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The Debye Model Professor Stephen Sweeney

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The Debye Model Professor Stephen Sweeney Solid State Physics Lecture 8 – The Debye model Professor Stephen Sweeney Advanced Technology Institute and Department of Physics University of Surrey, Guildford, GU2 7XH, UK [email protected] Solid State Physics - Lecture 8 Recap from Lecture 7 • Concepts of “temperature” and thermal Dulong-Petit equilibrium are based on the idea that individual particles in a system have some form of motion • Heat capacity can be determined by considering vibrational motion of atoms • We considered two models: • Dulong-Petit (classical) • Einstein (quantum mechanical) • Both models assume atoms act independently – this is made up for in the Debye model (today) Solid State Physics - Lecture 8 Summary of Dulong-Petit and Einstein models of heat capacity Dulong-Petit model (1819) Einstein model (1907) • Atoms on lattice vibrate • Atoms on lattice vibrate independently of each independently of each other other • Completely classical • Quantum mechanical • Heat capacity (vibrations are quantised) independent of • Agreement with temperature (3NkB) experiment good at very • Poor agreement with high (~3NkB) and very low experiment, except at (~0) temperatures, but high temperatures not inbetween Solid State Physics - Lecture 8 A more realistic model… • Both the Einstein and Dulong-Petit models treat each atom independently. This is not generally true. • When an atom vibrates, the force on adjacent atoms changes causing them to vibrate (and vice-versa) • Oscillations can be broken down into modes 1D case Nice animations here: http://www.phonon.fc.pl/index.php 3D case Java applet: http://dept.kent.edu/projects/ksuviz/leeviz/phonon/phonon.html Solid State Physics - Lecture 8 Debye model • Basic idea similar to Einstein model, with one key difference: Einstein: Energy of system = Phonon Energy x Average number of phonons Debye: Energy of system = Phonon Energy x Average number of phonons x number of modes The number and type of Einstein: number of modes = number of atoms modes are the key difference Debye: each mode has its own k value (and hence frequency) Solid State Physics - Lecture 8 Modes: lattice vibrations Modes exist in various areas of physics/nature Butterfly wing-beat Water Molecules Guitar modes Solid State Physics - Lecture 8 Modes: Quantum mechanics Modes are quantised in units of where the fundamental frequency of each mode is The Einstein model assumed that each oscillator has the same frequency Debye theory accounts for different possible modes (and therefore different ) Modes with low will be excited at low temperatures and will contribute to the heat capacity. Therefore heat capacity varies less abruptly at low T compared with Einstein model Low frequency modes correspond to multiple atoms vibrating together (sound or acoustic waves) Solid State Physics - Lecture 8 Standing waves: revision n = 4 Consider a vibrating string Lowest (fundamental) frequency L n = 3 2 n = 2 2L More generally n L λ n = 1 2 n L v vn vn Other results follow: f ω 2πf 2L L 2π n k L Solid State Physics - Lecture 8 Standing waves in a 1D crystal Consider solid as a continuous elastic medium: N atoms, 3 degrees of freedom 3N standing modes a 2π πn n 1D array of atoms: k L=Na λ L Na π max 2L 2Na kmin Fundamental mode Na (n=1) Highest order mode λ 2a k (n=N) min max a nmax N Therefore we get N modes for N atoms Solid State Physics - Lecture 8 Standing waves in 2D crystals Fundamental mode (2D) Each component of the wave is quantised separately and added in quadrature π x k ky x L y L L Magnitude of k-vector for mode 2 2 π k k ky 2 x L Corresponding angular frequency v ω v 2 ω 2πf 2 vk L Solid State Physics - Lecture 8 Standing waves in 2D crystals: Degeneracy 2π π k k π y L 2π y L k k x L x L x y L L L L 2 2 2 2 2 π 2 π k 5 k 5 L L L L L L In both cases ω v 5 so these two modes are degenerate L As frequency increases, more and more states share the same frequency & energy (called DEGENERACY) Solid State Physics - Lecture 8 Back to reciprocal space… (2D) • We can represent each mode as a point in reciprocal (k) space Q. How many modes are available at a particular k value? A. Need three pieces of information: 1. How “big” is an individual k-state 2. How much of