1. Introduction
Since the late 1990’s semiconductor NWs research is in expansion because of the huge number of potential applications of these one dimensional structures. This research effort is given expression as a vast number of research articles. Semiconductor NWs are the building blocks of advanced nanodevices; e. g. field effect transistors, interconnects, heterostructured devices, as vertically oriented nanoscale laser diodes and HBTs, thermoelectric converters, photovoltaic devices, and sensors [15].
The thermal transport in NWs is receiving a great deal of attention, because it constitutes a critical issue for the performance and reliability of NWs based devices. Because of the size reduction a large power density needs to be dissipated, especially in devices operating at high current and power densities; which induces a non negligible Joule heating; therefore, the control of the thermal conductivity is crucial to the thermal extraction from the active parts of these devices [6]. On the other hand, bulk semiconductors are usually good thermal and electrical conductors, which limits their potential performance in thermoelectric devices; however, it was demonstrated that surface engineered semiconductor NWs behave essentially as thermal insulators [2], being at the same time good electrical conductors, this means that semiconductor NWs can have a large themoelectric figure of merit [2,79]; which can make them very good thermoelectric materials for waste heat recovery and power generation over a large temperature range.
The experimental measurement of the thermal conductivity at the nanoscale is very challenging [1113], with very few available contrasted measurements. In this context of lack of experimental results, the development of confident models for describing the thermal conductivity of semiconductor NWs is very important in order to design the thermal management of devices, or the thermoelectric power conversion. All this has created the need for a great modeling effort permitting the design of nanodevices in which the thermal transport plays a capital role.
We present herein a review of the thermal transport in semiconductor NWs. The main experimental achievements are presented, both smooth and surface engineered NWs, also alloyed NWs are considered; these data evidence the relevance of the thermal transport problem in these one dimensional structures. These experimental data, though scarce, permit a certain degree of contrast of the theoretical models. According to this we present a critical overview of the main theoretical approaches to the problem of the thermal conductivity of semiconductor NWs following a historic sequence. Along this way, we review a large number of models, among others the classical approaches based on the Callaway formalism, atomistic models based on molecular dynamics simulations, or the hydrodynamic approaches. One revises the contribution of the different resistive phonon scattering processes. Also, the role of normal scattering processes is discussed, as well as the recent consideration of the scattering by optic phonons. Special attention is paid to the boundary scattering which is the main cause for the thermal conductivity suppression in semiconductor NWs. The role of engineered surfaces is also discussed.
The strengths and weakness of the main models are discussed. Finally, we introduce a predictive approach extended to the full temperature range, based on the CallawayHolland formalism, which permits an excellent fitting to the experimental data available. Using this approach empiric relations are deduced for the case of Si NWs, which permit to extract the thermal conductivity for a wide range of diameters and temperatures, which presents an unquestionable interest for practical purposes related to the nanodevice design and thermal managing.
2. Experimental approach to the thermal transport in semiconductor nanowires
The measurement of the thermal conductivity at the nanoscale is a challenging experiment. The first measurements of thermal transport of semiconductor NWs were carried out by D. Li et al. [11,13], they measured the thermal conductivity of a few individual Si NWs grown by the vapour liquid solid (VLS) method. The experiments were performed in a microdevice specially designed for measuring the thermal properties of onedimensional structures [10]. This device, called thermal bridge, consists of a nanowire suspended in between two plates of platinum, one of them acting as the heater, and the other as the thermometer. To guarantee accurate measurements with this device, the junction between the NW and the platinum plates must have an ideally null thermal resistance [10,12].
The results obtained by D. Li et al. [11] showed a strong suppression of the thermal conductivity of the Si NWs as compared with bulk Si (Figure 1A). Furthermore, a dependence of the thermal conductivity with the NWs diameter was revealed; the smaller the NW diameter the lower the thermal conductivity. On the other hand, the experimental data suggest a change in the temperature dependence of the thermal conductivity of Si NWs for diameters below 25 nm; the experimental data give a linear dependence with T for the thermal conductivity of a NW with 22 nm diameter, in which phonon confinement cannot be neglected [15], with respect to the dependence observed for larger diameter NWs, which follow a behavior similar to the bulk(Figure 1B).
Using the thermal bridge device developed by S. Li et al., the thermal conductivity of other semiconductor NWs has been measured [13,1620]. All the NWs measured, independently of their nature display a dramatic reduction of the thermal conductivity with the diameter, clearly revealing that the thermal conductivity of NWs is limited by the NWs dimensions. M. C. Wingert et al. carried out thermal conductivity measurements on Ge NWs [20], showing that below a critical diameter of the NW, the thermal conductivity follows a linear dependence with temperature (Figure 2), in agreement with the behavior of the 22 nm Si NW measured in [11,13]. Therefore, this phenomenon can be considered as an intrinsic property of the NWs with reduced dimension, suggesting changes in the phonon transport when phonon confinement effects start to be relevant.
