- Email: [email protected]
Phonon engineering in nanostructures for solid-state energy conversion
Materials Science and Engineering A292 (2000) 155 – 161 www.elsevier.com/locate/msea
Phonon engineering in nanostructures for solid-state energy conversion G. Chen *, T. Zeng, T. Borca-Tasciuc, D. Song Mechanical and Aerospace Engineering Department, 37 -132 Engineering IV, Uni6ersity of California, Box 951597, 420 Westwood Plaza, Los Angeles, CA 90095 -1597, USA
Abstract Solid-state energy conversion technologies such as thermoelectric and thermionic refrigeration and power generation require materials with low thermal conductivity but good electrical conductivity, which are difficult to realize in bulk semiconductors. Nanostructures such as quantum wires and quantum wells provide alternative approaches to improve the solid-state energy conversion efficiency through size effects on the electron and phonon transport. In this paper, we discuss the possibility of engineering the phonon transport in nanostructures, with emphases on the thermal conductivity of superlattices. Following a general discussion on the directions for reducing the lattice thermal conductivity in nanostructures, specific modeling results on the phonon transport in superlattices will be presented and compared with recent experimental studies to illustrate the potential approaches and remaining questions. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Phonon engineering; Nanostructures; Solid-state energy conversion
1. Introduction Solid-state energy conversion technologies such as thermoelectric and thermionic refrigeration and power generation require materials with low thermal conductivity but good electrical conductivity, which are difficult to realize in bulk semiconductors. Nanostructures such as quantum wires and quantum wells provide alternative approaches to improve the solidstate energy conversion efficiency through size effects on the electron and phonon transport [1 – 5]. Phonon heat conduction in nanostructures is a central issue for realizing a high energy conversion efficiency in these structures. Increasing experimental and theoretical studies have demonstrated that thermal conductivity of nanostructures can be significantly reduced below those of their corresponding bulk materials. It is generally believed that the boundary scattering and the phonon group velocity change can be responsible for the reduction in thermal conductivity. For example, an increasing number of experimental works have recently been reported on the reduced * Corresponding author. E-mail address: [email protected] (G. Chen).
thermal conductivity of superlattices in directions parallel and perpendicular to the superlattice plane [6–14]. Theoretically, several models have been developed to explain the observed thermal conductivity in superlattices. The first prediction of a reduced thermal conductivity in superlattices considered increased unklampp scattering due to zone folding and mini-phonon gap formation in superlattices [15]. The predicted thermal conductivity reduction due to such increased scattering mechanisms, however, was not enough to account for the orders of magnitude reduction observed experimentally. Over the last 2 years, several models have been established to explain the reduced thermal conductivity in superlatttices structures [16–22]. In this paper, we start from a qualitative discussion on the minimum thermal conductivity concept and points out the major differences between bulk materials and nanostructures. The discussion leads to possible alternative ways to reduce the lattice thermal conductivity in nanostructures as compared to the traditional methods for bulk materials, through engineering the propagating modes of phonon in these structures. Quantitative modeling results on phonon transport in superlattices will be given to amplify these possibilities.
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2. Phonon engineering for thermal conductivity reduction Ideal thermoelectric materials should have a low thermal conductivity and good electron transport properties (electrical conductivity and Seebeck coefficient). The so called phonon-glass electron-crystal behavior in which phonon thermal conductivity approaches that of glassy materials while the electrons behave as in crystal structures is a succinct representation of ideal thermoelectric materials [23]. In the past, the dominant way to reduce the thermal conductivity is through alloying. The mass difference in alloys scatters phonons more than electrons. In recent years, new ideas such as inserting phonon rattlers in cage structures have also been demonstrated to be effective in reducing the lattice thermal conductivity [24]. The pursuit in lowering the lattice thermal conductivity in bulk materials is to approach the so-called minimum thermal conductivity [25]. The key idea behind Slack’s minimum thermal conductivity theory starts from the following expression of the thermal conductivity K=
1 3
&
vmax
C(v)6(v)L(v) dv
(1)
0
where C is the modal specific heat, v the phonon group velocity, and L the phonon mean free path. Integration is over all allowable frequencies. Slack argued that the minimum thermal conductivity is reached when the phonon MFP equals to its wavelength and later Cahill et al. [26] further limited it to half the wavelength. For acoustic phonons, the speed of sound is used for estimating the minimum thermal conductivity and for optical phonons, the velocity is replaced with the interatomic spacing multiplied by the phonon frequency. While the minimum thermal conductivity is very intuitive for bulk materials, it may not be valid for low-dimensional structures because those structures are highly anisotropic. As we will see from the following discussion of phonon wave and particle transport in
superlattices, both the phonon group velocity and the MFP are directional-dependent quantities. We could rewrite the thermal conductivity expression to account for such directional dependence as K= ×
1 4p
#&
p
& & vmax
0
2p
sin2 8 d8
0
C(v)6(v,u,8)L(v,u,8) cos2 u sin u du
$n
dv
0
