tenths of a nanometer to the phonon wave vector. In the long wavelength limit, the mean free path at a few hundred nanometers. longitudinal vibrations are essentially identical The focus is largely on research from the past to sound waves in a solid. Quantum mechan- decade on silicon nanoelectronics, though ther- ically, the atomic displacement field may be mal modeling in gallium arsenide and other described either as an infinite number of dis- optoelectronic devices is also referenced where tinguishable, quantized oscillators, or as a gas appropriate. In particular, we have chosen of indistinguishable particles called phonons. to ignore more exotic nanoelectronic devices The particle description is particularly useful in such as molecular devices, quantum-effect de- treating interactions with other systems such as vices and single electron devices [10], the rea- electrons. In thermal equilibrium, phonons are son being that thermal aspects of these de- described by the Planck distribution, which ap- vices remain of secondary concern and have re- plies to bosons without rest mass and without ceived very little attention. We begin with an conservation of particle number. overview of subcontinuum heat conduction in The calculation of the heat capacity and the Section 2. In Section 3, we categorize differ- thermal conductivity of a solid requires knowl- ent approaches to modeling heat transport in edge of the phonon dispersion relationship. nanostructures. Investigations of electrother- The formalism for obtaining the dispersion re- mal transport in semiconductor devices are relationship employs the concept of a crystal po- viewed in Section 4. The issue of modeling cou- tential function to describe the chemical bind- pled length scales is discussed briefly in Sec- ing of atoms. The justification for this was tion 5. The review concludes in Section 6 with first provided by Born and Oppenheimer in the a summary of some of the outstanding issues context of molecular motion and is referred to in modeling thermal transport at nanoscales, as as the adiabatic approximation [11]. The ar- well as some of the new issues posed specifi- gument is based on the fact that the frequen- cally by research on nanoscale semiconductor cies of atomic motions are much smaller than devices. the electronic transition frequencies. The electronic system continuously adjusts itself with- 2. LATTICE HEAT CONDUCTION out changing its quantum number as atoms vi- brate about their mean positions. This allows Atoms located at the lattice sites of a dielec- the use of an effective crystal potential with- tric crystal undergo small oscillations about out solving for the electron system. In ad- their equilibrium positions at every tempera- dition to the adiabatic approximation, a sec- ture. The resulting atomic displacement field in ond approximation known as the harmonic ap- the crystal stores and transports energy. Clas- proximation is made in computing the disper- sically, this is described by a displacement vec- sion relationship. The crystal potential is ex- tor obeying a linear, homogeneous wave equa- panded in terms of the powers of the ampli- tion of second order in space and time. Thus, tudes of the vibrations and all terms beyond the field may be described as superpositions the second order are neglected, in effect mak-

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ing the interatomic “spring” harmonic. The phonon interactions and scattering. If the crys- atomic motions are now described by an eigen- tal potential were indeed harmonic, phonons system whose eigenvalues are related to the in an infinite and perfect crystal would have phonon frequencies. Solution of this eigensys- unlimited lifetimes, resulting in infinite ther- tem constitutes the problem of lattice dynam- mal conductivity. We know conclusively that ics [12]. A bulk crystal may be assumed to be this is not true even at very low temperatures. infinite in extent and the Born-von Karman pe- Additionally, a harmonic crystal would not riodic boundary conditions can be applied to show any thermal expansion. In a real crystal, further simplify the equations. Figure 3 shows atomic interactions are anharmonic, causing the phonon dispersion in silicon along the di- the phonons to scatter with each other, even in rections of high symmetry in the crystal. Sili- a defect-free crystal. The lowest-order scatter- con has a diatomic basis which leads to phonon ing process involves three phonons, as shown modes in which atoms in a primitive cell vi- in Fig. 4. A phonon traveling in the crystal can brate out of phase. These are referred to as opti- be annihilated to create two other phonons. cal phonons. The vibrations in which the basis The phonon is described in the schematic by atoms move in phase are referred to as acous- its angular frequency ω, which is a function of tic phonons. Transverse (T) and longitudinal its wave vector q and its polarization s. An (L) polarizations of optical (O) and acoustic (A) inverse process is equally possible from the phonon branches are indicated in the figure. principle of microscopic reversibility. Higher- The harmonic approximation, though useful order processes, such as one involving four in calculating the phonon dispersion relation- phonons, are also possible. All such processes ship, does not capture the physics of phonon- must ensure microscopic conservation of en-

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FIGURE 3. The phonon dispersion relationship is FIGURE 4. Anharmonic interactions between shown along directions of high symmetry in silicon. phonons leads to phonon-phonon scattering. The Longitudinal and transverse polarizations are indi- lowest-order process involves three phonons as cated by L and T respectively. The letters O and A shown schematically on top. Higher-order pro- refer to optical and acoustic branches, respectively. cesses, such as one involving four phonons (bot- The slope of the dispersion curve gives the group tom), are also possible velocity of the particular phonon mode

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ergy and crystal momentum. Anharmonicity is at higher temperatures. The length scale of the treated as a small perturbation to the harmonic problem determines whether phonons may be Hamiltonian for the crystal, and the lifetimes of modeled as particles or whether the wave na- phonons are computed using time-dependent ture of phonons is important. We note here that quantum mechanical perturbation theory [13]. the latter does not necessarily imply a quan- The level of detail that a heat conduction tum mechanical description. In fact, the clas- model must include depends on the length sical description of crystal vibrations involves scale of the problem and the temperature. The lattice waves. The particle behavior comes from phonon distribution obeys quantum statistics the quantum nature of these waves. The im- at low temperatures, but approaches the clas- portant length scales for comparison are the sical limit as the lattice temperature approaches phonon mean free path Λ, the phonon wave- the Debye temperature θD. Thus, quantum me- length λ, and the lattice spacing a0. The valid- chanical details that must be accounted for at ity of various modeling regimes is summarized very low temperatures become less important in Fig. 5. The diffusion length is included for

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FIGURE 5. Subcontinuum thermal modeling depends on the dimension of the system and the temperature. At dimensions comparable to the phonon wavelength and temperatures much smaller than the Debye temperature (bottom left), quantum mechanical nature is strongly manifest. At larger dimensions and room temperatures, a semiclassical approach is more pragmatic. PT in the ﬁgure stands for perturbation theory and α is the thermal diffusivity

