Fermi Statistics and Fermi Surface. Sommerfeld Theory. 2.1 Fermi Statistics and Fermi Surface

Transcription

1 erm Statstcs and erm Surface.1 erm Statstcs and erm Surface Snce Drude model, t too a quarter of a century for a breathrough to occur. That arose from the development of quantum mechancs and recognton that electrons are fermons. In the Sommerfeld ree Electron Model, electrons are treated as beng free and ndependent as n the Drude s model. However, the classcal Maxwell-Boltzmann (MB) dstrbuton s replaced by the erm-drac (D) dstrbuton: 1 f ( E), (.1) exp[( E ) / T ] 1 where f(e) s the probablty that a state wth energy E s occuped by an electron at temperature T, B s the Boltzmann s constant and s the chemcal potental of the system. Recall that the chemcal potental s defned by the followng normalzaton condton: B The no. of e- per unt volume, n def ( E) g( E), (.) where g(e) s the densty of electron levels at energy E per unt volume of the specmen. It s convenent to thn of the temperature, T 0, defned by T 0 = / B. or metals, t s typcally ~10 4 K. Because T 0 s much bgger than room temperature (~00 K), the D dstrbuton does not vary much from T = 0 K to typcal expermental temperatures. It s therefore of practcal nterest to examne the consequences of replacng the MB dstrbuton by the D dstrbuton at T = 0 K. () Sommerfeld Model at T = 0 At T = 0, the electron gas s at ts ground state. Snce the electrons are assumed to be ndependent, we may solve the wave functon (r) and energy for a sngle electron bounded nsde the volume of the metal and then extend the soluton to the N-electron case by smply wrtng the energy of the system to be N. Because the electrons are assumed to be free, the sngle-electron wave functon satsfes the Schrödnger s equaton of a free partcle: ( r ) ( r ). (.) m or mathematcal convenence, we choose the boundary condton as such that the metal s a cube of sde L = V 1/, where V s the volume of the specmen and assumed to be much bgger than the nter-onc separaton. We argue that ths assumpton about the boundary condton does not comprse the generalty of the soluton. It s because when the sze of a system s suffcently large, ts (bul) propertes should not depend on ts specfc outer dmensons as observed n real lfe. To solve eqn (.), we also need to specfy the boundary values of the wave functon. To ths end, we adopt the perodc boundary condton: 1

2 x, y, z + L) = x, y, z), x, y + L, z) = x, y, z) x + L, y, z) = x, y, z), (.4) where n x, n y, and n z are ntegers. Ths choce, as opposed to ones that leads to standng wave solutons (obtaned by settng specfc values for the wave functon at the boundares), s motvated by our ultmate goal to fnd the propertes of the electrons whle tnerant. It s straghtforward to show that the soluton to eqn.. subject to the boundary condton (.4) s: wth energy 1 r ( r ) e, V (.5) ( ), m (.6) where =. In eqn..5, use has been made of the normalzaton condton, d r ( r ) 1. Notce that n eqn..5, we have labeled the wavefuncton by the wavevector. By eqn. (.6), t s evdent that defnes the egen state energy. At the same tme, also defnes a momentum egen state as follows (You may notce that ths smultanety arses from the fact that the free-partcle Hamltonan commutes wth the momentum operator.): p ( r ) ( r ) ( r ). (.7) Ths reveals that ħ s the momentum observable of the electron. As n the Drude model, t s convenent to use the velocty of the electrons n fgurng the transport propertes. Gven that (r) s a smultaneous eght functon of the Hamltonan and momentum operator, the velocty of the electron s: Usng ths, () can be rewrtten n the famlar classcal form: v = p/m = / m. (.9) m 1 ( ) mv. (.10) Notce also from eqn. (.5) that the wave functon s an terant plane wave wth wavevector, or wavelength,.

3 In the Drude model, we had assumed average values as <> = B T/, <v x > = <v y > = <v z > = ( B T/m) 1/ and c v = n B for the electrons at equlbrum at temperature T. These values sprng from the MB dstrbuton assumed for the electrons. To adapt the dscussons n the framewor of the Sommerfeld model, we need to replace all these values by those approprate to the D dstrbuton of eqn. (.1), subject to the constrant of eqn. (.). In the followng, we shall dscuss how <>, the velocty relevant to the transport propertes, and c v can be derved. Typcally, one starts by thnng how the electrons fll themselves nto the allowed energy levels, (). At T = 0 K, the answer s straghtforward -- because electrons are fermons, they frst fll the lowest energy level untl all the degenerate states are flled, they begn to fll the next hgher energy level. Ths pcture brngs about the necessty to be able to count the number of degeneracy of an energy level. Ths s related to the densty of levels, g() mentoned above, whch we shall derve later. We frst derve the related quantty, g() - the densty of states, whch s the number of states contaned n a unt volume n the so-called -space that we shall next dscuss. By the perodc boundary condton, we have: x n n x y n z, y, z, n x, n y, n z ntegers. (.1) L L L The space, made up of the axes x, y and z s called the -space. Apparently, the allowed states of the electrons consttute the set of ponts defned by eqn..1 n the - space. Clearly, each of such ponts occupes the SAME volume (/L) d n the -space, where d s the dmensonalty of the system. or a dmensonal system, the densty of states s: g() = (L/) = V/8. (.1) But eqn..1 does not fully descrbe the densty of states. It s because each -state can accommodate two electrons because of the electron spns (allowed to be ether up or down). Therefore, the accurate expresson s: g() = V/4. (.1) Notce that g() ncreases wth V, the volume of the specmen. Physcally, t means that the number of avalable states ncreases n proporton to the sze of the system, whch s ntutve. To remove the non-ntrnsc factor, V, one commonly normalze the densty of states by V. Unfortunately, the resultant quantty s also commonly called the densty of states. Please beware of ths when you come across dscussons concernng the densty of states. As stated above, at T = 0 K the electrons fll the lowest-energy unoccuped states frst before fllng those n the next hgher energy level. Snce () ~, the states wth a smaller radus n the -space get occuped frst. So at T = 0 K, all the states enclosed by the sphere wth radus,, n the -space wll be flled, where

