Free electron gas. Nearly free electron model. Tight-binding model. Semiconductors

Transcription

1 Electroic Structure Drude theory Free electro gas Nearly free electro model Tight-bidig model Semicoductors Readig: A/M 1-3,8-10 G/S 7,11 Hoffma p

2 DC ELECTRICAL CONDUCTIVITY A costat electric field E results i a electrical curret per uit area J: Jr () = σ Er () where the proportioality costat σ is the electrical coductivity J (curret desity): A cm - E (field): V cm -1 σ (coductivity): A V -1 cm -1 (or Ω -1 cm -1 ) if the coductivity is a costat (field-idepedet) idepedet), we have Ohm s Law large σ = coductor (metal) moderate σ = semicoductor small σ = isulator the iverse of the coductivity is the resistivity, ρ 1 ρ = σ 107 V = IR

3 RESISTIVITY FORMULA L A I I = curret A = x-sectioal area of wire L = legth of wire J = σ E E ρj E ρi/a voltage drop: V=EL V/L ρi/a R ρl/a ρ is a materials property, idepedet d of geometry, with uits of ohm cm 108

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5 DRUDE MODEL (1900) classical (Newtoia) theory of electrical coductivity i metals Kietic theory of gases applied to electros i a metal. Metal treated as a gas of mobile valece electros movig agaist a backgroud of immobile ios. Coductivity it is determied d by electros collidig with scatterers. Assumptios: 1. o electro-electro forces (idepedet electro approximatio). betwee collisios, o electro-io forces (free electro approx.) ) 3. collisios are istataeous, leadig to abrupt chages i e - velocity 4. collisios radomize the e - velocity to a thermal distributio 5. there is a mea time betwee collisios of τ (collisio time) P dt Probability of a electro udergoig a collisio durig time iterval dt

6 If electros per uit volume move with velocity v across a area A i time dt, the charge crossig this area i time dt is J ev e v Adt This is the et curret desity (the et drift curret) Let s fid a expressio for the average electro velocity (the drift velocity). Without a field, < v > = 0. But with a field E, v = at = Ft/m = -e Et/m average time betwee collisios is τ, so: drift v = -e E /m e J=- e v= E = E ; m = e m 111

7 CARRIER MOBILITY The proportioality p costat betwee the absolute drift velocity ad the electric field is called the electro mobility, μ: e e vdrift = E= E ; m m ; = e Note: τ is the collisio time, ot the electro lifetime! 11

8 PREDICTIONS OF DRUDE THEORY predicted collisio time: predicted mea free path: m = 10 e sec l v Å average electroic speed: ~10 5 m/s from equipartitio theory but experimetal mea free paths ca be Å electros do ot simply bouce off the ios! Drude model provides good explaatios of: 1. DC ad AC coductivity of metals. Hall effect (magitude, ot sig) 3. Wiedema-Fraz Law ( ) / LT 4. thermal coductivity due to electros ear room temperature but, sice it uses classical statistical mechaics (Maxwell-Boltzma), it gets most other quatities wrog (heat capacity, thermopower, etc.).

9 SOMMERFELD THEORY the simplest quatum mechaical theory of electros i metals Applies Pauli Exclusio Priciple to kietic gas theory. Igores all forces except the cofiig surfaces of the solid, treatig electros as free & idy particles i a box. The quatum mechaical (QM) treatmet has two major effects: 1. oly electros with certai wave vectors (eergies) are allowed. quatum statistical mechaics (Fermi-Dirac distributio) the Pauli exclusio priciple must be obeyed (oe e - per state) The allowed eergy levels ls for a electro i a 3D box of volume V are foud by solvig the (time-idepedet) Schrödiger equatio where the Hamiltoia, (the eergy operator) is: H H E p m mx y z m

10 the Schrödiger equatio becomes: m E the geeral solutio is a plae wave: k () r 1 ik k r V e with eergy: E( k) 1 p k m m mv electros i the metal behave as plae waves of wave vector k p = m v = k ; k we ext see that the boudary coditios restrict k to discrete values

11 we require that the electros stay i the crystal, ad this places a costrait o the allowed values of k we apply Bor-vo Karma (periodic) boudary coditios to keep the electros i the metal. For a cubic crystal of edge legth L, ( x L, y, z) ( x, y, z) ( x, yl, z) ( x, y, z) ( xyz,, L) ( x, y, z) ik L 0 ik L this coditio is met oly whe: e x e y e z 1 i other words: L ik L x y z kx ; ky ; kz L L L where x, y, z are itegers 116

12 DENSITY OF k-space POINTS i D each electro level occupies a area i k-space of: L the umber of levels i a large the desity of k-space area of k-space Ω is: poits per uit area is: total area A A area per k-poit (4 / L )