The thermal conductivity of NWs is not only suppressed by the reduced diameter, but, A. Hochbaum et al. showed that the surface corrugation of the NWs is a major issue for the reduction of the thermal conductivity [2]. Using a similar microdevice to the one described in [10], they measured the thermal conductivity of Si NWs fabricated by an aqueous electroless etching method. These NWs present rough surfaces; in particular, the mean roughness height was estimated near 5 nm, with a lateral roughness correlation length of a few nanometers [2]. The experimental thermal conductivities of two NWs of 115 nm diameter and comparable length, but different surface roughness are represented in Figure 3, showing a five times reduction in the thermal conductivity for the rough NW. Furthermore, in NWs with engineered surfaces, one can expect that the thermal conductivity could be reduced down to almost the amorphous limit [2].
More recently, Y. Park et al. [21] carried out an experiment similar to that performed in [2], but using rough Si NW grown by an VLS method, by which they grew Si NWs with a mean roughness height close to 10 nm, even greater than the NWs used in [2]. However, the thermal conductivity was not so severely suppressed as it did for the electroless etched NWs prepared in [2]. This illustrates that the reduction of the thermal conductivity is not only propitiated by the roughness height, but as J. Lim et al. have recently shown it is the combination of the roughness height and its lateral correlation length, defined as the mean distance between surface hills/depressions, that limits the thermal conductivity in rough NWs [22]. They measured the thermal conductivity of several Si NWs with different surface morphologies, showing that the distance between surface hills/depressions plays a crucial role in the decrease of the thermal conductivity. The SEM images of the NWs measured in [21] compared to the SEM images of the NWs measured in [2], clearly show that the distance between hills/depressions of the NWs of reference [21] are significantly longer than for the NWs of reference [2], which should account for the lower thermal conductivity reduction observed by Lim [22] with respect to the strong reduction reported by Hoschbaum, despite the higher value of the mean roughness height measured in [21].
H. Kim et al. showed the effect of composition on the thermal conductivity of semiconductor NWs. They measured the thermal conductivity of alloyed Si_{1x}Ge_{x} NWs with a small Ge fraction [23]. Even for very diluted alloys the thermal conductivity of the NWs showed a dramatic reduction compared to pure Si NWs. Thereafter, H. Kim et al. grew rough Si_{1x}Ge_{x} NWs and showed the combined effect of the alloy composition and the surface roughness [24], which permits additional reductions of the thermal conductivity; thus, opening almost unlimited possibilities for designing nanodevices by its thermal conductivity, which is very attractive for thermoelectric devices. These results are summarized in Figure 4, where the experimental values for a smooth Si NW, and a smooth Si_{1x}Ge_{x} NW of comparable diameter and a very low Ge fraction (x=0. 004) are shown. The effect of the roughness is summarized in two NWs of the same composition and diameter but differing only by the surface roughness.
In a recent work, Y. Choongho reported additional thermal conductivity data of Si_{1x}Ge_{x} NWs for several compositions [18]; evidencing the relevant role played by the composition, but one cannot be conclusive about the composition bowing, because of the diameter dispersion.
3. Theoretical models
Before the first experimental measurement of the thermal conductivity in a semiconductor NW, a few theoretical approaches already pointed to a thermal conductivity decrease in low dimensional structures. Five years early with respect to the first experimental measurements of NW thermal conductivity, A. Balandin et al. in a pioneering work on the effect of low dimensionality on the thermal transport, presented a theoretical analysis of the lattice thermal conductivity of a semiconductor quantum well [25]. This analysis was built up on the basis that the spatial confinement involves a reduction of the phonon group velocities, therefore, of the phonon relaxation rate. This thermal transport reduction makes evident the increased relevance of the resistive phonon processes in reduced dimension structures. They calculated the thermal conductivity of Si quantum well [25] making use of a reduced Callaway formalism [26], in which only considered the resistive processes given by the Umpklapp scattering, the boundary scattering, and the processes related to the impurity scattering. A detailed description of the Callaway model and the different scattering mechanisms is addressed in paragraph 4. 2. The model proposed in [25] revealed that the thermal conductivity of a Si nanostructure must be substantially reduced with respect to its bulk counterpart. Shortly after, Zou and Balandin applied the same formalism to the study of the thermal conductivity of Si NWs [27]. They modified the simple model described on [25] under the basis of a phonon redistribution by the NW boundaries, which implies a modification of the reduced Callaway expression used in the previous work. On the other hand, the thermal conductivity dependence with the surface roughness of the NWs was also added. The scattering processes considered were a modification of the Umpklapp scattering, the electronphonon scattering, and a modified boundary scattering taking account of the effect of roughness, expressed in terms of the fraction of specularly to diffusively reflected phonons at the NW boundary [28]. With those modifications, the thermal conductivity of a Si NW with 20 nm diameter was calculated, showing a great dependence of the thermal conductivity reduction with the surface characteristics. Nevertheless, even in the case of the maximum surface contribution, the thermal conductivity calculated was higher, 14 W/mK at 300 K for 20 nm diameter, near twice, than the experimental value reported for a 22 nm diameter Si NW [11,13]. In a different approach, A. Voltz et al. made use of molecular dynamics simulations in thin Si NWs with diameters below 5 nm [29], reaching the same result of a dramatic reduction of the thermal conductivity.