4 4 V N V. (.14) 8 Here, N s the total number of valence electrons n the metal. Ths sphere, nown as the erm sphere, separates the occuped from the unoccuped states. Wrtng n = N/V, the electron densty, we have: 1/ ( n) (.15) To get an dea about the sze of, we mae use of the parameter, r s,= [/(4n)] 1/ ntroduced n eqn. 1.. Ths gves = 1.9/r s. (.16) Recall that r s vares from 1.87a 0 to 5.6a 0, where a 0 = / me = 0.59 Å s the Bohr radus. It follows that s of the order of Å -1. The correspondng debrogle wavelength, = / s thus of the order of Å or atomc spacng. The erm velocty, correspondng to the velocty of the most energetc electrons, s: v m. 6 1 r s 10 ms. (for r s n Å) (.17) Eqn..17 llustrates that the erm velocty s very large, about 1% the speed of lght even at T = 0 K. In contrast, the electron velocty n the Drude model s assumed to be thermal and s only ~10 5 ms -1 at room temperature, and decreases le ~T 1/ as T s decreased. Later we shall see that at expermental temperature (.e., T a few hundred K) only those electrons near the erm surface actvely partcpate n the transport phenomena and so v s the relevant velocty to use n descrbng them. Moreover, v, le, does not vary much wth temperature. These lead to notable revsons to the values of some quanttes prevously consdered by Drude. It s convenent to wrte the erm energy as: e a 0 m a (.18) 0 Here, e /a 0, nown as the rydberg (Ry), s the ground-state bndng energy of the hydrogen atom = 1.6 ev. Usng eqn..16, we can re-wrte Eqn..18 as: 50eV. (.19) ( r / ) s Ths shows that s typcally several ev,.e., ~100 tmes the thermal energy at room temperature, whch s ~0.05 ev. Correspondngly, the erm temperature, defned to be T = / B s ~10 4 K. As ponted out above, because the erm temperature s much bgger a 0 4

5 than room temperature, the D does not vary much wth T for typcal temperatures of nterest. The ground state energy, U, for the N-electrons system s just the sum of the energes of all the -states up to =, multpled by : U m. (.0) Because d ~ (/L) <<, the set of ponts representng the allowed electron states effectvely forms a contnuum nsde the space contaned by the erm sphere. In that case, we may approxmate the summaton n eqn. (.0) or other summatons over by replacng the summaton operator (1/V) by the followng ntegral operator: 1 lm V V Wth ths, eqn..0 can be re-wrtten as: d, (.) 8 U V d 4 d 4 m 4 m 10m 0. (.) Multplyng U/V by V/N, we obtan the average energy per electron. By further usng eqn..14 for N/V, we obtan: U/N = /5. (.4) Ths result says that the average energy of the electrons s ~ev even at T = 0 K. Ths should be contrasted wth the assumpton of the Drude that the average energy of the electrons vanshes as T approaches 0 K. The ree Electron Gas at nte Temperatures In the above, we have calculated that the chemcal potental, = ħ /m 1/ ( ( n) ) at T = 0 K. At fnte T, devates from (T = 0), and we need to use the D dstrbuton at fnte T to fnd the value of, defned by: f 1, ( ) / (.5) B e 1 T and N, (.6) f 5

6 where denotes the label for the allowed energy levels, whch s obvously related to.. The Electronc Heat Capacty The lnear T-dependence To evaluate the electronc specfc heat capacty, c v, we must frst calculate the total energy (U) per unt volume (V), u (= U/V): ( ) u d f ( ( )). (.8) 4 The normalzaton condton (eqn..6) n ntegral form or eqn.. s: all 1 n d f ( ( )). (.9) 4 all In evaluatng ntegrals le eqns..8 and.9 of the form 1 d ( ( )), (.0) 4 all one often maes use of the relaton ( ) symmetry n the -space and so we may wrte: m. Gven ths, (()) has sphercal d = 4 d (0< < ) (.1) Now, m ( ) d ( ) d and. Eqn..1 can be wrtten as: m m( ) md d 4. (.) Hence Eqn..0 becomes: 0 d g( ) ( ), where (.) ( m m( ) g ). (.4) 6

