17 Phonons and conduction electrons in solids (Hiroshi Matsuoka)

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1 7 Phoos ad coductio electros i solids Hiroshi Matsuoa I this chapter we will discuss a miimal microscopic model for phoos i a solid ad a miimal microscopic model for coductio electros i a simple metal. I both of these models the costituet quatum particles phoos or electros are regarded as o-iteractig idetical particles. We will apply the grad caoical esemble approach developed i Ch.6 to these models to calculate their state variables especially their molar heat capacities at costat volume. More specifically the models we will discuss below are:. The ideal phoo boso gas model for lattice waves i a solid Our mai goal i studyig this model is to calculate its molar heat capacity at costat volume: * 2 4 = 5 R T + T < < # # - R T > > # where R is the uiversal gas costat is the umber of atoms assiged to each crystallie lattice poit e.g. for NaCl = 2 ad is the characteristic temperature for the phoos or the Debye temperature that separates the low-temperature regime from the high-temperature regime for the phoos. As i the ideal phoo gas model is a fuctio of temperature T ad the molar volume v of the solid the Debye temperature must be a fuctio of the molar volume. We will also fid that depeds o the speed of soud w i the solid so that = vw. 2. The free electro fermio gas model for coductio electros i a simple metal Our mai goal i studyig this model is to calculate its molar heat capacity at costat volume: 2 = 2 R T * # T F * + 2R T >> T F T < < T F where T F is the characteristic temperature for the coductio electros or the Fermi temperature that separates the low-temperature regime from the high-temperature regime for the electros. As i the free electro gas model is a fuctio of temperature T ad the molar volume v of the

2 2 metal the Fermi temperature T F must be a fuctio of the molar volume. We will also fid that T F depeds o the mass of electro m so that T F = T F v m. The molar heat capacities for phoos ad electros at low temperatures Note that the heat capacities for the both models approach zero as the temperature is decreased toward the absolute zero temperature which is a cosequece of the third law of thermodyamics as discussed i Sec Note also that at low temperatures the heat capacities for the both models are expressed i uiversal scalig forms: for the phoos we fid = Rg # T p h where g p h is the uiversal scalig fuctio for the phoos g p h x = 2 4 x 5 while for the electros we fid T = Rg e # T F where g e is the uiversal scalig fuctio for the electros g p h x = 2 2 x. At low temperatures these heat capacities thus deped o the temperature quite differetly maily because the phoos are bosos whereas the electros are fermios. To show this differece betwee the phoos ad the electros is our mai goal i this chapter. The molar heat capacities for phoos ad electros at high temperatures At high temperatures both of the heat capacities approach differet costat values R = 6 2R for the phoos ad 2R for the electros. Accordig to the theorem of equipartitio of eergy i 2R for the electros correspods to three terms i the ietic eergy of each electro

3 2 m v 2 = 2 mv 2 x + 2 mv 2 y + 2 mv 2 z. Note that at high temperatures electros behave lie atoms i the moatomic ideal gas model. At T >> each atom i a solid behaves lie a harmoic oscillator so that 6 i R = 6 2R is a sum of ad which correspod to three terms i the ietic eergy of each atom ad three terms i its potetial eergy 2 m 2 r 2 = 2 m 2 x m 2 y m 2 z 2 where r is the displacemet vector for the atom. 7. Oe-particle states of quatum particles The mometum vector ad the wave vector of a quatum particle I this sectio we will show that each oe-particle state of o-iteractig bosos or fermios ca be labeled by a wave vector ad that the eergy eigevalue of the oe-particle state is determied by the wave vector: =. For o-iteractig quatum particles whether bosos or fermios the de Broglie wavelegth of each particle is determied by its mometum vector p through h p = h p where h is the Plac costat. We ca the defie the correspodig wave vector by havig poited i the directio of p ad its magitude directly related to by = 2. The wave vector is the related to its mometum vector p by p =.