The first experimental data about the thermal conductivity of Si NWs opened the possibility to contrast the theoretical models [11,13], allowing the possibility of developing predictive tools regarding the thermal conductivity of NWs. As mentioned above, the experimental measurements evidenced a strong dependence with the NWs morphology, including diameter and surface topography; as a matter of fact, the construction of a model capable of predicting the thermal conductivity of semiconductor NWS becomes a necessary tool for the design of nanodevices in which the thermal transport becomes a critical issue for performance and reliability. N.Mingo build up a model based on a modification of the ballistic phonon transport in a suspended NW [30]. The model takes account of the scattering events that undergo the phonons in a real semiconductor, and requires the calculation of the complete phonon dispersion relations in the NW. The phonon scattering considered by Mingo, as it was the case for the Balandin models [25,27], are the resistive processes given by Umpklapp, boundary, and impurity scattering mechanisms. At difference of the models reported on references [25,27], where exact expressions for the Umpklapp scattering were used, Mingo introduced two parameters which the values are obtained from the adjustment of their model to the thermal conductivity of bulk Si. [30] These parameters are fixed for the calculation of the thermal conductivity of the NWs; therefore, it can be considered as a free of parameters model. Unfortunately, this model underestimates the thermal conductivity value with respect to the experimental data, requiring of an additional adjustment parameter concerning the boundary scattering; this parameter is defined as an effective diameter, greater than the diameter of the NWS experimentally measured in the SEM [30]. Thereby, using effective diameters up to 30% higher than the experimentally measured ones, Mingo obtained a satisfactory fitting to the experimental values reported for NWs with diameters above the limit for phonon confinement. [30,31] On the other hand, Mingo showed that a reduced Callaway model as the one developed by Balandin et al. [25,27] can give a good fitting to the experimental values if one lowers the cutoff temperature with respect to the bulk [30].
Those models only deal with the resistive scattering processes; this means that the contribution of the normal threephonon events is neglected. However, it is well known that these processes contribute to the thermal transport resistance through the Herring mechanism [32]. Y. Chen developed a model based on Monte Carlo simulation considering normal and resistive processes [33]. Making use of a complex genetic algorithm and resorting to two free adjusting parameters, they showed a good fitting to the experimental data in the low temperature range (T<120K), but overestimated the thermal conductivity for higher temperatures. Once again the behavior exhibited by the 22 nm smooth Si NW was not reproduced by the model, which responded reasonably well above this limit.
Coming back to the first models developed by Zou et al. [27], M. Huang et al. carried out a more detailed treatment of the boundary scattering [34], presenting it as a rational function of the fraction of specularly reflected phonons at the boundary instead of the linear dependence originally used in [27]. Despite of being a model with a more realistic treatment of the boundary scattering the result was not satisfactory. On the other hand, they showed that not only the sound velocities decreased due the reduced diameter of the NWs, but the Debye temperatures had also be lower [34]; which, should account for the good agreement to the experimental data reached by Mingo when using cutoff temperatures lower than the bulk silicon cutoff temperatures in the Callaway model [30].
L. Liang et al. used a simple approach in which they established direct relations between the bulk semiconductor melting point and the nanowire melting point [35]; furthermore, they introduced an exponential expression for describing the boundary scattering due to surface roughness. This simple approach was able of predicting the thermal conductivity at 300 K for all the NWs measured in [11], even for the problematic 22 nm Silicon NW. The anomalous behavior of this thin NW, and thus of those NWs with diameter below the limit for phonon confinement,[11,20] was studied by P. Murphy et al. [36]. In contrast to previous works, they argued that in a strongly disordered surface not all the phonons are equally scattered, being these events dependent on the frequency of the incident phonons. Based on these assumptions, they developed a model splitting the behavior of the phonons in two different contributions, determined by the magnitude of the component of their wave vectors in the direction perpendicular to the NW axis. The behavior is nearly ballistic for phonons with a small transverse wave vector component, while for those with a large transverse wavevector component the phonon scattering was assumed to be purely diffusive, with a mean free path of the order of the NW diameter. Under these assumptions, they showed that for the low temperature range, the calculated thermal conductivity approaches to a linear dependence with the temperature for decreasing NW diameters [36]; further showing that the change from the T^{3} law for bulk and thick NWs to the linear dependence for the thinner NWs occurs between 25 nm and 20 nm in diameter, as the experimental measurements demonstrate [11,20].