(2) where u and 8 are the polar and azimuthal angles formed with the heat flux direction, as shown in Fig. 1. From the above equation, we can see the following alternative approaches to decrease the thermal conductivity value, which is possible through engineered nanostructures. First, the group velocity can be altered in nanostructures. The formation of standing waves in nanostructures means that the group velocity becomes smaller, thus reducing the thermal conductivity. In superlattices, the bulk acoustic phonons can be changed into optical phonons, drastically reducing their group velocity. Second, it is possible to induce anisotropic scattering in low-dimensional structures. For example, interface reflection and transmission are highly angular-dependent. As another example, the optical phonons in two materials have totally different energies. It is likely that the scattering of optical phonons at the interface will be highly directional, i.e. the optical phonons will be scattered backwards. Third, even the specific heat of nanostructures can be changed by changing (a) the density of states and (b) the degrees of freedom in the atomic vibrations. Theoretical studies on superlattices, however, suggest that these changes are not strong except at low temperatures [27,28]. Fourth, it is possible to change the limits of the integrals in Eq. (2). For example, in the angular integration, the phonon transmission above the critical angle is zero. Also, for the frequency integration, the phonon confinement effect may lead to a lower integration limit. Compared with the classical ways of reducing the thermal conductivity in bulk materials, the above discussion shows that nanostructures leads to additional flexibility in reducing the thermal conductivity values by other means. In the following section, we will use superlattice to demonstrate the possibility to engineer the phonon transport in nanostructures. 3. Thermal conductivity of superlattices
Fig. 1. Coordinates used for the thermal conductivity integral, Eq. (1).
We will use superlattices to amplify the discussion made in the above section. Two characteristics of superlattices are distinctly different from bulk materials:
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Fig. 2. (a) Schematic illustration of phonon reflection and transmission at an interface and (b) calculated reflectivity and transmissivity at an interface similar to Si/Ge for a transverse phonon polarized in the plane of incidence (SV phonon) incident from the Ge side, showing the mode conversion [from SV wave into a longitudinal wave (L)] and the total internal reflection phenomena.
the existence of many interfaces and the periodicity in the structure. We will start with a discussion of phonon transport at a single interface, followed by examining phonon transport in superlattices from a wave and a particle point of view.
3.1. Phonon reflection and thermal boundary resistance at an interface Phonons are the quantized normal modes of lattice vibrations. An example is the sound wave, which consists of low frequency phonons. Most of the wave characteristics of phonons can be obtained from the wave equations established under Newton’s second law. Intuitive understanding of heat transfer through an interface can be gained through studying the propagation of acoustic waves at interfaces and by making an analogy to photons [29]. Compared to an electromagnetic wave that is always transverse, a phonon wave can be transverse as well as longitudinal waves. For the long wavelength acoustic waves, the phonon reflection is due to the difference between the acoustic impedance across the interface of two materials. Here the acoustic impedance is defined as the product of the mass density and the speed of sound. In the most general case as illustrated in Fig. 2(a), an incident acoustic wave may excite three reflected and three transmitted phonon waves of different polarizations. The phonon reflectivity and transmissivity can be derived from the continuity requirements for the atom displacement and the traction force at the interface. Fig. 2(b) shows the calculated phonon reflectivity and transmissivity at an interface similar to that between Si and Ge, albeit with isotropic properties such that the two transverse phonon modes are degenerate. The incident phonon is from the Ge side with a transverse wave polarized in the plane of incidence. This figure shows (1) the mode conversion and (2) the total reflection phenomena.
Some of the transverse phonons are converted into longitudinal phonons. For incident angle higher than 33°, the critical angle, total internal reflection occurs. Beyond the critical angle, an evanescent wave exists in the Si side. Such an evanescent wave, however, does not carry net energy flow into the second medium. This means that the integration limit for u in Eq. (2) is reduced from [− 90°, 90°] to [− 33°, 33°] in the Ge side. In terms of heat transfer, one consequence of the phonon wave reflection is the thermal boundary resistance existing at a perfectly flat solid interface, or the Kapitza resistance for the interface between a solid and the liquid helium. Predictions for the thermal boundary resistance, based on the perfect acoustic-mismatch model, are reasonably good at low temperatures, but fail at high temperatures [30,31]. One possible reason is that at high temperatures, the dominant phonon wavelength is shorter and the interface roughness becomes more important, causing diffuse scattering of phonons. Another possible source of discrepancy lies in the mismatch of the phonon spectra between the two materials. Due to the mismatch in the phonon spectrum, the high frequency phonons must split into lower frequency phonons to transmit into the adjacent materials, or they will be limited inside the original medium, causing total reflection. Past work indicates that inelastic scattering should be taken into account in explaining the experimentally measured thermal boundary resistance at high temperature [32]. In general, the detailed interface phonon scattering mechanisms at room temperature are not clear and models are highly idealized.
