Classical gas (molecules) Phonon gas Number fixed Population depends on frequency of mode and temperature: 1. For each particle. For an N-particle gas

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1 Lecture 14: Thermal conductivity Review: honons as articles In chater 5, we have been considering quantized waves in solids to be articles and this becomes very imortant when we discuss thermal conductivity. In discussing thermal conductivity, we often think of a gas of honons, and the table below makes this more concrete via a comarison to a classical gas. Classical gas (molecules) Phonon gas Number fixed Poulation deends on frequency of mode and temerature: 1 n = e ħω/kbt 1 (What are the limits of this exression for very small T and for very large T)? Small T: e ħω/kbt 1 n ~e ħω/k BT Relationshi between energy and momentum Relationshi between temerature and total internal energy Collisions KE = 2 /2m For each article For an N-article gas U = 3Nk B T Molecules collide with each other and with walls of vessel ħω Large T: ek BT ~1 + ħω k B T n ~k B T/ħω Deends on which branch you are considering (see chater 4), but for acoustic honons near k=0, ω~v s k where k is crystal momentum and v s is seed of sound U = K < n K, > ħω K, = = ħω K, e ħω K,/k B T 1 K Phonons collide with each other, with surface of crystal, and with imurities Energy conserved in collisions? Yes Yes Momentum conserved in collisions? Yes, excet for at walls Yes, modulo a recirocal lattice vector G Number of articles conserved in collision? Yes No Thermal conductivity Thermal conductivity is defined as heat flow in resonse to a temerature gradient j U = K dt dx

2 j U is the energy transmitted across a unit area er unit time and K is the thermal conductivity coefficient. The negative sign reflects the intuitive fact that heat flows from hot regions into cold regions. We encounter thermal conductivity in many ways in the real world When we heat u a large an on a tiny electric burner and we have to wait for the edges to reach the same temerature as the center When we touch two different materials in thermal equilibrium with the same room, such as wood and marble, and the latter feels cooler to the touch (the higher thermal conductivity of marble draws heat away from your fingers to the cooler side which you are not touching) A challenge in modern electronics is to draw heat away from the rocessor, and aroriate materials with high thermal conductivity can serve this urose Temerature is an incoherent henomenon changes in temerature of a system are associated with changes in entroy. Thus, heat flow is not a ballistic rocess, but rather, involves a lot of collisions. As a starting oint, we continue with a article-like treatment of honons. As shown earlier, at higher temerature, there is higher occuation or oulation of honons modes with energy less than or comarable to the temerature. As these honons diffuse to the cooler side of a solid, they will scatter many times and the value of the thermal conductivity coefficient will deend on how often they scatter. K = 1 3 Cvl C is the heat caacity (at constant volume, er unit volume), v is the average article velocity, and l is the mean free ath between collisions. The factors of v and l are fairly intuitive (why?), but C needs some exlaining. When a article moves from a region with temerature T + ΔT to a region with temerature T it will give u cδt of energy. If this article moves a distance of one mean free ath in this rocess, ΔT = dt dx l x = dt dx v xτ Where τ is the average time between collisions. The subscrit x has been added because the temerature gradient is only in the x-direction for now The net flux of energy is: j u = nc < v 2 x > τ dt dx = 1 3 nc v2 τ dt dx There is a factor of v x 2 because the first factor of v x comes from the amount of energy the article gives u and the second factor comes from the rate of flow of these articles. The number of articles is denoted by n and nc=c is the heat caacity er unit volume (n is article density, number/volume). Thus, j U = 1 dt Cvl and the revious assertion for the exression for K is confirmed 3 dx Causes of thermal resistivity