7 As mentoned above, g() s nown as the densty of levels per unt volume or often 1/ smply called the densty of levels. Usng ( n) (Eqn..15) and, m Eqn..4 can be rewrtten: m g( ) n ( ) m ( ) 1 ( ) n 1/ 1/ 1/ (.6) n g( ) (.7) Now, go bac to the ntegrals Eqns..8 and.9, whch we shall now wrte as: u d g( ) f ( ) (.8) n dg( ) f ( ) (.9) Because actually only taes on postve values, n ths presentaton we should assume that g() = 0 for < 0, but resumes the form n Eqn..6 for > 0. To evaluate the ntegrals n eqns..8 and.9, we notce that T << T. Therefore, f() devates from the D dstrbuton at T=0 K only wthn a small regon wth wdth a few B T about (see gure.1). g. gure..1 The erm functon, f() = 1/{exp[() + 1]} versus e for gven, at (a) T = 0 and (b) T 0.01 (of order room temperature, at typcal metallc denstes). The two curves dffer only n a regon or order B T about. ( B T) 7

8 Therefore, the devaton I of the ntegral, I = H ( ) f ( ) d from ts zero-temperature value, H ( d, would be determned by the form of H() near =. If H() does not ) vary rapdly n the range ( B T, B T), I s obtanable, to a good approxmaton, by replacng H() by ts Taylor expanson about = : Contnued on next page 8

9 9

10 s 10

11 11

12 Physcally, one may perceve the result of the Sommerfeld model, eqn..5 as follows. The change n energy of the system u u 0 arses from shftng of the electrons from wthn B T below 0 to wthn B T above 0. The number of such electrons s ~ g( 0 ) B T, and each amounts to an average ncrease n energy of ~ B T. Ths leads to an estmate for u u 0 of about g( 0 )( B T), whch s qute close to the R.H.S. of eqn..5. f() B T B T 1

13 1

14 14

15 Inadequacy of the ree Electron Model.4 Inadequacy of the ree Electron Model The Hall Coeffcent, R H : ree electron theory predcts R H to be a negatve constant = 1/ne, ndependent of temperature or magnetc feld. Although the observed R H agrees n magntude wth ths predcton, t s generally dependent on both temperature and magnetc feld. In some cases, R H s postve, whch s ncomprehensble by a free-electron theory. Why are some elements not metallc? Ths s the most detrmental challenge to any free-electron model. Why, for example, s boron an nsulator whle ts vertcal neghbor n the perodc table, alumnum, an excellent metal? Why s carbon an nsulator when the atoms are arranged n the form of damond but a conductor when the atoms are arranged n the form of graphte? Why are the rare earth metals (characterzed by havng valence electrons n the 5d orbtals), bsmuth and antmony poor conductors? A major progress can be acheved by accountng for the facts that (1) the electrons are not free, but nteract wth the postve ons and addtonally the postve ons are arranged perodcally n space (as found n experment). The resultng propertes of the electrons ones that subject to a perodc potental caused by the postve ons - turn out to be not very dfferent from those found n the free electron approxmaton, except for some modfcatons to the energy egen states and energes. Ths explans why the free-electron models of Drude and Sommerfeld are successful n many ways. Another mportant change needed concerns the fact that the postve ons are not statonary, but undertae small oscllatory motons whose quantum mechancal analogs are called phonons. Incorporatng the resultant lattce dynamcs to the consderaton brngs about much mproved predcton of the specfc heat capacty. Later we shall also see that a sgnfcant mprovement to the theory of solds s acheved when we perceve the electron collsons to arse from collsons wth the quanta of lattce vbratons,.e., phonons, rather than the ndvdual ons presumed to be statonary. 15

16 Appendx.1 Dervaton of the erm-drac dstrbuton U TS At thermal equlbrum, s a mnmum. n 0 (1) n subject to the constrant that N = n = constant or 0 Snce eq. (1) must be satsfed for any combnaton of n, we can partcularly choose n nl for two arbtrary states and l, whle the rest of n 0. So n nl n nl 0 n n l for arbtrary and l n chemcal potental U n E, S ln P or each state, S S n n g P n!( g ( g ln g n, where P = no. of accessble states = g! n )! (ln g! ln n! ln( g E 1 E exp( ) 1 T n n E T( g ln n ln g T T ln n E n ln T g n E n exp( ) T g n n E n (1 )exp( ) g T g ln N! N ln N - B B B B N n )!) ( g n ln n T ln( g n / g 1 n / g f N s large n )ln( g ( g n )) n )ln( g E g n E 4 g 4 n 4 E g n E g n E 1 g 1 n 1 g : no. of states wth energy E n : no. of e - occupyng the state E n )) n ) T E B P g T ln n n Ths probablty of occupancy of an energy level (E ) s commonly referred to as the occupaton number. 16