13 3D CASE the allowed wave vectors are those whose Cartesia coordiates i k-space are itegral multiples of π/l each electro level occupies a volume i k-space of: L the umber of levels i a large volume of k-space Ω is: V (8 / L ) 8 the k-space desity of levels is: V

14 Assume that we have N electros i our solid. To build up the groud state of the solid (0 K), we add the electros oe by oe ito the allowed levels accordig to the Pauli exclusio priciple: each allowed wave vector (level) has two electroic states, spi up ad spi dow sice eergy is quadratic i wave vector: E ( k ) k m the lowest eergy level correspods to k = 0 ( electros) the ext lowest is k = π/l (6 levels, l 1 electros total), etc. whe N is eormous, the occupied regio of k-space will look like a sphere (the Fermi sphere). The radius of this sphere is labeled k F : volume of the Fermi sphere: 4 3 kf 3 3 F 119

15 k F at 0 Kelvi, the groud state of the N-electro system is formed by occupyig all sigle-particle levels with k less tha k F the umber of allowed values of k is: V kf kf V the total umber of electros is twice this: N 3 k F 3 V the free electro desity is the: N V 3 k F 3 10

16 FERMI SPHERE The surface of the Fermi sphere separates occupied ad uoccupied states i k-space. bouded by Fermi surface kf 3 1/3 radius is Fermi wave vector k F Fermi eergy: E k m F F / Fermi mometum: p F k F zero eergy Fermi velocity: F F / v k m Fermi eergy Fermi temperature: T E / k F F B

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18 i terms of the free electro desity, the eergy of the most eergetic electros is: E F (3 ) m /3 the total eergy of the groud-state electro gas is foud by addig up the eergies of all the levels withi the Fermi sphere. it s easiest to itegrate over cocetric shells: E tot k F V k 8 mm 0 (4 k dk) 3 E tot spi volume of shell of desity of eergy of width dk levels levels i this shell kf 5 V 4 kv F k dk m 10 m 0 13

19 3 N by substitutig for the volume i terms of N: V, 3 k F we ca fid the average eergy per electro: E tot N k 10 m 5 3 F 3 E F The average electro eergy at 0 K is 60% of the Fermi eergy. we ca also write this result as: E tot N 3 5 kt Typically, T F 5 x 10 4 K, while the eergy per electro i a classical electro gas (1.5k B T) vaishes (= zero) at 0 K. B F a classical gas achieves this E/N oly at T = (/5)T F 14

20 QUANTUM DEGENERACY PRESSURE the electro gas exerts a quatum mechaical pressure (called the degeeracy pressure) that keeps the free electro gas from collapsig at 0 K: P E E tot V N E tot kv (3 N) 10 m 10 m 5 5/3 F V /3 P 5/3 (3 N ) 5/3 E V tot 3 10 m 3 V The degeeracy pressure is a cosequece of the Pauli priciple. White dwarfs ad eutro stars are stabilized by this pressure.

21 3D DENSITY OF STATES the desity of states s g(e) is the umber of oe-electro states s (icludig spi multiplicity) per uit eergy ad volume g ( E ) 3D 1 dn V de N = Fermi sphere volume # levels l per uit volume ge ( ) N N 4 V k V * m E 3/ * 1 dn 1 m V de 3D 3/ g(e) E 1/ 16 E F

22 EFFECT OF REDUCED DIMENSIONALITY ON DOS D sheet quatum well 1D quatum wire ge ( ) m* L z 0D quatum dot ge ( ) m * e 1/ E LL x y g ( E ) discrete 17

23 si ( y) k x k i z e z k A y x 3D ( ) x y z i k x k y k z Ae ) ( 8 * * y x e z e z k k m L m h E Well * ( ) y x y z Ae E k k k si si z ik e y k x k A z ( ) x y z m e * * ) ( si si z y x y x k L L h E e y k x k A Wire * * ) ( 8 z e y x e m L L m Wire si si si z y x h z k y k x k A ) ( 8 * z z y y x x e L L L m h E Dot

24 Quatum Cofiemet ad Dimesioality 19

25 FERMI-DIRAC DISTRIBUTION FUNCTION At absolute zero the occupacy of states is 1 for E E F ad 0 for E > E F. At fiite temperatures, some electros ear E F have eough thermal eergy to be excited to empty states above E F, with the occupacy f(e k,t) give by the Fermi-Dirac distributio fuctio. Fermi fuctio: f( E, T) 1 k B e 11 k ( E )/ k T cosequece of Pauli exclusio priciple plays cetral role i solid state physics μ is the chemical potetial (μ = E 0 K) f( E) 1 ( E EF )/ kbt e 1 * f = 0.5 at E = E F 130