3 Phonons are not conducted ballistically from one end of the samle to another (if this were the case, thermal conductivity would solely be determined by sound velocity). Instead, a number of scattering rocesses introduce a thermal resistivity (resistivity is inverse of conductivity). These include: Imerfections in the crystal such as imurities, inhmogeneities, and crystal grain boundaries The edge or surface of the crystal Scattering off another honon (this requires an anharmonic otential, which tyically holds for real materials because the atomic otential is not erfectly quadratic) We begin by considering the latter: Imagine a rocess where two honons of momentum K 1 and K 2 collide and roduce a third honon with momentum K 3. Another way to say this is that two honons annihilate and roduce a new one with momentum K 3 = K 2 + K 1. Following this rocess, the occuation numbers of these three modes is as follows: Energy conservation requires that: n K1 n K1 1 = n K1 n K2 n K2 1 = n K2 n K3 n K3 + 1 = n K3 n K1 ħω(k 1 ) + n K2 ħω(k 2 ) + n K3 ħω(k 3 ) = n K1 ħω(k 1 ) + n K2 ħω(k 2 ) + n K3 ħω(k 3 ) = n K1 ħω(k 1 ) ħω(k 1 ) + n K2 ħω(k 2 ) ħω(k 2 ) + n K3 ħω(k 3 ) + ħω(k 3 ) Momentum conservation requires that: Where G is a recirocal lattice vector. 0 = ħω(k 1 ) ħω(k 2 ) + ħω(k 3 ) K 1 + K 2 = K 3 + G To make both of these exressions comletely generic, we can write: Energy conservation: ħω (K)n K,,initial ħω n K,,final = 0 Momentum conservation: Kn K,,initial = Kn K,,final + G Where the sum over denotes a sum over all honon branches ( olarization branches ).

4 The factor of G is needed because honon momentum is only defined modulo a recirocal lattice vector, so if the sum of two honon momenta is located outside the first Brillouin zone, it is maed onto an equivalent momentum in the first Brillouin zone by translating by a recirocal lattice vector. If G 0 is needed to make the equality above hold, the scattering is called an Umkla rocess. If G=0, it is called a normal scattering rocess. It should be noted that only Umkla rocesses contribute to a material s thermal resistivity. Comarison of normal (left) and umkla(right) rocesses. The grey square marks the first Brillouin zone in two dimensions, and for a square lattice, the intersection oints between the first Brillouin zone and the K x, K y axes are at ± π a To show that only umkla rocesses contribute to thermal resistivity, we consider the sum over all honon scattering rocesses K tot = Kn s (K) K 1st Brillouin zone The outer sum sums over all honon branches (e.g. all acoustic and otical disersions) and the inner sum counts all momenta in the first Brillouin zone. The terms being added together include the momentum (vector) of each honon mode being considered (K) multilied by how many of them there are (n (K)). If only normal scattering rocesses are included K tot = 0 because for every K, there is a K in the first Brillouin zone and n (K) = n s ( K) = 1 e ħω /k B T 1 In the context of thermal conductivity, what this means is that if you heat u one side of a rod, you will change the occuation numbers of honon modes, but if these honons can only scatter via normal rocesses, they will roagate down the length of the rod with no thermal resistance. Now consider a situation where Umkla rocesses are allowed. K tot = K initial K final = Kn K,,initial ( Kn K,,final + G) G = ν 1 b 1 + ν 2 b 2 + ν 3 b 3 is any recirocal lattice vector, so when we have many terms contributing to G, we accommodate this by making the integers ν 1, ν 2, ν 3 as large as needed

5 When we consider Umkla rocesses, we do not have the same cancellation as we did without and K tot = G. Even if we fold this back into the first Brillouin zone, we are still left with a non-zero net momentum (K ) which contributes a net drift velocity between collisions which is not zero. Consider two temerature regimes: 1) Temerature much larger than the Debye temerature (T θ) : at high temerature n (K) = order) 1 e ħω/k B T 1 k BT ħω (K) (from taylor exanding exonent to first Using K = 1 3 Cvl = v2 τ, C is indeendent of temerature in this high temerature regime, but τ is not. The collision frequency ( 1 ) should be roortional to the number of honons that a given τ honon can collide with and thus K τ 1 1. In reality, K n T 1/Tx where x is between 1 and 2. High temerature not only ensures that C is constant as a function of temerature, but it also ensures that Umkla rocesses have a reasonable robability of haening because it allows for sufficient thermal oulation of higher momentum (which are also higher frequency) acoustic honons. 2) T θ At low temerature, only low energy (and low momentum) acoustic modes will have substantial thermal oulation, so Umkla rocesses will be unlikely and honon-honon scattering will not contribute to thermal resistivity Imerfections Geometric effects can decrease the thermal conductivity if they increase a length scale shorter than l. For instance, if the material is in a reduced geometry (e.g. a thin film or a nanowire) with minimum dimension length D K CvD This also holds if the material consists of small crystallites of dimension D honons can scatter at the boundary between crystallites of different orientations and reduce the thermal conductivity Examle: Thermal conductivity of insulating solids at room temerature (insulating so we don t have to deal with electrons contribution to thermal conductivity) Material K (W/m K) v (m/s) Debye T Diamond , K Silicon K KCl K SiO K