3.2. Heat conduction in superlattices, treating phonons as wa6es To the accuracy of the harmonic oscillator approximation, the major differences between heat transfer at
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one interface and that in multilayer structures such as superlattices are the phonon interference and tunneling phenomena. Two distinct approaches have been developed for investigating phonon behavior in superlattices. One is based on the transfer matrix method for calculating the transmissivity of phonons and the other is to directly calculate the phonon dispersion as has been typically done in solid-state physics [33,34]. The two approaches lead to similar results. The phonon interference phenomenon in superlattices was demonstrated by directly measuring phonon transmissivity across a superlattice [35]. The measurement showed the familiar dip in the phonon transmissivity under certain wavelengths as observed in optical interference filters. In Fig. 3, we show the phonon transmissivity in a Ge/n(Si/Ge)/ Si/Ge superlattice structure calculated from the transfer matrix method, as a function of the incident phonon frequency. Here the n-period Si/Ge superlattice is sandwiched between two Ge substrates, with an additional Si layer between the superlattice and one Ge-susbtrate. Several stop bands is formed at this particular incident
angle. This translates into the formation of mini-gaps in the phonon spectra of superlattices and the corresponding reduced phonon group velocity. Similar to phonon reflection at one single interface, total internal reflection and mode conversion also happens at the interfaces in superlattices, as shown in Fig. 4. The major difference between a single interface and a superlattice is that tunneling can happen in superlattices above the critical angle, if the layer thickness is smaller than the penetration depth of the evanescent phonon waves. We use the phonon transmissivity results from the transfer matrix method to estimate the heat transfer over the Ge/n(Si/Ge)/Si/Ge superlattice structure [21]. The following assumptions are made: (1) each of the two cladding materials is at a uniform temperature; (2) all materials are isotropic; (3) the thermal phonons can be decomposed into plane waves; (4) internal scattering inside each layer is negligible; and (5) phonon scattering at interfaces is specular and elastic. In addition, the contribution of optical phonons is neglected, which is reasonable because of (1) the small group velocity of the optical phonons and (2) the large mismatch of the optical phonon energies between Si and Ge. Since the internal scattering is totally neglected, the model is not expected to capture all the heat conduction mechanisms occurring in superlattices. The emphasis here is on the impact of the phonon group velocity reduction and the mini-gap formation on the thermal conductivity. When the temperature difference between the two cladding media is small, the effective thermal conductance of the superlattice can be calculated from the following relation K= q/DT = (1/2) % ×
Fig. 3. Phonon transmissivity over a Ge/10(Si/Ge)/Si/Ge superlattice with each layer at 5 A, for a SV wave incident at 17.7°, showing the stop bands caused by the phonon interference.
Fig. 4. Phonon transmissivity over a Ge/10(Si/Ge)/Si/Ge superlattice as a function of the angle of incidence and the frequency.
&
p 1
& vpm
'v6pD(v) df/dT
0
tp (x,v,d)x dx dv
(3)
0
where f is the Bose–Einstein distribution at temperature T, 6p the phonon group velocity, v the phonon angular frequency, t the phonon transmissivity, and x(= cos u) the directional cosine. The summation is made over the three phonon polarizations. Fig. 5 shows the thermal conductance of superlattices made of different periods as a function of the layer thickness, calculated from the Debye model. In the same figure, we also give the results for a Ge/Si/Ge double heterojunction structure. Both the wave and the ray-tracing results are shown for this latter structure. The ray-tracing method neglects the interference and tunneling effects of the phonon waves. For the single layer structure (n= 0), the tunneling and interference effects become important if the layer thickness is thinner than 10 A, . Tunneling has a more profound impact on K than interference effects, and tunneling increases the thermal conductance as the film becomes thinner. For films thicker than 10 A, , both the wave and the ray
G. Chen et al. / Materials Science and Engineering A292 (2000) 155–161
Fig. 5. Thermal conductance of a Ge/n(Si/Ge)/Si/Ge superlattice structure calculated by the transfer matrix method, where the layer thickness is the thickness of each individual layer in the superlattice (equal thickness is assumed). The ray tracing results neglect the interference and tunneling effects.
tracing methods give the same results. As a comparison, we also calculated the thermal conductance for a transparent interface by assuming that the transmissivity equals one in Eq. (3). The conductance obtained under this circumstance is 0.1027 W m − 2 K − 1, which is a factor of 10 higher than that obtained by the ray tracing method after considering the interface reflection due to the acoustic and the frequency mismatches. Out of this factor of 10 reduction of the interface conductance due to the phonon reflection, about a factor of four of which is due to the acoustic mismatch (the difference in the acoustic impedance) and a factor of 2.5 due to the frequency mismatch. The simulation results in Fig. 5 show that for superlattices, the tunneling and interference effects become stronger for shorter periods. Compared with a single layer, the thermal conductance of superlattices is further reduced due to multiple reflections and the stop band formation, which translate into the phonon group velocity reduction. The value of the conductance changes little as the layer becomes thicker. The convergence of the conductance at large periods indicates that the superlattice approaches one with an infinite number of repetitions. The thermal conductivity for such an infinite repetition superlattice structure, however, would diverge to infinite because we neglected internal scattering.