Z. Tian et al. suggested that the role of the optical phonons must be taken into account for nanostructures [37]. Making use of the phononphonon scattering description derived by A. Ward et al. [38], they carried out the calculation of the thermal conductivity contribution of each phonon branch, showing that the optical phonon contribution to the thermal conductivity in NWs can represent near 20% of the total. Under the assumption of a diffusive regime for the boundary scattering, therefore in the Casimir limit [28], they calculated the thermal conductivity of smooth Si NWs in a good concordance with the experimental values. In a previous work, M. Kazan et al. had already considered the optical phonon contribution [39]. They showed how the consideration of the optical phonon decay into acoustic phonons in the frame of the Callaway formalism can cause an improvement in the prediction of the thermal conductivity of bulk Ge. Subsequently M. Kazan et al. applied this method to smooth Si NWs [40]. It should be noted, the Callaway formalism was conceived without the contribution of the optical modes [4143]; although, it is implicitly taken into account because of the increase of the cutoff temperatures necessary for fitting the experimental data [44]. Therefore, in the modified Callaway model introduced in [40],which explicitly considers the optical contribution, the cutoff temperatures are lower than the cutoff temperatures used when applying the Callaway formalism without considering the optical phonon contribution in a explicit way. Finally, working in the Casimir limit for boundary scattering and making use of a large number of adjustment parameters, they obtained a very good fitting to the thermal conductivity of Si NWs [40].
All those models, which give good fittings to the experimental values of the thermal conductivity for Si NWs work in the diffusive limit for the boundary scattering; this means that one cannot further reduce the thermal conductivity due the boundary scattering when considering rough surfaces. Thereby, when A. Hotchbaum et al. measured experimentally the thermal conductivity of rough Si NWs showing a great reduction in the thermal conductivity of rough NWs compared with the smooth Si NWs [2], none of those models could fit the experimental values.
F. Alvarez et al. showed that the thermal conductivity of semiconductor NWs can be described making use of an analogue hydrodynamic model for the phonon flux [45]. By solving the GuyerKrumhansl equation for the phonon flux in a NW [46, 47], they reached a very simple expression for the thermal conductivity depending on the bulk thermal conductivity, the dimension of the NWs, and the fraction of specularly reflected phonons at the boundary [48]. In a further study, A.Sellito et al. showed that in a hydrodynamic framework, the roughness of the boundaries may cause a local inversion of the phononic flux, thus extra limiting the thermal conductivity of the NW [49]. This result is consistent with a previous work by A. Moore, who showed how the surface roughness of a NW may cause backscattering events resulting in a more aggressive limitation of the thermal conductivity than the diffusive limit considered in the previous models [50]. On this basis, A. Sellito et al. [51] returned to the hydrodynamic model described in [48], which gave a simple expression for the thermal conductivity, and added the contribution of the backscattering events. With this modification, they attained a good fitting to the thermal conductivity of the smooth Silicon NWs below 150 K [51], besides, obtaining values for the thermal conductivity of the rough NWs close to those reported in reference [2] in the same temperature range.
The effect of the surface roughness in the boundary scattering was studied by P. Martin et al. using perturbation theory [52]; the perturbation due to surface roughness was introduced in the Hamiltonian; also, the existence of a thin layer of native SiO_{2} layer surrounding the NW was considered. This model shows values close to the experimental measurements for smooth Si NW, but without a satisfactory fitting to the experimental data of reference [11]. More interesting is the capability shown by the model when one evaluates the thermal conductivity of rough NWs, insomuch as the values are close to those measured by Hochbaum et al. in [2]. On the other hand, they showed that the effect of the boundaries in the thermal conductivity reduction increases the more the smaller the diameter [52]; which suggests a dependence of the thermal conductivity with the diameter and the mean roughness height (
Other approaches making use of molecular dynamics simulations intended to assess the effect of the roughness in the phonon transport [53, 54], showing that the reduction of the thermal conductivity is strongly dependent on the morphology of the surface. On the other hand, D. Donadio using atomistic simulations [55], suggested that this reduction is due to the flattened phonon dispersion curves causing nonpropagating phonon modes. Afterwards, M. Luisier on the basis of the atomistic quantum transport corroborated these results [56]. Moreover, he suggested that there is no need of large mean roughness height to enter in the pure diffusive limit in which the mean free path (MFP) is comparable to the diameter of NW; thus, giving an explanation of why the models that work in this hypothesis for the smooth NWs measured in [11] were getting a good fitting.