k-space is covered at a particular k 3. Account for degeneracy kl2 gkdk dk 2 Solid State Physics - Lecture 8 Number of States in 3D In 3D we consider the number of states k within a sphere of radius k z k 4 3 Sphere “volume” = k 3 ky 3 “volume” of k-state = kx l 3 l Vk 2 gkdk 2 dk 2 k-state Solid State Physics - Lecture 8 Number of States in 3D kz k We know that ω vk dω vdk ky 2 V k Hence gd d x 2 2v3 i.e. the number of standing waves (modes) l increases as 2 Vk 2 gkdk 2 dk Sound can propagate with2 2 transverse and 1 longitudinal k-state wave in a solid total no. of states = 3g()d Solid State Physics - Lecture 8 Debye frequency For any one wavelength of oscillation there are shorter wavelength oscillations that will also have the atoms in the same position on the lattice (c.f. aliasing in electronics) There is a minimum wavelength which can oscillate which corresponds to a maximum frequency, max (Debye frequency) We can calculate max since we know (from earlier) that the maximum number of states = 3N max 3N 3gd 0 1 3 2 N ωmax 2 3V V So, ωmax v6π dω ω3 V 2 3 2 3 max 0 2π v 2π v Solid State Physics - Lecture 8 Some crystal modes of vibration Phonon animations here: http://www.phonon.fc.pl/index.php Solid State Physics - Lecture 8 Debye model: Total average energy of System Energy of Phonon Average no. No. of From earlier: = x x system energy of phonons modes max 1 max 3g E 3gd d 0 0 exp 1 exp 1 kBT kBT Integrate over all modes (NB: Ignoring zero-point energy) Solid State Physics - Lecture 8 Debye model: Total average energy of System max 3g E d 0 exp 1 3V max 3 E d kBT 2 3 2 v 0 exp 1 kBT From before: V 2 gd 2 3 d 2ω v ωmax Make substitution: x and define Debye temperature: θD kBT kB D 3Vk 4T 4 T x3 E B dx From which (finally) we can 2 3 3 extract the heat capacity, C 2 v 0 expx1 Solid State Physics - Lecture 8 The Debye Temperature D This is perhaps the most useful parameter in the Debye theory • It allows us to predict the heat capacity at any temperature • It provides an indication of the temperature at which we approach the classical limit of the Dulong-Petit theory From earlier, we know that and ωmax θD 1 3 kB max DkB 2 N Therefore, v 1 6 3 2 N V 6 1 3 V 2 N ωmax v6π So if we know N/V then we can predictV the speed of sound in a solid Solid State Physics - Lecture 8 The Debye Temperature D: examples High D corresponds to a large max Large max implies large forces, low max implies weak bonds Diamond D = 2230K Dulong-Petit poor fit at room temperature. Strongly bonded Iron D = 457K Dulong-Petit reasonable fit at room temperature. Lead D = 100K Dulong-Petit good fit at room temperature. Weakly bonded Solid State Physics - Lecture 8 Debye model: Heat Capacity D 3Vk 4T 4 T x3 ω E B dx where x 2 3 3 2 v 0 expx1 kBT At high T: x is small exp(x) 1 x ... so… D D D T x3 T x3 T 3 dx dx x2dx D 3 0 expx1 0 1 x ...1 0 3T 1 4 3 3 Vk T k 2 N E B D and since v D B 6 2 2v33 V Dulong-Petit ! dE dE E 3Nk T Heat capacity, C 3Nk C 3N k B dT B molar dT A B Solid State Physics - Lecture 8 Debye model: Heat Capacity D 3Vk 4T 4 T x3 ω E B dx where x 2 3 3 2 v 0 expx1 kBT At low T: x3 4 Take limit that /T and use identity dx D 0 expx1 15 3Vk 4T 4 4 3 4 Nk T 4 so… E B B 2 3 3 3 Debye T3 law 2 v 15 5D 1 3 k 2 N 3 v D B 6 dE 12 4 T Heat capacity, C V NkB dT 5 D Solid State Physics - Lecture 8 Debye T3 law Heat capacity for solid Argon (from Kittel) ) 1 - K 1 - Heat Capacity (mJ mol T3 (K3) Solid State Physics - Lecture 8 Comparison of Dulong-Petit, Einstein and Debye models of heat capacity Dulong-Petit Solid State Physics - Lecture 8 Thermal Conductivity • Thermal conduction is a measure of how much heat energy is transported through a material per unit time • In metals conduction is due to free electrons (a later lecture) • In non-metals conduction is largely due to phonons • most hard insulators have a low thermal conductivity • Phonons have energy and can therefore conduct heat • Scattering mechanisms limit the thermal conductivity of non-metals, due to • Imperfections (grain boundaries, point defects, dislocations) • Phonons themselves can scatter other phonons (Umklapp processes – we won’t cover that here) Solid State Physics - Lecture 8 Thermal Conductivity Thermal conductivity T T high low 1 dE dT Area, A Q A dt dx 1 dE dx 1 dE dx Energy flow along x A dt dT A dT dt C v i.e.
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