4 4 sice p = h = h 2 = where = h 2. The wave vector eigevalues for oe-particle states with the periodic boudary coditio As discussed above the wave vector of oe-particle state is related to its mometum vector p by p =. I quatum mechaics the classical mometum vector p is replaced by a operator p i so that the classical wave vector must be also replaced by the followig operator: = p i. We ca the fid the eigevalues of this wave vector operator for a particle i a cubic box of volume V = L by solvig the followig eigevalue equatio: i r = r together with a boudary coditio for the wave fuctio r. As log as the boudary coditio does ot affect the wave fuctio deep iside the cubic box we ca choose almost ay coditio. I codesed matter physics the most widely used boudary coditio for phoos ad electros is the periodic boudary coditios where r. satisfies = y z L yz # x L z = x z xyl = x y L I other words the wave fuctio taes the same values o each pair of the opposig faces of the cube. This boudary coditio miimizes the effect of the boudary or the surface of the cube o the wave fuctio sice it is equivalet to divide the etire three-dimesioal space ito a

5 5 array of cubes each with volume V = L same way i every cube. The eigestates of the wave vector are the plae waves r = exp i ad to assume that the wave fuctio behaves i the r V ad the correspodig eigevalues are = 2 L x y z = 2 L where each eigestate ad eigevalue are labeled by three itegers or three quatum umbers whose rage icludes both positive ad egative itegers as well as zero. = x y z. = ± ± 2 ±... Note that = is also a eigevalue of the wave vector ad the correspodig eigestate is costat throughout the volume V: = r = V. I the -space all the possible wave vectors form a discrete lattice where each lattice poit is located at a corer of a small cube of volume = # 2 L = 2 V. The itrisic quatum umber: polarizatio idex ad spi Besides the three quatum umbers a quatum particle may have a itrisic quatum umber such as the polarizatio idex for phoos = 2 ad which labels two trasverse polarizatio directios ad oe logitudial polarizatio directio for lattice waves ad the spi quatum umber for electros = ± / 2.

6 6 The eergy eigevalue for a oe-particle state Geerally the eergy eigevalue of a oe-particle state for o-iteractig quatum particles depeds o the wave vector : =.. For quatum particles such as phoos that we observe as waves o the macroscopic level the eergy eigevalue is give by = where is the agular frequecy of the wave ad is related to by what is called the dispersio relatio : =. Note that as approaches zero at the log wave legth limit the agular frequecy ad the eergy eigevalue also goes to zero. 2. For quatum particles such as electros that we observe as particles o the macroscopic level the eergy eigevalue is give by = 2 p 2 2m = 2 2m where m is the mass of each particle. For both phoos i solids ad electros i simple metals the lowest eergy eigevalue for their oe-particle states is set at zero. I other words we set their iteral eergy scales so that their iteral eergies tae the value of zero whe a sigle phoo or electro is foud i the lowest eergy eigestate.

7 7 7.2 The ideal phoo gas model I this sectio we discuss the ideal phoo gas model where phoos are assumed to be o-iteractig quatum particles. The umber of phoos depeds o T ad V: the chemical potetial of a phoo gas is zero The chemical potetial of a phoo gas is idetically zero. Physically it is because phoos are cotiuously created ad destroyed i a solid so that the umber N p h of phoos is ot a idepedet variable ad is rather a fuctio of T ad V of the solid. The Helmholtz free eergy of the phoo gas is the a fuctio of T ad V: F = F TV. O the other had usig the fudametal equatio of thermodyamics du = TdS PdV + µdn p h we fid df = du d TS = TdS PdV + µdn p h = SdT PdV + µdn p h TdS + SdT so that F µ = #N p h T V =. The iteral eergy ad the umber of phoos deped o T ad V As the Helmholtz free eergy depeds oly o T ad V so do all other state variables of the phoo gas. proportioal to the volume: Beig extesive the iteral eergy ad the umber of phoos should be U = Vu T ad N p h = V p h T where u ad p h are the iteral eergy ad the umber of phoos per uit volume ad they deped oly o the temperature. As the volume V is related to the mole umber of the solid by V = v where v is the molar volume of the solid the molar iteral eergy u of the solid due to the phoos is give by u = vu T so that the molar heat capacity at costat volume of the solid is the give by