J. Shadhu et al developed a model in which they introduced different frequencydependent regimes for the scattering rates and a complex analysis of the roughness contribution [57]. They found that the correlation length plays a major role in the reduction of the thermal conductivity. Short correlation lengths leads to a larger number of phonons suffering diffusive surface scattering events; even more, a large roughness correlation length implies that only the phonons with long wavelengths, and thus phonons that do not significantly contribute to the thermal transport, are scattered by the surface. The correlation length also determines the incident angles for which the phonons suffer a scattering event. On the other hand, they indicate that the short wavelength modes are strongly attenuated in rough boundaries with short correlation lengths, resulting in a MFP for these phonons shorter than the NW diameter, which can be interpreted as a breakdown of the Casimir limit. With this complex model, they obtained a very good fitting to the experimental thermal conductivities for both rough and smooth Si NWs. Even more, the results achieved in this work are consistent with the recent work of J. Lin that demonstrated experimentally the importance of the correlation length together with the roughness mean height on the thermal conductivity reduction in semiconductor NWs [22].
The idea of a breakdown in the Casimir limit for rough NWs was previously pointed out, but without a physical explanation accounting for it. When C. Guthly et al. measured the thermal conductivity of rough gallium nitride NWs [17], they showed that a very good fitting to the experimental values can be achieved using the Callaway model with a modified boundary scattering. Taking into account that the diffusive limit, characterized by a phonon MFP equal to the diameter of the NW, cannot take account of the roughness effect, they used the MFP as an adjusting parameter. A very good fitting to the experimental values was achieved for values of MFP below the NW diameter, therefore, below the Casimir limit. The result of this paper, together with the confirmation of the violation of the Casimir limit for rough NWs [57], allow the use of simple models based on the Callaway formalism for calculating the thermal conductivity of rough NWs, making use of a reduced effective diameter for the boundary scattering.
When considering alloyed NWs, e. g. Si_{1x}Ge_{x} NWs, the thermal conductivity modeling must address the effect of the alloy scattering, besides all the above mentioned scattering terms. N. Yang et al. carried out a first approximation in NWs composed of a mixture of Si isotopes making use of a molecular dynamic simulation [58]. They showed a large reduction in thermal conductivity, due to the isotopic scattering, with a maximum reduction for a composition of 50% between the constituent specimens. Afterwards, J. Chen et al. performed an equivalent molecular dynamics simulation on Si_{1x}Ge_{x} NWs [59], showing a strong dependence of the thermal conductivity on the composition of the NW. Furthermore, they found that the thermal conductivity is severely reduced even for small fractions of Ge, what was later on experimentally demonstrated [23]. They estimated a reduction in the thermal conductivity of 50% with respect to pure Si NWs for a Ge fraction of only 5%; however, one experimentally observes a much greater reduction, even for a smaller fraction of Ge in the NW [23]. In another attempt to modeling the thermal conductivity of Si_{1x}Ge_{x} NWs, Z. Wang [60] suggested a model on the basis of the previous works made by N. Mingo for Si NWs [30]. They assumed that all the scattering relaxation times, except the impurity relaxation time, can be viewed as independent processes for Si and Ge; therefore, they were added according to the Mathisessen’s rule weighted by the composition, while the impurity scattering related to the alloy, is assumed to depend of the composition by a quadratic bowing function. In this way, they obtained a simple model able to achieve a prediction of the thermal conductivity of the order of magnitude of the experimental values obtained in [23]. Unfortunately, although they estimate thermal conductivity values of the order of magnitude of the experimental results, the fitting to the experimental results was not satisfactory, especially in the low temperature range.
Additional work is still needed to supply a good model accounting for the experimental thermal conductivity of Si_{1x}Ge_{x} NWs, and in general of alloyed, or doped NWs. If one observes the historical development of the models elaborated for describing the thermal conductivity of pure semiconductor NWs, one can see that the major advances were achieved once reliable experimental data were available. Very recently, new measurements of Si_{1x}Ge_{x} NWs have been published by Y. Choongho in a wide range of compositions [18], suggesting that more new models concerning alloyed NWs will arrive in the next future.
4. A Predictive model for the thermal conductivity of NWs
4.1. Preliminary remarks
As discussed in the previous paragraph different approaches have been attempted for the description of the thermal conductivity of semiconductor NWs. The measurement of the thermal conductivity at the nanometer scale is a rather complex experimental exercise; usually, the experimental data are hindered by the incertitude introduced by the NW/ heat sink contact. Therefore, the experimental data providing the thermal conductivity of NWs are scarce, and only a few experiments have supplied reliable thermal conductivity data; these data have been used to check the validity of the thermal transport models allowing a significant advance of the modeling formalisms. Concerning the different approaches used for modeling the thermal conductivity of NWs we have already mentioned that, some of them attained a good fitting to the experimental data using complex calculations, some other required of too many fitting parameters; while some other provided a good fitting in a reduced temperature range. The lack of generalized experimental results is a serious problem for both the design of thermoelectric devices based on NWs, and the thermal management of NW based devices. Therefore, a comprehensive model giving a predictive estimation of the thermal conductivity with the factors that contribute to limit it, e. g. the NW diameter, and the surface roughness, would be of high technological interest. On the other hand, experimental data are restricted to temperatures below room temperature, the data at higher temperatures are not available; however, this information is required for many applications that can operate at higher temperatures. We present herein a predictive model based in the wellknown CallawayHolland formalism that substantially simplifies the estimation of the thermal conductivity, provides an excellent fit to the experimental data, and can be used as a tool for all the technical situations in which the thermal conductivity of NWs is required for the design of NW based devices.