3.3. Heat conduction in superlattices, treating phonons as particles One major difficulty in the wave approach of heat conduction in superlattices lies in the treatment of interface conditions. The wave calculation is most suitable for specular interfaces. Past experimental results on the thermal boundary resistance suggest that at
159
room temperature, diffuse scattering at interfaces is possible and may even be dominant. We have taken an approach based on treating phonons as particles and solving the Boltzmann transport equation (BTE) for phonon transport with heat flow in both the in-plane and the cross-plane directions [17,19,20]. The input parameters for these models include the phonon mean free path (MFP), specific heat, mass density, and group velocity in the each bulk material constituent of the superlattices. Previously developed models for the interface thermal boundary resistance [30,31] have been extended to include phonon dispersion and the possibility of inelastic scattering. Because the mechanism for phonon scattering at interfaces is not clear, we assumed that the interfaces are partially diffuse and partially specular and used a fitting parameter p to represent the fraction of specularly scattered (reflected and transmitted) phonons at the interface. Interface scattering is treated as a boundary condition in solving the BTE, rather than as a volumetric scattering process. Results of the modeling depend strongly on the assumed interface scattering processes. Fig. 6(a) and (b) compare the cross-plane experimental data with modeling results obtained from two different interface scattering models: one assuming totally elastic interface scattering while the other totally inelastic. The elastic scattering model overpredicts the thermal conductivity reduction while the inelastic scattering model leads to reasonable agreement with experimental data, indicating that inelastic scattering must occur at the interface. We should point out that phonon zone folding and quantum confinement phenomena have been observed experimentally using for example, Raman spectroscopy [36]. There exist, however, no quantitative assessments presently of the magnitude of the phonon confinement. The source of thermal conductivity reduction in the cross-plane direction as reveal by the particle models is the thermal boundary resistance. Fig. 7 shows the temperature drop inside each layer and across the interfaces of one period of a superlattice. It shows that most of the temperature drop occurs at the interfaces due to the thermal boundary resistance phenomenon we explained before, although for superlattices, the multiple reflection at interfaces alter the expression for calculating this interfacial thermal boundary resistance. When the phonon MFP is larger than the film thickness, it is not the bulk thermal conductivity, but the mismatch of the density, group velocity, phonon spectrum, and the specific heat of the two adjacent materials that controls the thermal conductivity of the superlattice structure. Fig. 6 indicates that the thermal conductivity depends on strongly whether the interface scatters phonons specularly or diffusely, i.e. the interface scattering parameter p, which represents the fraction of specularly scattered phonons. The dependence is even more drastic in the in-plane thermal conductivity, as
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Fig. 6. Comparison of models with reported thermal conductivity of Si/Ge superlattices based on the assumption of (a) elastic interface scattering and (b) inelastic interface scattering.
shown in Fig. 8. If the interface is totally specular, little reduction in the thermal conductivity occurs. On the other hand, slight diffuse phonon scattering causes a very large reduction in the thermal conductivity. This suggests the possibility of engineering the interfaces to scatter phonons selectively, rather than scattering electrons. Both the wave and the particle models seem to be able to predict an order of magnitude reduction in the thermal conductivity in superlattices in the cross-plane direction, this is not surprising since one major source of thermal conductivity reduction — the total internal reflection — is correctly treated in both models. The phonon frequency mismatch can also be approximately included in the particle model. The group velocity reduction in the wave model is partially included in the particle model through the consideration of multiple reflection. The stop bands (minigaps in the phonon spectrum) and the tunneling, however, cannot be treated through the particle model. In theory, it is always better to start from the wave approach if such a model could be established to take into account for the lose of coherence, the bulk and interface scattering processes. It seems that the key is whether the phonon scattering is specularly or diffuse, or in between. If the interfaces are totally specular, the wave models are more suitable. So far there has no prediction on the thermal conductivity in the in-plane direction based on the wave model. The particle model results suggest that the in-plane thermal conductivity reduction is due to mainly the diffuse scattering of phonons at interfaces. It will be very interesting to check whether the in-plane
thermal conductivity reduction could be explained by the wave model.
4. Concluding remarks Solid-state energy conversion technologies such as thermoelectric and thermionic refrigeration and power generation require materials with good electrical properties but low thermal conductivities. It has been proven that band engineering for electrons can lead to better electron transport properties for energy conversion in nanostructures. In this work, we discuss possibilities to engineer phonon transport in nanostructures
Fig. 7. Temperature distribution in one period of a superlattice for heat conduction in the cross-plane direction, demonstrating that most of temperature drop occurs at the interface.
G. Chen et al. / Materials Science and Engineering A292 (2000) 155–161
Fig. 8. The in-plane thermal conductivity of GaAs/AlAs superlattices.
for reaching low thermal conductivity values. Nanostructures provide alternative ways to reduce the lattice thermal conductivity compared to bulk materials through the anisotropy of the structure, the phonon spectrum change, and interface scattering. Some of these possibilities are illustrated through modeling of phonon transport in superlattices. This is clearly a subject at its infancy and intensive collaborative effort is needed from different disciplines to bring the field into fruition.
Acknowledgements This work is supported by a DOD MURI program on thermoelectrics and an NSF young investigator award to G.C.