8 8 = u # T v = v d u. # dt The iteral eergy ad the umber of phoos i the ideal phoo gas model Because µ = the thermodyamic potetial of a phoo gas is idetical to its Helmholtz free eergy: = U TS µn = U TS = F. This implies that the grad caoical esemble theory of the ideal phoo gas is the same as its caoical esemble theory. I fact the Gibbs sum of the ideal phoo gas is othig but its partitio fuctio: p h TV = exp E { # # } = + * [ { }] = Z p h T V 6 = exp / 2 5 = e / = 2 45 =. - exp / + * e / where the factor of is the umber of polarizatios for each. We ca the derive the Helmholtz free eergy of the ideal phoo gas as [ ] F T V = T l Z p h = T l e / # T.. - We ca fid the iteral eergy either by settig the chemical potetial to zero i the expressio derived i Ch.6: e # µ # U = = e # or by usig the Helmholtz free eergy just derived: U = T 2 # F / * - = + T T. V e /. Comparig this expressio for the iteral eergy with

9 9 we fid U = # = # = e # so that we ca fid the average umber of the phoos by N p h = = e #. The eergy eigevalue for the phoo oe-particles state For phoos i a solid at log wavelegths >> a or small waveumbers << 2 a where a is the distace betwee a pair of earest-eighbor atoms iside the solid the dispersio relatio is well approximated by w for << 2 a where w is the speed of soud so that w. for << 2 a Derivatio where = 2f # 2 w f w = w which is valid oly at log wavelegths. As the waveumber icreases away from the log wavelegth limit the dispersio relatio starts to deviate from the liear relatio w obtaied above. As we will see to fid the molar heat capacity at both low ad high temperatures we do ot eed to ow exactly how the dispersio relatio behaves outside the log wavelegth limit.

10 The first Brilloui zoe: the allowed regio for the phoo wave vectors The allowed regio for the wave vectors for the phoos i the -space is actually bouded sice the wavelegth of a phoo caot be less tha the lattice costat a or the distace betwee a pair of earest-eighbor atoms: which implies a. = 2 # 2 a. This bouded regio for is called the first Brilloui zoe whose shape i the -space depeds o the crystallie structure of the atoms i the real space. The maximum phoo eergy ad the characteristic temperature for phoos As the phoo eergy is a smooth fuctio of the wave vector there must be a maximum eergy max for the phoo oe-particle states. The maximum phoo eergy thus sets the phoo eergy scale ad we ca roughly estimate its order-of-magitude as follows. The maximum waveumber max i the first Brilloui zoe must be o the order of a ad the maximum phoo eergy max must be o the order of w max so that m s max ~ w max ~ w a ~ O 4 J# s m * ~ O 2 J J ev /. ~ O 2 ev ~ O 2 J where a ~ O m ad w ~ O m s. A characteristic temperature T p h for the phoos which separates their low-temperature regime from their high-temperature regime ca be the defied by T p h max ~ O #2 J #2 ~ O K. J K I the Debye approximatio of the ideal phoo gas model we will defie a slightly differet characteristic temperature for the phoos called the Debye temperature by

11 w 6# 2 N a V / where the volume per atom V N a is o the order of a so that ~ w # N a V / ~ w a ~ w max ~ T p h. The total umber of the phoo oe-particle states i the first Brilloui zoe At high temperatures N a atoms i the solid must be vibratig fast eough that the atoms behave lie N a three-dimesioal harmoic oscillators which must be described by N a quatum umbers. This implies that the total umber of the allowed phoo oe-particle states must be also N a so that N a = BZ where the summatio is over the wave vectors i the first Brilloui zoe. This equatio becomes useful below whe we discuss the high-temperature behavior of the iteral eergy of the phoo gas. The iteral eergy ad the average umber of phoos As the eergy eigevalue of the phoo oe-particle state depeds o its wave vector = : we ca express the iteral eergy ad the average umber of the phoos as ad U = N p h = e = # BZ BZ f Plac T e = f Plac # T BZ BZ where the summatio is over the wave vectors i the first Brilloui zoe ad we have defied the Plac distributio fuctio or the Bose-Eistei distributio fuctio with µ = by

12 2 f Plac T e #. The Plac distributio fuctio The Plac distributio fuctio is show below as a fuctio of. For << or << T we ca approximate e i the Plac distributio fuctio by + or e # + so that f Plac +# = # = T B # << T whereas for >> or >> T we fid e >> so that f Plac e # >> T. Note that for each phoo oe-particle state whose eergy is much lower tha T its average occupatio umber or the value of the Plac distributio fuctio becomes large as f Plac T T >> but the cotributio to the iteral eergy by all these phoos i this oe-particle state altogether amouts to just T sice f Plac T T which is idepedet of the oe-particle state eergy.