4.2. The CallawayHolland formalism
The CallawayHolland formalism has been successfully used for the description of the thermal conductivity of bulk semiconductors [40,42,61]; therefore, it must be conveniently adapted to the nanoscale requirements. It is well known that one of the main consequences of the reduced dimension of the NWs is the appearance of quantum confinement effects; in particular, the phonon confinement is relevant for diameters below 2025 nm [15]. Many of the applications with NWs are built up with NWs with diameters above this phonon confinement limit; therefore, for these NWs one can reasonably assume a bulk like behavior.
First, we will introduce the main features of the CallawayHolland formalism, initially developed for the low temperature range [41,43]. Despite the original limitation in temperature, the formalism was demonstrated to be a very useful tool in the calculation of thermal conductivity of semiconductor materials over the full range of temperatures, by the addition of some modifications.
The thermal conductivity is calculated by solving the Boltzmann equation
where N is the phonon density, and
Where
On the other hand, because a small deviation from equilibrium was assumed one can approximate
By introducing a global relaxation time,
allowing to transform the Boltzmann equation in the more tractable form:
Where
Using this relaxation time, one can estimate the thermal conductivity through the following equation:
Where
The thermal conductivity transforms into the expression
where the subindex refers to the different acoustic branches, the longitudinal and the two transversal branches. The explicit form of the
Replacing the term
And making use of
From this expression and taking account of the independence between
Finally, taking into account that
(16) 
Were the integral limits
4.3. The phonon scattering terms
The application of the CallawayHolland formalism to a particular problem in an extended temperature range resumes in the appropriate characterization of the different scattering terms. Therefore, a detailed analysis of the different scattering processes is required.
4.3.1. Three phonon scattering
This kind of process is determined by the exclusive interaction between phonons and follows two ways. In the first one, a phonon of wave vector q_{1} combines with a phonon of wave vector q_{2} given another phonon of wave vector q_{3}. In the second process a phonon of wave vector q_{1} decays in two phonons of wave vectors q_{2} and q_{3} respectively.
In these processes the energy is conserved; but such conservation does not necessarily hold for the momentum. The processes in which the momentum is not conserved are called Umpkklap, in this process the initial and final moment differs by a non zero reciprocal lattice vector, and the resultant phonons are out of the Brillouin zone; the momentum selection rule for these processes reads as:
Where q’_{3} is the “reflected” phonon of wave vector q’_{3}, and G the reciprocal lattice vector [28]. The phonon wave vector is opposite to q_{1} and q_{2} phonons, resulting in a process resistive to the heat flow.
Peierls proposed for the Umpklapp processes a scattering rate of the form
Regarding the Normal processes, it has been claimed that they do not contribute to the thermal resistance [25, 27, 30, 31, 33]. However, they cannot be neglected because the CallawayHolland formalism requires of the normal process relaxation time [4143]. Furthermore, the normal processes can contribute indirectly to the thermal resistance by redistributing the phonon population, which can further participate in resistive scattering processes. These can be explained by the Herring mechanism [32], which redistributes the phonon momentum among the different phonon branches. In this mechanism, the relaxation of transverse phonons consists of its interaction with a longitudinal acoustic phonon, resulting in another longitudinal phonon. On the other hand, the relaxation of the longitudinal phonons occurs via two different processes; they decay in two transverse phonons; or a longitudinal phonon arises from two transverse phonons. Taking account of these processes, the relaxation time suggested by Herring for each normal process can be written as [32]:
Where the values of the constants
where V is the unit cell volume.