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the sake of completeness. Typical silicon semi- the Monte Carlo (MC) [17] technique to sta- conductor device dimensions lie in between tistically simulate phonon transport. Another the phonon mean-free path and the dominant choice is molecular dynamics [18], which is phonon wavelength in room-temperature sili- classical in origin and involves the use of sta- con. As shown in the figure, semiclassical or tistical mechanics to compute transport coeffi- classical models are applicable in this regime. cients. Molecular dynamics does not assume When domain dimensions are comparable to the validity of the BTE, but can be used to com- the phonon wavelength and the temperature pute physical parameters needed by the BTE. is small compared to the Debye temperature, Figure 6 summarizes the hierarchy of semiclas- a quantum mechanical model is necessary. Fi- sical thermal modeling and highlights the main nally, the extent of the crystal can alter the features of each approach. We focus on the phonon dispersion relation [14] and the density methods in this section, and discuss their ap- of phonon states. As dimensions of the crys- plication to semiconductor device problems in tal are reduced, surface effects become impor- the next section. tant and the assumption of an infinite crystal does not hold along one or more dimensions. We review the different approaches indicated in 3.1 Boltzmann Transport Equation Fig. 5 in the following sections. The phonon Boltzmann transport equation describes the rate of change of a statistical distri- 3. MODELING PHONON TRANSPORT IN bution function for phonons. The fundamental NANOSTRUCTURES assumption in deriving the phonon BTE is that there exists a distribution function, Nq,s(r, t), At nanoscale dimensions, it becomes possi- which describes the average occupation of the ble to engineer electron and phonon transport phonon mode (q, s) in the neighborhood of by controlling the physical dimensions of the a location r at a time t. The equation as- structures, which influences available states. sumes the simultaneous prescription of phonon Examples of nanostructures include quantum position and momentum with arbitrary preci- wells, quantum dots, and superlattices, which sion. However, in quantum mechanics these are common in optoelectronics and are cur- quantities correspond to noncommuting oper- rently being investigated for thermoelectric ap- ators and, hence, obey the uncertainty princi- plications [15]. Silicon transistors, in con- ple. The semiclassical BTE for phonons resolves trast, did not start out by exploiting quantum this contradiction by treating phonons as wave mechanical length scales but are nevertheless packets [19] (see Fig. 7 for example) formed approaching such dimensions through device from a superposition of normal modes of the scaling [7]. Since phonon wavelengths domi- crystal. Each wave packet has a small spread nating heat conduction in silicon are approx- δq in the wave vector and is localized in space imately 1–2 nm at room temperature, a full in a region of size δr such that δqδr ∼ 1. The quantum treatment of phonons is rarely war- wave packet is constructed such that its group ranted. In this section, we review general semi- velocity corresponds to the slope of the phonon classical and classical methods for modeling dispersion relationship. For a detailed discus- phonon transport. Semiclassical modeling in- sion on the validity of the wave packet picture, volves either the use of the phonon Boltzmann we refer the reader to Ref. [19]. In the semiclas- transport equation (BTE) [16] with varying de- sical model, the number of phonons becomes a tails of the phonon dispersion relationship, or function not only of the wave vector q and po-

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FIGURE 6. Sub-continuum thermal modeling uses either the phonon BTE or the MC/MD methods. Though all approaches include the role of phonon dispersion, the exact dispersion relationship taken into account differs between various approaches

FIGURE 7. A phonon wave packet [3] created from a linear combination of the eigenmodes is shown. The longitudinal displacements of individual atoms at different axial locations are shown on the left and the corresponding energy distribution is shown on the right

International Journal for Multiscale Computational Engineering REVIEW: MULTISCALE THERMAL MODELING IN NANOELECTRONICS 115

larization s, but also of the spatial coordinate r. of operators is cumbersome and has not been The distribution function evolves in time in a adopted in solving device problems so far. The six-dimensional phase space due to motion in relaxation time approximation has emerged as real space, due to anharmonic scattering with the default choice in almost all of the literature other phonons, and due to scattering with im- on microscale heat conduction. The derivations purities and boundaries. The BTE is formally of the relaxation times involve time-dependent written as perturbation theory from quantum mechanics ¸ (see Ref. [13], for example). Klemens [20] and ∂N ∂N q,s + v · ∇N = q,s (1) others [22,23] have developed semiempirical ∂t q,s q,s ∂t c expressions for the relaxation rates that model thermal conductivity data over a wide range of where (q,s) refers to the phonon mode with temperatures. It is important to note that in wave vector q and polarization s, v is the group the original work [23], the group velocity has velocity, N is the phonon occupation number, been used interchangeably with the phase ve- and the term on the right-hand side is the rate locity. The expressions must be corrected when of change due to collisions. using a realistic nonlinear dispersion relation- The solution to the phonon BTE requires ship [25,26]. evaluation of the collision term. Although A key conceptual problem in using the re- Peierls [19] obtained the formal expression as laxation time approximation is the requirement an integral in wave vector space over all al- for a thermodynamic temperature that governs lowed scattering processes, it is too compli- the scattering rate. Since phonons are not in an cated to be directly evaluated in practice. In- equilibrium distribution, there is no tempera- stead, the relaxation time approximation has ture to strictly speak of. The usual practice in been widely used to model the collision term. such nonequilibrium problems is to define an Under this approximation, the BTE is rewritten ad hoc “equivalent” temperature based on the as local energy [27–30]. We note that a Planck distribution based on the equivalent temperature ∂Nq,s Nq,s − N does not represent the local phonon population + vq,s · ∇Nq,s = − (2) ∂t τq,s in most cases. where N is the Planck distribution at the lo- 3.1.1 Equation of Phonon Radiative Transfer cal temperature and τ is the relaxation time of (EPRT) the mode (q,s). Thermal conductivity models The phonon occupation can be solved to first based on the solution to the phonon BTE under order by linearizing the gradient term in the the relaxation time approximation are available BTE. The solution is of the form in the literature [20–23]. In a more rigorous approach, Guyer and Krumhansl [24] used colli- ∂N sion operators to represent normal (elastic) and N = N(T ) − τq,svq,s · ∇T (3) ∂T umklapp (inelastic) scattering and obtained the formal solution for thermal conductivity in The second term in the solution is the first- terms of the matrix elements of the operators. order departure from equilibrium and con- The relaxation times derived in Refs. [20–23] forms to the Fourier heat flux law. The above appear as special limiting cases in the full op- first-order approximation works well, provided erator solution. However, the computation the temperature gradient is small enough so

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that the temperature does not vary appreciably plays an important role in determining thermal over the relaxation length [20]. This is formally conductivity. expressed as