6 This table illustrates the general scaling between thermal conductivity, sound velocity, and Debye temerature, which makes sense since both K and θ deend on v Diamond is characterized by a high thermal conductivity, and this is how one tests to see if a material is a diamond. Physically, the large sound velocity, which roduces high thermal conductivity and Debye temerature, originates from very light atoms (carbon) connected by very stiff covalent bonds Overview of Chater 5 It is common knowledge that atoms vibrate more at higher temerature, and in this chater we quantified this ahorism using the quantum nature of atomic lattice vibrations. A key ste in doing this is recognizing/believing/decreeing that honons are bosons, and thus any number of articles can occuy a given state which is defined by a momentum, frequency, and olarization (a olarization is a distinct disersion branch). For a state with a given frequency, its exected occuation number is given by the Planck distribution n = 1 e ħω/k BT 1 This occuation number allows us to derive heat caacity, thermal exansion (if we were to do a more rigorous derivation than the textbook), and thermal conductivity. A state with a given frequency also has an associated crystal momentum because elastic waves in crystalline solids have secific relationshis between frequency and momentum called disersion relations. Using boundary conditions, we can show that crystal momentum is quantized, with ermissible momenta (in 1 dimension) given by: 0, ± 2π, ± 4π, ± Nπ where N is the number of atoms in the chain, L L and L is the length of the chain. Because any given K only allows secific frequencies to satisfy the equations of motion (this is what we solved for in Ch. 4), the statement above also means that only certain secific ω are allowed for each K. The number of ermissible ω for each K is determined by how many honon branches exist in the material. For a D dimensional solid with atoms in the basis, there are D acoustic branches and D(-1) otical branches. Each unique combination of ω, K, and olarization defines a normal mode or a state that can be occuied by n honons. Although K and olarization (again, olarization is a term the book uses, and it is not exactly the same as light olarization it is just an index to count all the honon branches you have) hel define the state you are considering, only the frequency, ω, relative to the temerature will determine the occuation of each state. L

7 The density of states, D(ω) is a concet that we use to convert ω(k) disersion relations into information we can use to determine occuation numbers. The density of states, as the name imlies, is the number of states available between frequency ω and frequency ω + dω. Density of states deends on the details of the disersion relation and the dimensionality, but it does not deend on temerature (unless temerature changed the disersion relations). Occuation number is a searate issue from the density of states. In three dimensions, a generic exression for the density of states is: D(ω) = K2 V 2π 2 dk dω If we had an analytic or comutational exression for ω(k) we could invert it to get K(ω) and take a derivative to get dk/dω and the density of states. But sometimes it is equally useful to make a hysically motivated aroximation, and the Debye model is such an aroximation which works well for redicting behavior of real materials. The Debye model only considers acoustic honons and aroximates all acoustic branches as having the same grou velocity v. There is also a cutoff frequency, ω D, which defines the highest frequency honon which is allowed in the debye model. This frequency also defines a momentum k D = ω D /v and a Debye temerature θ = ħω D /k B. The Debye temerature is hysically significant because it reresents roughly the temerature at which a honon gas starts acting like a classical gas. The Debye model gives a closed-form aroximation for heat caacity in the limit of low temerature and high temerature (low and high relative to θ). Low T: C V 12π4 5 Nk B ( T θ )3 (This is consistent with exeriments) High T: C V 3Nk B (This is also consistent with exeriments and with a classical gas) In almost every concet we have discussed in this chater, a article velocity has come into lay, and this is essentially the seed of sound. Thus, the seed of sound in a solid, technically given by ω/ k near k=0, is a crucial arameter determining how non-electromagnetic (e.g. thermal, but also vibrational and elastic wave) information is transmitted from one end of a solid to another.

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