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Phonon engineering in nanostructures for solid-state energy conversion G. Chen *, T. Zeng, T. Borca-Tasciuc, D. Song Mechanical and Aerospace Engineering Department, 37 -132 Engineering IV, Uni6ersity of California, Box 951597, 420 Westwood Plaza, Los Angeles, CA 90095 -1597, USA
Abstract Solid-state energy conversion technologies such as thermoelectric and thermionic refrigeration and power generation require materials with low thermal conductivity but good electrical conductivity, which are difficult to realize in bulk semiconductors. Nanostructures such as quantum wires and quantum wells provide alternative approaches to improve the solid-state energy conversion efficiency through size effects on the electron and phonon transport. In this paper, we discuss the possibility of engineering the phonon transport in nanostructures, with emphases on the thermal conductivity of superlattices. Following a general discussion on the directions for reducing the lattice thermal conductivity in nanostructures, specific modeling results on the phonon transport in superlattices will be presented and compared with recent experimental studies to illustrate the potential approaches and remaining questions. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Phonon engineering; Nanostructures; Solid-state energy conversion
1. Introduction Solid-state energy conversion technologies such as thermoelectric and thermionic refrigeration and power generation require materials with low thermal conductivity but good electrical conductivity, which are difficult to realize in bulk semiconductors. Nanostructures such as quantum wires and quantum wells provide alternative approaches to improve the solidstate energy conversion efficiency through size effects on the electron and phonon transport [1 – 5]. Phonon heat conduction in nanostructures is a central issue for realizing a high energy conversion efficiency in these structures. Increasing experimental and theoretical studies have demonstrated that thermal conductivity of nanostructures can be significantly reduced below those of their corresponding bulk materials. It is generally believed that the boundary scattering and the phonon group velocity change can be responsible for the reduction in thermal conductivity. For example, an increasing number of experimental works have recently been reported on the reduced * Corresponding author. E-mail address: [email protected] (G. Chen).
thermal conductivity of superlattices in directions parallel and perpendicular to the superlattice plane [6–14]. Theoretically, several models have been developed to explain the observed thermal conductivity in superlattices. The first prediction of a reduced thermal conductivity in superlattices considered increased unklampp scattering due to zone folding and mini-phonon gap formation in superlattices [15]. The predicted thermal conductivity reduction due to such increased scattering mechanisms, however, was not enough to account for the orders of magnitude reduction observed experimentally. Over the last 2 years, several models have been established to explain the reduced thermal conductivity in superlatttices structures [16–22]. In this paper, we start from a qualitative discussion on the minimum thermal conductivity concept and points out the major differences between bulk materials and nanostructures. The discussion leads to possible alternative ways to reduce the lattice thermal conductivity in nanostructures as compared to the traditional methods for bulk materials, through engineering the propagating modes of phonon in these structures. Quantitative modeling results on phonon transport in superlattices will be given to amplify these possibilities.
0921-5093/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved. PII: S0921-5093(00)00999-0
G. Chen et al. / Materials Science and Engineering A292 (2000) 155–161
156
2. Phonon engineering for thermal conductivity reduction Ideal thermoelectric materials should have a low thermal conductivity and good electron transport properties (electrical conductivity and Seebeck coefficient). The so called phonon-glass electron-crystal behavior in which phonon thermal conductivity approaches that of glassy materials while the electrons behave as in crystal structures is a succinct representation of ideal thermoelectric materials [23]. In the past, the dominant way to reduce the thermal conductivity is through alloying. The mass difference in alloys scatters phonons more than electrons. In recent years, new ideas such as inserting phonon rattlers in cage structures have also been demonstrated to be effective in reducing the lattice thermal conductivity [24]. The pursuit in lowering the lattice thermal conductivity in bulk materials is to approach the so-called minimum thermal conductivity [25]. The key idea behind Slack’s minimum thermal conductivity theory starts from the following expression of the thermal conductivity K=
1 3
&
vmax
C(v)6(v)L(v) dv
(1)
0
where C is the modal specific heat, v the phonon group velocity, and L the phonon mean free path. Integration is over all allowable frequencies. Slack argued that the minimum thermal conductivity is reached when the phonon MFP equals to its wavelength and later Cahill et al. [26] further limited it to half the wavelength. For acoustic phonons, the speed of sound is used for estimating the minimum thermal conductivity and for optical phonons, the velocity is replaced with the interatomic spacing multiplied by the phonon frequency. While the minimum thermal conductivity is very intuitive for bulk materials, it may not be valid for low-dimensional structures because those structures are highly anisotropic. As we will see from the following discussion of phonon wave and particle transport in
superlattices, both the phonon group velocity and the MFP are directional-dependent quantities. We could rewrite the thermal conductivity expression to account for such directional dependence as K= ×
1 4p
#&
p
& & vmax
0
2p
sin2 8 d8
0
C(v)6(v,u,8)L(v,u,8) cos2 u sin u du
$n
dv
0