∂T 3.1.2 Energy Moment of the Phonon BTE vτ ¿ T (4) ∂x An energy moment formulation of the BTE [30] and is usually valid except at length scales com- involves taking a density-of-states-weighted parable to the phonon mean free path. Ma- frequency moment of the BTE. Using the Debye jumdar [27] pointed out the limited usability model, the moment equation is written as of the above solution in modeling transport in ∂e00 e00 − e00 thin films and instead proposed the equation of + v · ∇e00 = − EQ (7) phonon radiative transfer (EPRT). The EPRT is ∂t τ written in terms of the phonon intensity Iω with 00 00 the one-dimensional form being where eEQ and e are the equilibrium energy and the excess energy, respectively, per unit 1 ∂I ∂I I − I0(T (x)) volume per unit solid angle. The excess energy ω + µ ω = − ω ω (5) v ∂t ∂x vτ is given by Z where µ is the direction cosine and v is the De- £ ¤ e00 = ~ω N − N(T ) g(ω)dω (8) bye velocity. The phonon intensity is given as ref X Iω = vN(ω(q, s), x, t)~ωg(ω) (6) Equation 7 can be further simplified by as- s suming an isotropic distribution and integrat- ing over the solid angle. The resulting local en- where g(ω) is the phonon density of states. ergy density can be expressed in terms of the In order to rigorously derive the EPRT from volumetric heat capacity C and a nonequilib- Eq. (2), the nonlinear phonon dispersion rela- rium lattice temperature TL, such that Eq. (7) tionship of Fig. 3 must be replaced by the lin- reduces to [36,37] ear Debye model. The EPRT reproduces the expected radiative behavior in the acoustically ∂T 1 T − T L + v · ∇T = − L EQ (9) thin limit and conforms to the Fourier heat flux ∂t C L τ law in the continuum limit. Besides thin films [27,31,32], radiative 3.1.3 Ballistic-Diffusive Equations (BDE) phonon transport has been used to investigate interfacial transport [33], subcontinuum heat The ballistic-diffusive equations (BDE) [29] are sources [34], and ballistic conduction from based on the phonon BTE and involve an a pri- nanoparticles [28]. Prasher [35] recently ex- ori description of the phonon distribution func- tended the EPRT to include phase information tion. The BDE divides the distribution func- in the in-scattering term on the right-hand side. tion N at any point into a ballistic, Nb, and However, the use of the Debye model limits a diffusive part, Nm. The ballistic part origi- the accuracy of the EPRT. While the Debye nates from the boundary and experiences out- model is very accurate in predicting the specific scattering only while the diffusive part origi- heat capacity of a crystal, it is less accurate nates inside the domain and evolves through in describing transport. Nonlinearity in the in-scattering or through a source term. The phonon dispersion relationship of real crystals equations are formally written as

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50 NON−LOCAL 1 ∂N N FORMULATION b + Ωˆ · ∇N = − b |v| ∂t b |v|τ

∂Nm Nm − N 40 + v · ∇Nm = − (10) HEAT ∂t τ DIFFUSION T (K) ∆ 30 where Ωˆ is the unit vector along the direction of propagation. The BDE has been shown to reasonably ap- 20 0 10 20 30 40 50 proximate the phonon BTE while requiring less x (nm) computations. However, the comparisons were made using an energy moment formulation un- FIGURE 8. The temperature rise near a 40 nm long der a gray-body approximation for phonons. It sheetlike symmetric heat source [38] is computed will be interesting to compare the BDE with the using the nonlocal formulation and compared to phonon BTE for a realistic phonon dispersion that from heat diffusion using Fourier law. The dif- relationship. Another formulation that is conference in the peak temperatures scales as the power ceptually close to the BDE but involves the non- density at the source local formulation described below, is the split- flux model [38]. This is discussed in Section 4. nanoscale phonon source [38] using the nonlocal formulation and the heat diffusion equa- 3.1.4 Nonlocal Formulation tion, respectively. While the nonlocal formulation provides more information than moments The nonlocal formulation [39,40] was devel- of the BTE by solving directly for the phonon oped for heat conduction problems where the distribution function, it becomes computation- phonon mean free path is comparable to do- ally expensive if the geometry is complex. It main dimensions or where the time scale of in- may be advantageous in such cases to opt for a terest is comparable to the phonon relaxation Monte Carlo solution instead. This is discussed times. Under such conditions, Eq. (4) is not sat- in the next subsection. isfied and it is necessary to retain higher-order terms in the departure function. For simple 3.2 Monte Carlo Simulations domains, the departure function can be solved by direct integration of the BTE under the re- The Monte Carlo (MC) method is a statistical laxation time approximation. The solution in- sampling technique [17] that can be applied to volves a nonlocal integral in space and a retar- a wide variety of problems, from simulating dation function in time. The nonlocal formula- neutron trajectories in a reactor to solving an tion has the advantage that it does not require integration in multidimensional space. In the simplifying assumptions about the phonon dis- MC method, a physical process may be simu- persion relationship. The departure function lated directly without requiring knowledge of can be integrated with full phonon dispersion the underlying governing equations. The sole information to yield the heat flux, the local en- requirement is that the process be describable in ergy density, and a nonequilibrium equivalent terms of probability density functions (PDFs). temperature field. Figure 8 compares the tem- The simulation proceeds by sampling from the perature fields computed for a one-dimensional PDFs using a computer generated sequence of