13 The uiversal iteral eergy of the phoo gas at high temperatures ad the Dulog- Petit law Whe the temperature T of the phoo gas is much higher tha max so that T >> max or max << all the phoo oe-particle states iside the first Brilloui zoe have their eergies much lower tha T so that as we foud i the previous sub-sectio we ca approximate the Plac distributio fuctio for each phoo oe-particle state with eergy by f Plac T T ad cosequetly every oe-particle state cotribute a eergy of T to the iteral eergy of the phoo: U = # f Plac T T # = N a T = RT BZ BZ where is the mole umber of the solid ad N a = N Avogadro where is the umber of atoms assiged to each crystallie lattice poit e.g. for NaCl = 2. We have also used N a = BZ The molar iteral eergy at high temperatures the becomes u = U a RT The molar heat capacity at costat volume at high temperatures is the give by = u # T v R which is the Dulog-Petit law we have discussed i Sec Note that this result does ot deped o the specific detail of the dispersio relatio which varies from oe solid to aother ad the Dulog-Petit law is therefore a uiversal property of the ideal phoo gas. The two mai igrediets for this uiversal property are the behavior of the Plac distributio fuctio at phoo eergies much lower tha T ad the presece of the cutoff eergy max. At high temperatures where T >> max the Plac distributio fuctio for all the phoo oe-particle states i the first Brilloui zoe is approximately T >> so that all the phoo oe-particle states below the cutoff eergy are excited ad cotribute to the iteral eergy. Moreover all the phoos i a phoo oe-particle state with eergy together

14 4 cotribute f Plac T of eergy to the iteral eergy. As this eergy due to the phoos i each oe-particle state is idepedet of the phoo eergy the iteral eergy is simply the product of the total umber of the phoo oe-particle states N a ad T. The umber of phoos at high temperatures We ca also calculate the umber of phoos by N p h = # f Plac T B T BZ # T BZ which icreases liearly with the temperature but is ot uiversal because the sum of does deped o the detail of the dispersio relatio. Approximatig the summatio over phoo oe-particle states by a itegral To calculate the iteral eergy ad the average umber of the phoos at low temperatures it is coveiet to approximate the summatio over the wave vector i the equatios for U ad N p h by a itegral. For two eighborig wave vectors that satisfy we fid = 2# L ~ w # # ~ O 8 K + T * = 2 L m s w - T + ~ O/ 4 J s *. / m 2 J K T 2 2 where we have used the expressio for the phoo eergy for the log wavelegths. For temperatures ragig from mk to K we therefore fid # < <. We ca the approximate the summatio over the wave vector i the above equatios for U ad N p h by a itegral as follows. = # # = V d 2 BZ BZ Z where we have used

15 5 so that ad = # 2 L = 2 V U = V d 2 e # = V d Z 2 Z f Plac T N p h = V d 2 e # = V d f Z 2 Plac # T. Z The uiversal temperature depedece of the iteral eergy of the phoo gas at low temperatures ad the T law for the molar heat capacity at costat volume Whe the temperature T of the phoo gas is much lower tha max so that T << max or max >> for phoo eergy that satisfies that satisfy >> T or >> the expoetial factor e becomes so large that the Plac distributio fuctio becomes egligible. Physically oly the oe-particle states with eergy w less tha T cotribute to the iteral eergy. Roughly the umber of these oe-particle states is N p h # T w 4 T = # w = V T 2 2 # w ad o the average each of these oe-particle states cotribute eergy o the order of T to the iteral eergy so that = O U ~ O N p h T 2 2 w V T # 4 T 4. More precisely i the itegral for U we ca itegrate over the etire -space istead of the iside the first Brilloui zoe: U = V d 2 e # V d Z 2 e # Etire space without affectig the value of the itegral. We ca also use the log-wavelegth expressio for the phoo eergy to obtai