(2) where u and 8 are the polar and azimuthal angles formed with the heat flux direction, as shown in Fig. 1. From the above equation, we can see the following alternative approaches to decrease the thermal conductivity value, which is possible through engineered nanostructures. First, the group velocity can be altered in nanostructures. The formation of standing waves in nanostructures means that the group velocity becomes smaller, thus reducing the thermal conductivity. In superlattices, the bulk acoustic phonons can be changed into optical phonons, drastically reducing their group velocity. Second, it is possible to induce anisotropic scattering in low-dimensional structures. For example, interface reflection and transmission are highly angular-dependent. As another example, the optical phonons in two materials have totally different energies. It is likely that the scattering of optical phonons at the interface will be highly directional, i.e. the optical phonons will be scattered backwards. Third, even the specific heat of nanostructures can be changed by changing (a) the density of states and (b) the degrees of freedom in the atomic vibrations. Theoretical studies on superlattices, however, suggest that these changes are not strong except at low temperatures [27,28]. Fourth, it is possible to change the limits of the integrals in Eq. (2). For example, in the angular integration, the phonon transmission above the critical angle is zero. Also, for the frequency integration, the phonon confinement effect may lead to a lower integration limit. Compared with the classical ways of reducing the thermal conductivity in bulk materials, the above discussion shows that nanostructures leads to additional flexibility in reducing the thermal conductivity values by other means. In the following section, we will use superlattice to demonstrate the possibility to engineer the phonon transport in nanostructures. 3. Thermal conductivity of superlattices
Fig. 1. Coordinates used for the thermal conductivity integral, Eq. (1).
We will use superlattices to amplify the discussion made in the above section. Two characteristics of superlattices are distinctly different from bulk materials:
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157
Fig. 2. (a) Schematic illustration of phonon reflection and transmission at an interface and (b) calculated reflectivity and transmissivity at an interface similar to Si/Ge for a transverse phonon polarized in the plane of incidence (SV phonon) incident from the Ge side, showing the mode conversion [from SV wave into a longitudinal wave (L)] and the total internal reflection phenomena.
the existence of many interfaces and the periodicity in the structure. We will start with a discussion of phonon transport at a single interface, followed by examining phonon transport in superlattices from a wave and a particle point of view.
3.1. Phonon reflection and thermal boundary resistance at an interface Phonons are the quantized normal modes of lattice vibrations. An example is the sound wave, which consists of low frequency phonons. Most of the wave characteristics of phonons can be obtained from the wave equations established under Newton’s second law. Intuitive understanding of heat transfer through an interface can be gained through studying the propagation of acoustic waves at interfaces and by making an analogy to photons [29]. Compared to an electromagnetic wave that is always transverse, a phonon wave can be transverse as well as longitudinal waves. For the long wavelength acoustic waves, the phonon reflection is due to the difference between the acoustic impedance across the interface of two materials. Here the acoustic impedance is defined as the product of the mass density and the speed of sound. In the most general case as illustrated in Fig. 2(a), an incident acoustic wave may excite three reflected and three transmitted phonon waves of different polarizations. The phonon reflectivity and transmissivity can be derived from the continuity requirements for the atom displacement and the traction force at the interface. Fig. 2(b) shows the calculated phonon reflectivity and transmissivity at an interface similar to that between Si and Ge, albeit with isotropic properties such that the two transverse phonon modes are degenerate. The incident phonon is from the Ge side with a transverse wave polarized in the plane of incidence. This figure shows (1) the mode conversion and (2) the total reflection phenomena.
Some of the transverse phonons are converted into longitudinal phonons. For incident angle higher than 33°, the critical angle, total internal reflection occurs. Beyond the critical angle, an evanescent wave exists in the Si side. Such an evanescent wave, however, does not carry net energy flow into the second medium. This means that the integration limit for u in Eq. (2) is reduced from [− 90°, 90°] to [− 33°, 33°] in the Ge side. In terms of heat transfer, one consequence of the phonon wave reflection is the thermal boundary resistance existing at a perfectly flat solid interface, or the Kapitza resistance for the interface between a solid and the liquid helium. Predictions for the thermal boundary resistance, based on the perfect acoustic-mismatch model, are reasonably good at low temperatures, but fail at high temperatures [30,31]. One possible reason is that at high temperatures, the dominant phonon wavelength is shorter and the interface roughness becomes more important, causing diffuse scattering of phonons. Another possible source of discrepancy lies in the mismatch of the phonon spectra between the two materials. Due to the mismatch in the phonon spectrum, the high frequency phonons must split into lower frequency phonons to transmit into the adjacent materials, or they will be limited inside the original medium, causing total reflection. Past work indicates that inelastic scattering should be taken into account in explaining the experimentally measured thermal boundary resistance at high temperature [32]. In general, the detailed interface phonon scattering mechanisms at room temperature are not clear and models are highly idealized.