4.3.2. Isotope scattering
The impurities are centers for phonon scattering. This mechanism is described as the isotope scattering and is related to the mass difference between the host atoms and the impurity atoms. Even in the case of pure crystals, without impurities, is necessary to consider the existence of isotopes, which efficiently scatter high frequency phonons. The perturbation in the lattice due to a substitutional atom was evaluated by Klemens using perturbation theory [44,64], and the expression for the relaxation time due to isotope mixing can be expressed as
Where
4.3.3. Boundary scattering
A real crystal has a finite size; therefore, the scattering due to the crystal boundaries cannot be neglected. The boundaries act as highly efficient scattering regions, confining the MFP of the phonons to the order of the dimension of the sample. This problem was discussed by Ziman on the basis of the Casimir theory [28], and for the nanoscaled samples is the dominant scattering mechanism accounting for the reduction of the thermal conductivity. This process can be phenomenologically explained by considering two types of phonon boundary scattering processes; one of them corresponding to the specular phonon scattering, which mainly happens in smooth surfaces; while the other one corresponds to the pure diffusive phonon scattering, which mainly occurs on rough surfaces. Taking account of both processes, which occur with a probability p for the pure specular scattering, and 1p for purely diffusive scattering, Ziman derived an expression for the MFP of phonons in a finite sample as
The key issue of this scattering lies in the form of the probability function, p, which has been addressed in different ways in the literature. Commonly, it has been taken equal to zero [25, 30, 33, 37] or as a constant independent of temperature and frequency [27, 34, 35], thereby simplifying the boundary scattering relaxation time. However, a more realistic approximation can be achieved if one eliminates such oversimplifications. Ziman showed that the probability function p depends on the wavelength of the incident phonons [28], being the surface characterized by the mean value of the height deviation over a surface reference level; this parameter is called the asperity parameter,
Moreover, based on the dependence of the probability function p with frequency and roughness, Soffer addressed a different solution in the presence of a temperature gradient along the sample [68], according to which the probability function takes the form:
where
4.4. The thermal conductivity of NWs
Using this relaxation time, the thermal conductivity can be calculated over the full temperature range. The solution of the complex equation 16 was carried out by the GaussKronrod local adaptive method, fixing a tolerance <10^{9} for enhancing the accuracy. This method provides calculation several times faster than other common numerical methods applied to this kind of problems. In order to test the model, first we carried out a prediction of the thermal conductivity of bulk silicon and germanium, obtaining an excellent fit to the experimental measurements over the full temperature range (Figure 5A and 5B).
The parameters used for the calculations are given in table 1. The values for
The values of the cutoff temperatures,
L (m) 








 
Si  6x10^{3}  6x10^{3}  1 x10^{9}  2. 16 x10^{4}  627. 7  435. 8  8430  5840  1. 1  0. 66 
Ge  4x10^{3}  4x10^{3}  1x10^{9}  6. 08x10^{4}  387. 1  278. 3  4920  3540  1. 1  0. 66 
The equivalent cutoff temperature obtained as
Finally, the equivalent sample diameter, being of the order of millimeters, does not have a big impact in the thermal conductivity for temperatures above 300 K. This allows to adjust the temperature cutoffs for matching the experimental values in this temperature range. Therefore, the equivalent sample diameter is commonly used to matching the thermal conductivity in the low temperature range [61].
On the other hand, one can assume that the phonon confinement effects only show relevant effects below 25 nm in diameter [15]; this means that one can treat the NWs with diameter larger than 25 nm in a bulklike approach.
Having established the excellent agreement with the experimental data for bulk Si and Ge, the same procedure was applied to Si NWs, and the results were compared to those measured in [11]. The parameters used for these calculations are given in Table 2. The agreement with the experimental results was excellent over the full temperature range for NWs without phonon confinement (D>25nm). While the results, as expected, show imprecise below the phonon confinement threshold (D< 25nm). The pseudolinear experimental behavior of the 22 nm NW cannot be predicted by this bulklike model; which suggests that alternative approaches such as the one conducted in [36] are required for describing the thermal transport in NWs of reduced diameter, exhibiting confinement effects. These results are summarized in Figure 6.
Several authors have shown that the speed of sound of the different phonon branches is lower in semiconductor NWs than in their corresponding bulk counterparts, even for NWs with diameters larger than the one for phonon confinement [25, 27, 34, 40, 69]. The values of the sound velocities present differences depending upon the method followed to obtaining them; therefore, the sound velocities are left as free parameters when adjusting the experimental thermal conductivity data. The cutoff temperatures for semiconductor NWs must be also lower than their bulk counterparts [30, 34, 40, 69]. This is consistent with the values of the cutoff temperatures and the sound velocities found by fitting the experimental thermal conductivity data, Table 2.
On the other hand, if one assumes that the crystallinity and the impurity concentration are the same for rough and smooth NWs, the only relevant difference between both types of NWs is the surface roughness; therefore, the thermal conductivity reduction in rough NWs must be exclusively related to the contribution of the roughness to the boundary scattering. The roughness is included in the boundary scattering through the asperity parameter
This large MFP reduction in rough NWs cannot be explained in terms of the surface scattering mechanism described by Ziman [28], and must be considered in a first instance as an admissible calculation tool for obtaining good experimental data approach as was shown in [17]. The thermal conductivities calculated for NWs with the same geometric characteristics as those measured in [2] are shown in Figure 7B, evidencing a very good fitting to the experimental data, using the parameters listed in Table 3. The effective Casimir lengths obtained by fitting the experimental values show a significant reduction with respect to the NW diameters, the ideal Casimir lengths; similar reductions have been also reported by other authors [17].