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random numbers. The MC technique can be reintroducing phonons sampled from a Planck used to simulate thermal transport by a distri- distribution based on the local energy density bution of phonons. The method involves track- before annihilation. Despite this shortcoming, ing the trajectories of individual phonons in the scheme shows good agreement with data phase space, given the PDFs associated with for thin-film thermal conductivity and is a very various scattering events. promising start to more detailed MC simula- While there has been rapid development in tions. The power of the method lies in its abil- the MC method for electron transport in semi- ity to study the underlying physics of phonon conductor devices, phonon MC is still in its transport, which cannot be accessed through infancy. This may be attributed partly to a the BTE methods described above. lack of a compelling need to model subcontin- Phonon-phonon scattering events are partic- uum transport in engineering problems. An- ularly difficult to model rigorously in a phonon other explanation is the difficulty of using MC MC scheme. To appreciate the complexity, to simulate particles whose number is not con- one may consider the probability for the sim- served. With the first reason fast vanishing, plest phonon-phonon scattering event involv- phonon MC is likely to draw more attention in ing three phonons. Assuming a momentum the near future. Klistner et al. [41] employed the conserving normal process [16], a phonon with MC method to study radiative phonon trans- frequency ω, wave vector q, and occupation port in a crystal. Phonons were assumed to N can be created from a combination of two travel ballistically between the crystal surfaces phonons with frequencies ω0, ω00; wave vectors and all scattering occurred at the interfaces. Pe- q0, q − q0; and occupations N 0,N 00, respectively. terson [42] reported phonon MC studies that in- Using the time-dependent perturbation theory cluded scattering within the crystal, but used a from quantum mechanics, the rate of transition linear Debye dispersion. from the initial to the final state is Mazumder and Majumdar [43] developed a 2π W = |hf|Φ |ii|2 δ (E − E ) (11) more comprehensive MC scheme for acoustic ~ 3 f i phonons that includes isotropic phonon dispersion. In this scheme, phonons are introduced where f represents the final state, i represents into the crystal based on a probability dictated the initial state, E is the energy, and Φ3 is the by the product of the phonon density of states third-order term in the expansion of the crystal and the Planck distribution function at a lo- potential. The sum total of all three-phonon in- cal temperature. Heat conduction in the crys- teractions yields an expression for the net tran- tal occurs due to the motion of the phonons sition rate as follows [44]: at their individual group velocities. A bound- ¯ ¯ Z 0 0 2 ary may specularly or diffusely scatter phonons ~ X ¯b 0 00 (q, q , q − q )¯ W = dq jj j as well as radiatively emit or absorb phonons. 128π2M 3 ωω0ω00 Phonon-phonon and phonon-impurity scatter- j0j00s × {δ(ω − ω0 − ω00) [(N +1)N 0N 00 ing within the bulk can annihilate a phonon. Energy conservation is the weak point in the − N(N 0 +1)(N 00 +1)] + 2δ(ω + ω0 − ω00) scheme since details of microscopic interactions [(N +1)(N 0 +1)N 00 −NN 0(N 00 +1)]} are overlooked. In the absence of microscopic energy conservation, an ad hoc global conserva- (12) tion scheme is used. In this scheme, the energy where M is the atomic mass, j, j0, j00 represent lost due to phonon annihilation is recovered by indices for the three phonons, and b is a matrix

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element obtained from the third-order deriva- where v1 is the self potential, v2 is the contribu- tive of the potential. The integration is carried tion from pairs of atoms, and v3 is the contri- out over the Brillouin zone of the crystal. To bution from triplets of atoms. A potential that use a rigorous probability for scattering that in- includes terms only up to v2 is called a pair po- cludes the microscopic conservation of energy tential. A three-body potential includes terms and crystal momentum, transition rates such up to v3. Finite difference algorithms [18] such as Eq. (12) need to be evaluated at each time as the Verlet or the Gear predictor-corrector are step in an MC calculation, which is clearly in- used to solve the system at each time step, feasible. The use of semiempirical relaxation which is usually less than a femtosecond. Com- rates [22,23] derived from thermal conductiv- putational constraints dictate that a trade-off be ity modeling is the current method in phonon made between the size of the system and the MC. However, as pointed out in Section 2, these time for the simulation. Since the maximum rates need to be carefully reexamined to remove phonon wavelength is given by the size of the some of the simplifying assumptions regarding system, long wavelength phonons are cut off in the phonon dispersion relationship. An alter- an MD simulation. Use of periodic boundary nate method for evaluating relaxation rates is to conditions and a careful choice of the system use detailed molecular dynamics simulations. size offsets this limitation. Parallelization per- This is discussed in the next subsection. mits simulations of systems comprising tens of thousands of atoms. To provide an idea of computational requirements, a 50 ps simulation of 3.3 Molecular Dynamics Simulations a 10,000 atom silicon system with a three-body potential requires approximately two days of Molecular dynamics (MD) refers to the solu- computer time on 48 nodes of a Pentium-class tion of classical equations of motion (Newton’s cluster. laws) for a set of molecules. In the context of phonon transport in solids, this amounts to At the core of the MD calculation is the inter- solving the dynamics of atoms in a crystal lat- atomic potential used. The nature of the chemi- tice. As described in Section 2, atoms at a lattice cal bonds decides the type of forces that must site are free to undergo small oscillations driven be taken into account by the potential. The by interatomic forces. For a system of p atoms simplest potential is the well-known Lennard- interacting through a potential V , the equation Jones potential that models weak dipole-dipole of motion for the i-th atom is van der Waals bonding. Silicon, however, has highly directional covalent bonds and cannot be described by a pair potential. Three-body m r¨ = f = −∇ V (13) i i i ri potentials [45,46] account for bond directional- ity by including the effect of bond bending in where r is the position, m is the mass, and f is addition to bond stretching, and can describe a the force on the atom. The interatomic potential silicon crystal with reasonable accuracy. For an has the general form excellent comparison of silicon interatomic potentials, we refer the reader to Ref. [47]. X X X The literature on MD simulations of heat V = v1(ri) + v2(ri, rj) transport may be divided into two broad cat- i i j>i egories: the calculation of thermal conductivity X X X and the calculation of phonon relaxation times. + v3(ri, rj, rk) + ... (14) i j>i k>j>i The output from an MD simulation provides

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the position and momenta of the atoms at each ment in thermal conductivity and phonon life- time step. A thermodynamic property may be times. McGaughey and Kaviany [66] have used obtained from this information using statisti- a Lennard-Jone argon crystal to investigate the cal mechanics [18]. Calculation of a transport validity of single-mode relaxation time (SMRT) coefﬁcient such as the thermal conductivity re- approximation, commonly used to solve the quires use of the Green-Kubo relation derived phonon BTE. They compute the thermal con- from the linear response theory [48]. Using lin- ductivity of the crystal using the BTE, with the ear response theory, the thermal conductivity relaxation times derived from MD simulations can be expressed as an autocorrelation function instead of empirical ﬁts. In comparing these of the heat current [48,18] as follows calculations with those based on the Green-