3.2. Heat conduction in superlattices, treating phonons as wa6es To the accuracy of the harmonic oscillator approximation, the major differences between heat transfer at
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one interface and that in multilayer structures such as superlattices are the phonon interference and tunneling phenomena. Two distinct approaches have been developed for investigating phonon behavior in superlattices. One is based on the transfer matrix method for calculating the transmissivity of phonons and the other is to directly calculate the phonon dispersion as has been typically done in solid-state physics [33,34]. The two approaches lead to similar results. The phonon interference phenomenon in superlattices was demonstrated by directly measuring phonon transmissivity across a superlattice [35]. The measurement showed the familiar dip in the phonon transmissivity under certain wavelengths as observed in optical interference filters. In Fig. 3, we show the phonon transmissivity in a Ge/n(Si/Ge)/ Si/Ge superlattice structure calculated from the transfer matrix method, as a function of the incident phonon frequency. Here the n-period Si/Ge superlattice is sandwiched between two Ge substrates, with an additional Si layer between the superlattice and one Ge-susbtrate. Several stop bands is formed at this particular incident
angle. This translates into the formation of mini-gaps in the phonon spectra of superlattices and the corresponding reduced phonon group velocity. Similar to phonon reflection at one single interface, total internal reflection and mode conversion also happens at the interfaces in superlattices, as shown in Fig. 4. The major difference between a single interface and a superlattice is that tunneling can happen in superlattices above the critical angle, if the layer thickness is smaller than the penetration depth of the evanescent phonon waves. We use the phonon transmissivity results from the transfer matrix method to estimate the heat transfer over the Ge/n(Si/Ge)/Si/Ge superlattice structure [21]. The following assumptions are made: (1) each of the two cladding materials is at a uniform temperature; (2) all materials are isotropic; (3) the thermal phonons can be decomposed into plane waves; (4) internal scattering inside each layer is negligible; and (5) phonon scattering at interfaces is specular and elastic. In addition, the contribution of optical phonons is neglected, which is reasonable because of (1) the small group velocity of the optical phonons and (2) the large mismatch of the optical phonon energies between Si and Ge. Since the internal scattering is totally neglected, the model is not expected to capture all the heat conduction mechanisms occurring in superlattices. The emphasis here is on the impact of the phonon group velocity reduction and the mini-gap formation on the thermal conductivity. When the temperature difference between the two cladding media is small, the effective thermal conductance of the superlattice can be calculated from the following relation K= q/DT = (1/2) % ×
Fig. 3. Phonon transmissivity over a Ge/10(Si/Ge)/Si/Ge superlattice with each layer at 5 A, for a SV wave incident at 17.7°, showing the stop bands caused by the phonon interference.
Fig. 4. Phonon transmissivity over a Ge/10(Si/Ge)/Si/Ge superlattice as a function of the angle of incidence and the frequency.
&
p 1
& vpm
'v6pD(v) df/dT
0
tp (x,v,d)x dx dv
(3)
0
where f is the Bose–Einstein distribution at temperature T, 6p the phonon group velocity, v the phonon angular frequency, t the phonon transmissivity, and x(= cos u) the directional cosine. The summation is made over the three phonon polarizations. Fig. 5 shows the thermal conductance of superlattices made of different periods as a function of the layer thickness, calculated from the Debye model. In the same figure, we also give the results for a Ge/Si/Ge double heterojunction structure. Both the wave and the ray-tracing results are shown for this latter structure. The ray-tracing method neglects the interference and tunneling effects of the phonon waves. For the single layer structure (n= 0), the tunneling and interference effects become important if the layer thickness is thinner than 10 A, . Tunneling has a more profound impact on K than interference effects, and tunneling increases the thermal conductance as the film becomes thinner. For films thicker than 10 A, , both the wave and the ray
G. Chen et al. / Materials Science and Engineering A292 (2000) 155–161
Fig. 5. Thermal conductance of a Ge/n(Si/Ge)/Si/Ge superlattice structure calculated by the transfer matrix method, where the layer thickness is the thickness of each individual layer in the superlattice (equal thickness is assumed). The ray tracing results neglect the interference and tunneling effects.
tracing methods give the same results. As a comparison, we also calculated the thermal conductance for a transparent interface by assuming that the transmissivity equals one in Eq. (3). The conductance obtained under this circumstance is 0.1027 W m − 2 K − 1, which is a factor of 10 higher than that obtained by the ray tracing method after considering the interface reflection due to the acoustic and the frequency mismatches. Out of this factor of 10 reduction of the interface conductance due to the phonon reflection, about a factor of four of which is due to the acoustic mismatch (the difference in the acoustic impedance) and a factor of 2.5 due to the frequency mismatch. The simulation results in Fig. 5 show that for superlattices, the tunneling and interference effects become stronger for shorter periods. Compared with a single layer, the thermal conductance of superlattices is further reduced due to multiple reflections and the stop band formation, which translate into the phonon group velocity reduction. The value of the conductance changes little as the layer becomes thicker. The convergence of the conductance at large periods indicates that the superlattice approaches one with an infinite number of repetitions. The thermal conductivity for such an infinite repetition superlattice structure, however, would diverge to infinite because we neglected internal scattering.