16 6 U 2 w V T 4 so that the molar iteral eergy at low temperatures becomes u = U 2 w v T 4 where v is the molar volume of the solid. Derivatio U V d # 2 e # V 4 2 w d Etire space 2 e w = = V 2 2 w * 4 V 2 2 w B T + - d w w = e w + - = 5 4 * 4 2 w V T B 4 V 2 2 w T 4 dq q e q where # dq q = 4 e q 5. The uiversal molar heat capacity at costat volume at low temperatures The molar heat capacity at costat volume at low temperatures is the give by = u # T v v T B B # w * T. Note that the oly quatities i this equatio that are specific to a particular solid are the molar volume v ad the speed of soud w. We also ow that the molar heat capacity should be expressed i uits of the uiversal gas costat R so that we should expect to have the followig scalig form:

17 7 T = Rg# # T p h v w where T p h is the characteristic temperature for the phoos defied earlier by T p h max ~ w # N a V / ad g is a scalig fuctio that assigs a dimesioless umber for a dimesioless umber. Substitutig v = V = VN Avogadro N Avogadro = N Avogadro V N a where N a = N Avogadro ito the above expressio for we ideed fid i the scalig form: v # T B B w V T N a = N Avogadro # w = 22 * 5 R T + w N a V / - /. / which implies that is a uiversal fuctio of T T p h. R T # T. p h I the ext sub-sectio we will defie a characteristic temperature for the phoos called the Debye temperature by w 6# 2 N a V / ~ w N a V / ~ T p h ad express i terms of T as 2# 4 5 R T *

18 8 The umber of phoos ad the iteral eergy at low temperatures The umber of phoos ca be also foud to be N p h = V d 2 e # V d 2 e # V 4 Etire space 2 2 d e w = = BZ V 2 2 w + * V 2 2 w B T 2. d w w - = e w { 2/ } =.22 V 2 2 w T dq q 2 e q 2 w V T B =.66 2 w V T B so that U 2 w V T 4 = 4 # N T = 4 p h B 6.6 N T p h B which implies that the iteral eergy of the phoos is roughly the umber of phoos times the typical value of phoo eergy T. The Debye approximatio I the Debye model for the ideal phoo gas we exted the log-wavelegth expressio for the phoo eergy to the maximum cutoff eergy max which we ow call the Debye eergy D max so that for D = max w. The first Brilloui zoe is the replaced by a sphere of radius D defied by so that D D w N a = = V 2# d = V 4# 2# D+ = V * 2# 2 D BZ D from which we fid D = 6 2 N a # V /

19 9 ad # D = w D = w 6 2 N a V /. We the defie the Debye temperature by # D = w 6 2 N a * V / = w +N 6 2 Avogadro * v / where we have used N a = N Avogadro. Note that depeds o the molar volume v of the solid ad the speed of soud w. To calculate the iteral eergy i terms of it will be coveiet to use the followig expressio for the volume i terms of : so that # V = 6 2 w N a V 2 2 w = 9N a. The iteral eergy of the phoo gas i the Debye model is the give by U V d # 2 e # = V D 4 2 w V d 2 = e w 2 2 w * 4 D +. T - dq q e q. T = 9N a. B T 4 dq q e q = 9/R. Ṭ + * - 4. T dq q e q so that ad u = U = 9R # T = u # T v 4 T * = 9R 4 T. -. # dq q e q + / T dq q * /. e q * # T e / T * which implies that is a uiversal fuctio of T /.

20 2 The umber of the phoos is also give by N p h V d 2 e # = V D V d = e #w 2 2 w * # D. T = 9N a. B T dq q 2 T + = 9N e q a * T dq q 2 e q. T + - dq q 2 e q a At low T T << : by replacig the itegratio iterval T U ad N p h we fid [ ] by [ ] i the itegral for u = U 9R# T # 4 + dq q = - 4 e q * 5 R# T # 4 = u # T v *R T # + # T N p h 9N a * U 4 5 N T T a B # dq q 2 # T + =8 N e q a = 4 * N T p h B b At high T T >> : by usig e q + q i the itegral for U ad N p h we fid u = U 9R# T # 4 # T q + dq = 9R# T + q # * # T 4 = RT = u # T v R the Dulog-Petit law # T N p h 9N a T q 2 # T * dq = 9N + q a 2 # T 2 = 9 2 N # T a U N a T = 2 N p h