Z ∞ Kubo formulation, they found that the agree- V0 ment depended on the degree to which dis- K = 2 dt hjα(t)jα(0)i (15) kBT 0 persion is taken into account. The general ap- plicability of the semiempirical relaxation time where jα(t) is the heat current at time t (the sub- models [22,23], which employ several fitting script refers to a Cartesian component) and V0 is the volume. Thus, thermal conductivity can be parameters, is thus doubtful. Molecular dy- calculated from the fluctuations in the thermal namics simulations can greatly help in this case current in a system under thermal equilibrium. by providing a methodology to extract relax- The nonequilibrium method involves imposing ation rates for any novel material in a sys- a temperature gradient on the system and com- tematic manner. More general topics on MD puting the resulting heat current. In an alter- simulations are covered in a recent review by nate approach, the temperature gradient may Poulikakos et al. [67]. also be computed in response to an imposed heat current [49]. For a comprehensive compar- 4. ELECTROTHERMAL TRANSPORT ison of equilibrium and nonequilibrium methods to compute the thermal conductivity, we re- Thermal transport in a semiconductor device is fer the reader to Ref. [50]. strongly coupled with charge transport through The literature on thermal conductivity cal- electron-phonon interactions. The electrons in culation using MD covers a variety of mate- the channel gain energy from the applied elec- rials. We point out studies on materials and tric field and subsequently lose it to the lat- structures of current interest: thin films [51], tice while restoring thermodynamic equilib- nanowires [52], superlattices [53,54], strained rium through scattering. Electrons may also materials [55,56], diamond [57], and carbon absorb phonons, thereby gaining energy from nanotubes [58–61]. The use of MD to study the lattice. This occurs near the device source phonon lifetimes and phonon-phonon scatter- where phonon absorption enables the less en- ing is much more limited. The MD method ergetic electrons to overcome the potential bar- has been used to study the lifetimes of vibrons rier. At the drain terminal, electrons shed en- (analogous to phonons) in glasses by Fabian ergy gained from the field by emitting phonons. and Allen [62], and later by Bickham and Feld- In addition, charge carriers also interact with man [63]. Oligschleger and Schön [64] have in- photons during generation and recombination. vestigated the decay of single phonon modes in The overall thermodynamic system compris- selenium and quartz. Ladd et al. [65] have coming electrons, holes, phonons, and photons is pared MD calculations with classical perturba- shown schematically in Fig. 9, along with the tion theory calculations to show good agree- paths for energy exchange.

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4.1 Modeling of Charge Transport

Semiclassical electron transport in a device is governed primarily by three sets of equations [68]. The first is a set of equations for conservation of particles. The second set re- lates the particle currents to the field and the particle gradients. Finally, the Poisson equation provides the electrostatic potential inside the device due to charged immobile dopant atoms and mobile carriers. This basic descrip- FIGURE 9. The schematic, adapted from Ref. [70], tion comprises what is known as the drift- shows the different thermodynamic systems in a semiconductor device and their energy interaction diffusion model [68,69]. The drift-diffusion pathways model does not include an energy conservation equation, but instead assumes isothermal con- Electrothermal modeling in devices may be ditions. This leads to inaccuracies in the dif- split into a two-step problem. The first step fusion term. The next model in the hierarchy involves modeling the transport of charge car- is the hydrodynamic model, which includes riers to extract the heat source distribution in an energy equation and models temperature- the device. The second step is to model the dependent diffusion in the equation for the transport of phonons and calculate the result- particle current. In both the drift-diffusion ing temperature distribution in the lattice. The and the hydrodynamic models, the constitu- interaction of charge carriers and phonons is tive equations for charge continuity and parti- a many-body problem in quantum mechanics, cle momentum are derived by taking moments and a holistic treatment ensuring rigorous self- of the Boltzmann equation for electrons. In consistency is not practical. We classify elec- the hydrodynamic model, heat generation is trothermal modeling in literature as belonging treated rigorously using irreversible thermody- to one of two categories. The first develops namics [70,71] and Onsager’s relations. How- fully coupled models, but makes simplifying ever, the energy transport equation does not in- assumptions regarding transport by electrons clude a subcontinuum description of heat flow. and phonons and their interactions. The second Higher in the hierarchy of electron transport philosophy is to consider heat generation and models are Monte Carlo (MC) device simula- transport as two separate problems and treat tions [72–74], which track the trajectory of elec- each as rigorously as possible. trons inside the device. The MC method includes a quantum mechanical description of In this section, we begin with a brief electron-phonon scattering, but the electrons overview of electron transport modeling in the behave as classical particles, obeying Newton’s context of a transistor. Although we do not dis- laws of motion, in between the collisions. This cuss hole transport explicitly, the approach for semiclassical picture in effect treats the periodic holes is similar to that for electrons. We next field of the crystal quantum mechanically and discuss the physics of heat generation through the externally imposed field classically. Hence, electron-phonon scattering. Finally, we review the electronic states are quantum mechanical, the literature on the two classes of electrother- but the motion of electrons in response to the mal models as discussed above. field is classical. For a rigorous justification of

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the semiclassical model in the context of elec- along these directions [80]. Electron-phonon tron transport, we refer the reader to Ref. [75]. scattering can be categorized as intervalley and In devices with ultrathin bodies such as the one intravalley, depending on whether the scatter- shown in Fig. 2, long-range Coulomb interac- ing moves the electrons within a valley or from tions and quantum effects become important. one valley to another in wave vector space. The MC scheme must be modified in this case Intravalley scattering in silicon is entirely due to include a charge distribution based on the to acoustic phonons, since optical phonons are electronic wave function, which is given by the forbidden from energy and symmetry consid- Schrodinger equation. [76,77]. For a more com- erations [81]. Intervalley scattering, however, prehensive review of charge transport in de- involves mainly optical and zone-edge acous- vices, we refer the reader to Refs. [78] and [79]. tic phonons. Energetic electrons tend to scatter more with optical phonons, since such processes result in a higher energy loss [82]. An 4.2 Electron-Phonon Scattering and Heat electron must have energy in excess of about 51 Generation meV, the minimum energy of optical phonons Whatever model one may choose to describe in silicon, in order to emit an optical phonon. electron transport, the influence of the lattice Phonon modes that conserve crystal momen- cannot be ignored. Electron-phonon scatter- tum in intervalley processes have been calcu- ing limits the electron’s velocity, leading to ve- lated by Long [83]. Intervalley scattering re- locity saturation [68]. A rigorous treatment quires a large change in the wave vector as of electron-phonon scattering is too compli- shown by the arrows in Fig. 10. The two cated to implement in practice. A common ap- types of electron transitions, labeled g- and f- proach is to make the rigid ion approximation, scattering, involve scattering between valleys in which the potential field surrounding each along the same axis and along different axes, ion is assumed to be rigidly attached to it. The respectively. In the reduced zone scheme, both change in potential due to the motion of ions is are umklapp processes as shown in Fig 11. The then described in terms of a deformation poten- g process¡ requires¢ a phonon with wave vec- tial that is related to the volume change of each 2π tor 0.3 × a h001i. The f process requires a cell. For a discussion on the limitations of this model as well as a comparison with other models, we refer the reader to Ref. [16]. We now discuss electron-phonon interactions in silicon from the viewpoint of momentum conserva- g−type f−type tion. Electron transport in solids is analyzed in the wave vector or the reciprocal space of the crystal. In n-type silicon, the electrons involved in transport are located at the bottom of the conduction band [11]. The constant-energy surface at the bottom of the conduction bands in silicon consists of six equivalent ellipsoids along the h100i directions, as shown in the Bril- louin zone representation of Fig. 10. The con- FIGURE 10. The f and g intervalley electronic tran- duction band minima occur at about 0.85 of the sitions [83] are shown in the Brillouin zone of the distance from the zone center to the zone edge crystal