4.5. A predictive expression for the thermal conductivity of Si NWs
A big constraint for the applications of NWs in which the thermal conductivity plays a major role is the lack of experimental data over a large range of diameters and temperatures, or in its defect the absence of a model supplying such information. In view of the difficulties inherent to thermal conductivity measurements at the nanoscale, and the variety of diameters and surface roughness, the development of a model capable of predicting the thermal conductivity in a wide range of diameters and surface roughness acquires a capital interest.
We have shown that one can achieve an excellent fit of the experimental thermal conductivity of semiconductor NWs using a modified CallawayHolland formalism. This agreement should permit us the prediction of the thermal conductivity for NW diametersand the temperature ranges for which experimental data are not available. Particularly relevant is the case of the high temperature range, above 400 K, for which there are not experimental measurements available. Since the model for bulk Si provides a very good result in the high temperature range, it is reasonable to assume that the same model for NWs without phonon confinement works well in the high temperature range.
In our model, the fitting to the thermal conductivity is achieved using the speeds of sound and the cutoff temperatures for transverse and longitudinal modes, as free parameters. By relating those parameters to the NW diameter, one could predict the thermal conductivity for any NW diameter of a given roughness. In the case of the smooth NWs, a very good result was obtained for NWs with a diameter above the phonon confinement limit. Unfortunately, there are only three diameters for which reliable experimental data are available. However, additional data exist when one considers rough NWs; assuming that the reduction in the thermal conductivity of these NWs with respect to smooth NWs is exclusively related to the contribution of the roughness to the boundary scattering, which does not modify the sound velocities, nor the cutoff temperatures, one can extend the range of available experimental data. This hypothesis was confirmed with rough NWs of 115 nm diameter, for which an excellent fitting was achieved by acting only on the boundary relaxation time, keeping all the other parameters the same as the ones used for the smooth 115 nm NW (see section 4. 4.). Therefore, one can use the values of the adjustable parameters, speed of sound, and cut offtemperature, of both smooth and rough NWs to build up the predictive expression for the thermal conductivity.
The values of the speed of sound and cutoff temperature estimated for each NW diameter for which experimental data are available are plotted in Figure 8 (dots). In this figure one can observe that the behavior of the both velocities, the speed of sound of the longitudinal acoustic branch and the transverse acoustic branch respectively, have the same smooth form. Similar behavior is observed for the cutoff temperatures for each branch. This allows us to assume that those free parameters can be phenomenologically expressed as a function of the NW diameter. In the same figure, we show the nonlinear regression for each free parameter, showing in all the cases an adjustment of
Therefore, with these phenomenological relations one is allowed to interpolate the values of the free parameters used in the calculation of the thermal conductivity of Si NWs in a range of diameters between the phonon confinement limit to slightly above the hundred of nms in diameter, using for that the adjustment parameters given in Table 4. Note that these parameters only represent numbers suitable for describing the sound velocities and the temperature cutoffs as a function of the NW diameter.
a  b  c  
2,362x10^{3}  1,478x10^{4}  4,91310^{7}  
2,563x10^{3}  2,566x10^{4}  9,41410^{7}  
1,668 x10^{1}  3,671x10^{3}  2,969x10^{4}  
5,153 x10^{2}  2,395x10^{3}  1,621x10^{4} 
Using these parameters one can construct the thermal conductivity vs T plots for any NW diameter. These plots are shown in Figure 9A. The thermal conductivity vs the NW diameter for two fixed temperatures, 300K and 600K, are represented in Figure 9B; one observes that at high temperature the influence of the diameter on the thermal conductivity is flattened out with respect to what happens at lower temperatures; which roughly can be interpreted in terms of the balance between boundary scattering processes and three phonon processes. In the low temperature range (T<300K) the boundary scattering is dominant, while in the high temperature range the three phonon scattering processes increase its weight.
5. Summary and perspectives
An overview of the thermal transport in semiconductor NWs has been presented. Both experimental and theoretical aspects have been discussed, evidencing that the theoretical models need of reliable thermal conductivity measurements to contrast the models, in order to advance in the understanding of the thermal transport in semiconductor NWs. Several clues need to be addressed in the close future; in particular, the role of optical phonons scattering, the scattering mechanisms in low diameter NWs, in connection with the phonon confinement phenomena; and a comprehensive view of the boundary scattering in rough surfaces, for which a breakdown of the Casimir limit occurs; also, the modeling of the thermal transport in alloyed NWs has to be addressed because of the importance of alloys for engineering the thermal conductivity. We have developed a model based on a modified CallawayHolland formalism, which agglomerates the contributed ideas provided by different authors. This model permits an excellent fitting of the experimental results over an extended temperature range. Using the fitting of this model to the available experimental results one established empiric relations that permit to build up a predictive tool, which opens the access to the thermal conductivity of any Si NW diameter above the phonon confinement limit. This formalism can be extended to any other semiconductor without loss of generality, provided that reliable experimental thermal conductivity data for a few NWs diameters are available.