3.3. Heat conduction in superlattices, treating phonons as particles One major difficulty in the wave approach of heat conduction in superlattices lies in the treatment of interface conditions. The wave calculation is most suitable for specular interfaces. Past experimental results on the thermal boundary resistance suggest that at
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room temperature, diffuse scattering at interfaces is possible and may even be dominant. We have taken an approach based on treating phonons as particles and solving the Boltzmann transport equation (BTE) for phonon transport with heat flow in both the in-plane and the cross-plane directions [17,19,20]. The input parameters for these models include the phonon mean free path (MFP), specific heat, mass density, and group velocity in the each bulk material constituent of the superlattices. Previously developed models for the interface thermal boundary resistance [30,31] have been extended to include phonon dispersion and the possibility of inelastic scattering. Because the mechanism for phonon scattering at interfaces is not clear, we assumed that the interfaces are partially diffuse and partially specular and used a fitting parameter p to represent the fraction of specularly scattered (reflected and transmitted) phonons at the interface. Interface scattering is treated as a boundary condition in solving the BTE, rather than as a volumetric scattering process. Results of the modeling depend strongly on the assumed interface scattering processes. Fig. 6(a) and (b) compare the cross-plane experimental data with modeling results obtained from two different interface scattering models: one assuming totally elastic interface scattering while the other totally inelastic. The elastic scattering model overpredicts the thermal conductivity reduction while the inelastic scattering model leads to reasonable agreement with experimental data, indicating that inelastic scattering must occur at the interface. We should point out that phonon zone folding and quantum confinement phenomena have been observed experimentally using for example, Raman spectroscopy [36]. There exist, however, no quantitative assessments presently of the magnitude of the phonon confinement. The source of thermal conductivity reduction in the cross-plane direction as reveal by the particle models is the thermal boundary resistance. Fig. 7 shows the temperature drop inside each layer and across the interfaces of one period of a superlattice. It shows that most of the temperature drop occurs at the interfaces due to the thermal boundary resistance phenomenon we explained before, although for superlattices, the multiple reflection at interfaces alter the expression for calculating this interfacial thermal boundary resistance. When the phonon MFP is larger than the film thickness, it is not the bulk thermal conductivity, but the mismatch of the density, group velocity, phonon spectrum, and the specific heat of the two adjacent materials that controls the thermal conductivity of the superlattice structure. Fig. 6 indicates that the thermal conductivity depends on strongly whether the interface scatters phonons specularly or diffusely, i.e. the interface scattering parameter p, which represents the fraction of specularly scattered phonons. The dependence is even more drastic in the in-plane thermal conductivity, as
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Fig. 6. Comparison of models with reported thermal conductivity of Si/Ge superlattices based on the assumption of (a) elastic interface scattering and (b) inelastic interface scattering.
shown in Fig. 8. If the interface is totally specular, little reduction in the thermal conductivity occurs. On the other hand, slight diffuse phonon scattering causes a very large reduction in the thermal conductivity. This suggests the possibility of engineering the interfaces to scatter phonons selectively, rather than scattering electrons. Both the wave and the particle models seem to be able to predict an order of magnitude reduction in the thermal conductivity in superlattices in the cross-plane direction, this is not surprising since one major source of thermal conductivity reduction — the total internal reflection — is correctly treated in both models. The phonon frequency mismatch can also be approximately included in the particle model. The group velocity reduction in the wave model is partially included in the particle model through the consideration of multiple reflection. The stop bands (minigaps in the phonon spectrum) and the tunneling, however, cannot be treated through the particle model. In theory, it is always better to start from the wave approach if such a model could be established to take into account for the lose of coherence, the bulk and interface scattering processes. It seems that the key is whether the phonon scattering is specularly or diffuse, or in between. If the interfaces are totally specular, the wave models are more suitable. So far there has no prediction on the thermal conductivity in the in-plane direction based on the wave model. The particle model results suggest that the in-plane thermal conductivity reduction is due to mainly the diffuse scattering of phonons at interfaces. It will be very interesting to check whether the in-plane
thermal conductivity reduction could be explained by the wave model.
4. Concluding remarks Solid-state energy conversion technologies such as thermoelectric and thermionic refrigeration and power generation require materials with good electrical properties but low thermal conductivities. It has been proven that band engineering for electrons can lead to better electron transport properties for energy conversion in nanostructures. In this work, we discuss possibilities to engineer phonon transport in nanostructures
Fig. 7. Temperature distribution in one period of a superlattice for heat conduction in the cross-plane direction, demonstrating that most of temperature drop occurs at the interface.
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Fig. 8. The in-plane thermal conductivity of GaAs/AlAs superlattices.
for reaching low thermal conductivity values. Nanostructures provide alternative ways to reduce the lattice thermal conductivity compared to bulk materials through the anisotropy of the structure, the phonon spectrum change, and interface scattering. Some of these possibilities are illustrated through modeling of phonon transport in superlattices. This is clearly a subject at its infancy and intensive collaborative effort is needed from different disciplines to bring the field into fruition.
Acknowledgements This work is supported by a DOD MURI program on thermoelectrics and an NSF young investigator award to G.C.
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