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q : 2π /a x f 3.5 [−0.15, −0.15, −1] 3

q f q : 2π /a x g q 0.3[0,0,−1] 2 g

[001] 1 Phonon Generation (A.U.) [110]

0 FIGURE 11. The f and g processes are shown 0 2 4 6 8 10 ω 13 in a (110)-plane to identify the wave vectors of the (rad/s) x 10 involved phonons. The f-process phonon has the wave vector q and the g-process phonon has the f FIGURE 12. Phonon emission from electron scat- wave vector q purely from geometry considera- g tering at 4 MV/m [82] is shown. The source term in tions the BTE may be extracted from this emission spectrum phonon with wave vector that is 11◦ off h100i. In the sketch of Fig. 11, the wave vector is ap- MC framework. We discuss this approach in ¡ 2π ¢ £ ¤ proximately a 0.15, 0.15, 1 . Further, sym- greater detail in the subsection on detailed ther- metry considerations [84] based on a zeroth or- mal models. der expansion dictate that only LO phonons are allowed in a g process and LA and TO are allowed in an f process. However, it has 4.3 Coupled Subcontinuum Electrothermal been shown by a first-order expansion that low- Modeling frequency TA and LA phonons are also in- The first of the two categories of electrother- volved in the g process [81]. In fact, scattering mal simulations discussed above uses the hy- with these modes must be considered in order drodynamic model for electron transport and to match the mobility data in silicon [80]. The f- an energy moment of the BTE for phonons. and g-transition picture holds up to moderately Fushinobo et al. [36,85] calculated the electron high electric fields (< ≈105 V/cm). and the phonon temperatures in submicrome- Details of electron-phonon scattering, as de- ter gallium arsenide devices. They used a sys- scribed above, are not captured by the drift- tem of coupled energy equations for the lattice. diffusion and hydrodynamic models. The so- Treatment of phonon dispersion was limited phistication of a Monte Carlo simulation is to distinguishing between longitudinal optical needed to model such microscopic physics. and acoustic phonons, with the optical phonons Pop et al. [82] have computed the spectral dis- assumed to be purely capacitive and all heat be- tribution of phonons emitted by hot electrons ing conducted by the acoustic phonons alone. at different electric fields. Figure 12 shows the The nonequilibrium phonon distribution was distribution for a field of 4 MV/m. This de- assumed to be isotropic and the model equa- tailed information is particularly useful in de- tions resembled Eq. (9). Each of the three scribing phonon transport, since it can be used systems, electrons, LO phonons, and acoustic to specify the source term in a phonon BTE or phonons was assumed to remain in local ther-

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modynamic equilibrium with well-defined in- how the phonon distribution emitted by hot dividual temperatures. Further, electrons were electrons in the on-state evolves during the off- assumed to couple only with longitudinal opti- state. The thermal problem remains coupled to cal (LO) phonons over a single relaxation time electron transport in the sense that the obtained of 100 fs and the energy in LO phonons was temperature or phonon distribution is fed back subsequently passed onto a gray distribution of to the electron transport problem to estimate its acoustic phonons over a relaxation period of 10 impact on device characteristics. ps. Device simulations showed that while the 4.4.1 Phonon Hot Spot in a Device electron temperature in the device reached more than 1000 K, the lattice temperature rise A detailed consideration of phonon transport was only about 10 K. Further, the difference be- in a device using the BTE-based models de- tween the LO and the acoustic temperature was scribed in Section 3 shows that the heat gener- less than 10% of the LO temperature. An in- ation region in the device behaves as a phonon teresting observation was that if the gate volt- hot spot. A phonon hot spot is defined as a lo- age was modulated on a time scale of 10 ps, calized region with a nonequilibrium popula- electron-LO relaxation would be unaffected but tion of high-frequency phonons [86]. The heat LO-acoustic relaxation would be retarded. This source is spatially localized to within 20 nm would lead to an energy bottleneck in the LO in a bulk device and to nearly 4 nm in a nan- phonons. Lai and Majumdar [37] adapted the otransistor, as shown in Fig. 2. Additionally, above model for a bulk submicrometer silicon the relaxation rate is approximately 0.1 ps for device. While their study did not show any sig- electrons and on the order of 10 ps for optical nificant rise in the lattice temperature, the au- phonons [81,87]. This disparity, coupled with a thors found that altering the substrate temper- strong excitation of optical phonon modes, re- ature by 100 K decreased the drain current by sults in the heat source having a power density 17% and the electron temperature by 8%. on the order of 5 W/µm3 in a bulk device. The small extent and the high rate of phonon gener- 4.4 Detailed Thermal Modeling ation in the higher frequencies combine to give rise to hot spot effects. These are described be- In the coupled treatment described above, low. the complexity in the phonon dispersion re- Chen [28] used the equation for phonon ra- lationship and the anisotropy in the nonequi- diative transfer under a gray-body approxima- librium distribution are ignored. However, tion to show that the thermal conductivity in both of these are crucial in accurately model- the region surrounding a nanoscale heat source ing phonon transport. In this subsection, we decreases as the ratio of the source size to the review electrothermal models where phonon phonon mean free path. The magnitude of this transport is solved in isolation from electron size effect depends on the degree to which ther- transport. In order to extract the heat source mal transport in the device is truly ballistic. The term, electron transport is modeled during the spread in phonon free paths in room tempera- dynamic period lasting tens of picoseconds, ture silicon is shown in Fig. 13 as a function of when the device is being switched from the on- phonon frequency and polarization. The mean- state to the off-state and vice versa. Thermal free paths for phonons emitted at the hot spot transport is modeled during the off-state when are also provided. Since the mean-free paths electrical conditions are static. The focus is on of the emitted optical and acoustic phonons are

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will scatter inside the contact and create a hot spot. However, if the contacts are metallic, such as in carbon nanotube FETs, thermal transport 4 10 from the hot spot will be dominated by electrons instead of phonons. The impact of the hot spot will partly depend on the current density ) [nm] ω ( LA in the device, which can be quite high in nan- Λ 2 168 nm 10 LO otube FETs [88]. This needs further investiga- TA 18 nm tion. 37 nm TO 1.5 nm 5 10 15 4.4.2 Phonon BTE Modeling ν [THz] Of the three different semiclassical models de- FIGURE 13. The distribution of free paths in scribed in Section 3, phonon BTE-based models room-temperature silicon as a function of phonon have been exclusively used in modeling trans- frequency and polarization is shown. The mean- port in devices. The reason stems from the free path of phonons emitted by hot electrons in a device are also given for comparison. The use of a relatively low computational cost of the BTE gray-body approximation for the heat source leads compared to MC or MD. Sverdrup et al. [30] to large errors in predicting the magnitude of the hot solved a steady-state two-step energy moment spot effect of the BTE for a 400 nm gate length silicon- on-insulator (SOI) device. Phonons were sepa- comparable to the source size, transport is bal- rated into two ﬂuids, one serving as a purely ca- listic in the vicinity of the hot spot. In addi- pacitive reservoir in which all electronic energy tion to the size, the power density at the hot was dissipated, and the other solely responsible spot is also important. If the rate of gener- for heat conduction. The two-ﬂuid model in- ation of phonons that have high frequencies dicated that the temperature rise in the device but relatively low group velocities exceeds the was 160% higher as compared to that predicted rate of their decay into phonons with lower fre- by solving the heat diffusion equation. Naru- quencies but higher group velocities, there is manchi et al. [34] considered the unsteady prob- a bottleneck to energy propagation from the lem under a gray-body approximation to show hot spot. This leads to severe nonequilibrium that the difference between the solutions to the in the high frequency modes. Sinha et al. [38] BTE and the heat diffusion equation depended have shown that this effect is more important strongly on the boundary conditions. Diffuse in nanoscale devices than the above-mentioned scattering at the boundaries caused a larger de- size effect. Since the equivalent temperatures viation from the Fourier law. In a separate arti- are locally high in the vicinity of the hot spot, cle, Narumanchi et al. [89] extended the energy the size effect is reduced due to the shorter moment formulation to include phonon disper- mean free path of phonons outside the hot spot. sion and polarization in the acoustic branches. Finally, we note that though we have discussed As in the above models, the optical branch was hot spots only in the context of silicon, they are assumed to have a zero group velocity. Al- likely to be created in any ballistic device. Elec- though the model accounted for selection rules trons traversing the channel of such a device that govern phonon-phonon scattering, it did

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8 not ensure microscopic energy and momentum x 10 8 conservation. A macroscopic energy conservation equation was developed in terms of equiv- 7 CUMULATIVE HEAT FLUX alent phonon temperatures. The model exhib- 6 LO ited expected behavior in the ballistic and the LA 5 ) TO continuum limits and was able to predict bulk 2 TA silicon thermal conductivity above 300 K with 4 (W/m x reasonable accuracy. J 3

All of the models described above are based 2 on the energy moment of the BTE and do not include a phonon polarization and frequency- 1 dependent heat source term. Electron-optical- 0 0 50 100 150 200 250 300 phonon coupling is assumed to dominate and x (nm) the heat source term is included only in the equation for optical phonons. Since the mod- FIGURE 14. A branchwise breakup of the far- els further assume optical phonons to be purely from-equilibrium heat flux near a hot spot in a bulk capacitive, they tend to exaggerate the tem- device [38] shows the dominance of LO phonons perature rise in the device. Recently, Sinha et al. [38] have proposed a split-flux BTE model the simulations were for polar semiconductors, that solves directly for phonon occupation in- only LO phonons were simulated. Their results stead of phonon energy density. The general indicated that that the phonon population is approach is similar to a nonlocal formulation driven into severe nonequilibrium during pho- with the phonon departure from equilibrium toexcitation and impedes the relaxation of elec- having a second-order term. An isotropic dis- trons and holes. persion relationship that includes all branches and polarizations is taken into account. The 4.4.3 Evaluation of Phonon Relaxation Times source term is obtained through detailed Monte Using MD Carlo simulations [82]. The temperature rise computed from this model is much lower than A critical parameter in all BTE models is the the temperature rise from previous BTE mod- relaxation time for phonon-phonon scattering. els. This is attributed to the dominance of Since the contribution of optical phonons to phonons with nonzero group velocities in the thermal conductivity is much less than that of emission spectrum. This reduces the thermal acoustic phonons, the evaluation of relaxation resistance arising from ballistic transport. The rates for optical phonons has not received much flux due to various phonon branches is shown attention in the literature. However, these rates in Fig. 14. We note that like previous BTE mod- are critical in modeling transport near device els, the split-flux model does not enforce micro- hot spots since the hot spot has a higher pop- scopic energy conservation. ulation of optical phonons. Sinha et al. [3] Although the Monte Carlo method is yet to have used parallel MD to investigate anhar- be used for phonons in silicon devices, Lugli monic scattering of optical phonons at device- and coworkers [90–92] have performed self- like hot spots. The hot spot is modeled as consistent electron-phonon Monte Carlo sim- a wave packet, shown in Fig. 7. Figures 15 ulations in bulk GaAs, bulk InP, and GaAs- and 16 show the evolution of hot spots phonons AlxGa1−xAs heterostructures. However, since observed using MD. The energy in different

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phonon modes may be computed by Fourier analysis of the atomic displacements at any given time. This method allows identiﬁcation 40 of decay channels for hot spot phonons and the 20 computation of relaxation rates. In another re- E [eV] cent MD study, McGaughey and Kaviany [66] 0 used a solid argon system to investigate the validity of the single mode relaxation time ap- 200 proximation used in BTE models. They showed 150 15 10 Time [ps] 5 100 0 ν [THz]

FIGURE 16. The distribution of energy in the phonon spectrum is shown at various times during the decay of phonons at a hot spot with 50 eV energy

that relaxation rates may be directly obtained from MD calculations without requiring ﬁtting parameters used in thermal conductivity modeling.

4.5 Outstanding Issues in Electrothermal Modeling

Compared to the maturity of electron transport modeling in devices, phonon transport modeling is still in the process of development. Key contributions in this area are summarized in Ta- ble 1. Despite the progress made in the past decade or so, there remain numerous outstanding issues in electrothermal modeling, espe- FIGURE 15. Snapshots of anharmonic phonon cially in the context of nanotransistors. The first scattering at a 2 eV hot spot observed in parallelized concerns the use of realistic phonon dispersion MD simulations are shown. The axial displace- relationships in BTE-based models and phonon ments are plotted on the left and the correspond- MC. The phonon dispersion relationship in ing energy distribution in real space is plotted on strained materials and ultrathin films [14] has the right. The x-axis shows the axial location. The been shown to differ from that of the bulk. snapshot at the top is taken just before scattering is The impact of phonon confinement in thin films initiated. As scattering proceeds, phonons of various modes are created, corresponding to the lumps and nanowires [93] on electrothermal transport on the left and the peaks on the right, and propagate needs further investigation. Another important from the hotspot at different group velocities issue is the impact of surface phonon modes or surfons [94] in highly confined device geome-

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TABLE 1. Key contributions on subcontinuum electrothermal simulations are summarized. The nomenclature used is as follows. BC: Boundary condition; T: Temperature; QW: Quantum well; Combination 1: Adiabatic top surface and isothermal substrate; Combination 2: Layout speciﬁc heat loss to contacts and substrate; h: heat transfer coefﬁcient

Approach Dispersion Approx. Device/Material Key Result Sample Ref. Thermal BCs Electron Transport Energy moment Two-fluid 0.2 µm GaAs MESFET Toptical ≈Tacoustic 3 2 BTE [85] h=10 W/m K Coupled, hydrodynamic Telectron weakly influenced by h Energy moment Two-fluid 0.3 µm bulk Si Increase in substrate temperature BTE [37] Combination 1 Coupled, hydrodynamic reduces drain current

Energy moment Two-fluid 0.4 µm SOI Tlattice, peak is 160% higher BTE [30] Combination 1 Decoupled, hydrodynamic than Tdiffusion, peak Non-local Isotropic 90 nm bulk Si LO phonons with non-zero group velocity BTE [38] Combination 2 Decoupled, MC dominate transport near hot spot Monte Carlo Gray LO GaAs, InP, QWs Non-equilibrium phonon distribution [91, 92] – Coupled, MC impedes cooling of photoexcited electrons tries such as the FINFET. Interfacial phonon 5. LINKING MULTIPLE SCALES IN transport [95,96] remains another area of con- THERMAL TRANSPORT cern, especially in three-dimensional structures with numerous interfaces between novel mate- While thermal transport in devices and nanos- rials. tructures involves multiple length scales, the coupling of length scales has not received much attention in electrothermal modeling. The fo- Quantum interference effects become impor- cus has been mostly on the development of tant at the nanometer length scale. As shown methods for subcontinuum length scales. The in Fig. 5, transport in such mesoscopic sys- boundary conditions in electrothermal simu- tems is modeled in terms of either the Lan- lations are typically obtained using simplify- dauer formalism [97] or using the nonequilib- ing assumptions about the geometry for heat rium Green’s function (NEGF) [98] approach. conduction outside the device [101]. How- The former approach applies to ballistic trans- ever, boundary conditions have been shown to port between two reservoirs, which results in a strongly affect the lattice temperature inside the quantized thermal conductance. The more gen- device [37,38]. An accurate treatment of bound- eral NEGF formalism includes scattering and ary conditions requires linking subcontinuum may be thought of as the quantum analogue of thermal transport by phonons inside the de- the BTE. There is very little literature on heat vice to the continuum heat diffusion outside the transport in mesoscopic systems, and this is an device. A multiscale approach that links a fi- important area for future research. As a start, nite volume or finite element method with the the Landauer and the NEGF formalisms have phonon BTE or MC needs development in the been used to investigate heat transport in meso- near future. Such a framework already exists scopic systems in Refs. [99] and [100], respec- for MD in the context of mechanics problems tively. such as crack propagation [102]. The extension

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of methods such as the coupling of length scales ods are fast gaining importance. These ap- (CLS) [103] and the coarse-grained MD [104] to proaches avoid the many simplifying assump- thermal transport needs investigation. tions in a BTE formulation, but are computa- tionally prohibitive for current devices. As devices continue to diminish in size in future, 6. CONCLUDING REMARKS these methods will become increasingly feasi- In this paper, we have reviewed the emerg- ble. While the MD method is fairly well devel- ing field of subcontinuum thermal simulations oped, phonon MC needs more theoretical de- of semiconductor devices. As device dimen- velopment, particularly in terms of improved sions are increasingly reduced below 100 nm in treatment of phonon-phonon scattering and en- the coming decade, thermal modeling will as- ergy/momentum conservation schemes. An sume more importance in device design. Start- important assumption in all thermal modeling ing from a basic overview of heat conduction of semiconductor devices so far has been the by phonons, we have discussed three principal use of a bulk dispersion relationship. Clearly, methods for investigating phonon transport: in devices whose body thickness is only sev- the phonon Boltzmann transport equation, sta- eral atomic layers, phonon dispersion changes tistical Monte Carlo methods, and molecular dramatically [14] and surface effects become dynamics. Most of the past modeling ef- important. This poses a significant challenge, fort at the device level has revolved around since data on phonon dispersion in silicon thin the phonon BTE. Device simulations using the films is not available in the literature. Measure- phonon BTE show that the small size of the heat ment of phonon dispersion in ultrathin films is, source and the high rate of volumetric energy thus, an important step for the future. Addi- dissipation in the lattice leads to the formation tionally, lattice dynamical models for ultrathin of a phonon hot spot. Transport in the vicin- films need to be developed in parallel. ity of the hot spot is ballistic, leading to impeded heat conduction. The resulting temper- ACKNOWLEDGMENTS ature rise exceeds that calculated from the heat diffusion equation. However, the deviation be- S.S. acknowledges financial support from the tween the two depends on the degree to which Stanford Graduate and the Intel Fellowship the phonon dispersion relationship is modeled. programs. This work was supported through Energy moment formulations that assume all Semiconducting Research Corporation Task heat to be generated as optical phonons with 1043. S.S. thanks Eric Pop for providing data zero group velocities predict the difference to for Fig. 12 and for many helpful discussions. be as high as 160%. However, detailed Monte Carlo simulations of electron-phonon scatter- REFERENCES ing show that the hot spot is dominated by op- 1. Dennard, R. H., Gaensslen, F. H., Yu, H. W., tical phonons with nonzero group velocities. If Rideout, V. L., Bassous, E., and LeBlanc, A. R., this nonequilibrium source distribution is in- Design of ion-implanted mosfets with very cluded in a BTE formulation, LO phonons are small physical dimensions, IEEE J. Solid-State found to dominate the heat flux close to the hot Circuits SC-9:256–268, 1974. spot and the deviation is reduced to as low as 2. Moore, G. E., Cramming more components 13% in a bulk device. onto integrated circuits, Electronics 38:114, 1965. While past effort focused on the phonon BTE, 3. Sinha, S., Schelling, P. K., Phillpot, S. R., and molecular dynamics and Monte Carlo meth- Goodson, K. E., Atomistic simulations of g-

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