26 STATISTICAL DISTRIBUTION FUNCTIONS Boltzma distributio: for idepedet, distiguishable classical particles (high T, low desity, so quatum effects egligible) f B ( E) 1 exp[( E ) / kt] 0 < f(e) < N B Fermi-Dirac distributio: for idepedet, idetical Fermios (particles of half-iteger spi) f FD ( E) 1 exp[( E ) / k T] 1 0 < f(e) < 1 B Bose-Eistei distributio: for idepedet, idetical Bosos (particles of iteger spi) f BE ( E) 1 exp[( E ) / k T] 1 0 < f(e) < N B 131

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28 Fermi fuctio f(e) vs. eergy, with E F = 0.55 ev ad for various temperatures i the rage 50K T 375K. 133

29 f ( E ) 1 ( E EF )/ kb T e 1 For E E F >> kt, f(e) reduces to the classical Boltzma fuctio: 1 1 f( E) e e ( EE )/ k T ( EE )/ k T e F B 1 e F B ( EE )/ k T E/ k T F B B The Boltzma tail of the distributio 134

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32 f(e) for METALS At ormal temperatures, kt is small compared with E F. Sice oly those electros withi ~kt of E F ca be thermally promoted or participate i electrical coductio, most of the electros are froze out. 137

33 f(e) for SEMICONDUCTORS 138

34 CARRIER DENSITY i SEMICONDUCTORS f( E) g( E) de 139

35 IMPORTANCE OF g(e) ad f(e) electros holes f(e)g(e) product for a bulk semicoductor, showig the pools of free electros ad holes at the bad edges 140

36 ELECTRONIC STRUCTURE METHODS 141

37 NEARLY-FREE ELECTRON MODEL Starts from the free electro perspective (V = 0) ad adds a very weak periodic potetial to represet the ios. Electros still idepedet. sice the potetial is very weak, we ca use perturbatio theory to calculate how the free electro wavefuctio ad eergies are chaged. Bloch s Theorem: The wavefuctio of a electro i a periodic potetial ca be writte as a plae wave times a fuctio with the periodicity of the Bravais lattice. () r u () r e i k k kr with: u k () r u ( r R) k mai result: The periodic potetial deforms the parabolic E(k) of free electros oly ear the edge of the Brilloui zoe; this results i a eergy gap at the Brilloui zoe boudary (i.e., whe k π/a). most useful for s- ad p-block metals (e.g., alkali metals) 14

38 1D BLOCH WAVE ( x) u ( x) e i k k kx 143

39 TIGHT-BINDING MODEL Molecular l orbital ( chemical ) approach to the electroic structure t of ifiite 3D solids. starts from basis of liear combiatios of atomic orbitals (LCAOs), ad cosiders iteractios betwee atomic sites as perturbatios opposite simplificatio of the free electro models () r c () r coefficiet i atomic orbital particularly useful for isulators, d bads of trasitio metals, polymers, some semicoductors, ad other tightly-boud systems 144

40 Molecular Orbital Theory 145

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42 Diatomic Molecules 147

43 MO Theory for Solids Qualitative Expectatios Iteractio of two atoms: Iteractio of may atoms: bad N atomic orbitals give N molecular orbitals. whe N is eormous, the MOs form bads ( E << kt) 148

44 bads form oly whe there is sufficiet spatial overlap betwee atomic orbitals to form delocalized states depeds o iteratomic distace valece electros delocalized over etire crystal Sodium metal 1s s p 6 3s 1 localized atomic (core) states greater orbital overlap wider bads at high eough pressures, may solids become metallic149

45 Orbital overlaps for sodium: r plots 150

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47 The differeces betwee metals, semimetals, semicoductors, ad isulators deped o: the bad structure whether the valece bad is full or oly partly full the magitude of ay eergy gap betwee full ad empty bads filled valece bad, empty coductio bad semicoductor/isulator 15

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51 Miimizatio with respect to coefficiets c 1 ad c gives a system of two simultaeous equatios 156

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54 Overlap Itegral S: proportioal to degree of spatial overlap betwee two orbitals. It is the product of wave fuctios cetered o differet lattice sites. Varies from 0 (o overlap) to 1 (perfect overlap). Coulomb Itegral α: It is the kietic ad potetial eergy of a electro i a atomic orbital experiecig iteractios with all the other electros ad all the positive uclei Resoace Itegral β 1 : Gives the eergy of a electro i the regio of space where orbitals 1 ad overlap. The value is fiite for orbitals o adjacet atoms, ad assumed to be zero otherwise. 159

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60 For a 1D solid with lattice costat a ad = atom idex: 165

61 k e ika = Γ X rage of uique k e i0a 1 i e ( 1) 166

62 large umber of MOs form bad of states t Bad structure 167

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65 WIDTH OF THE BANDS ev Note: as always, the bodig orbitals are less stabilized tha the atibodig orbitals are destabilized a cosequece of overlap: e.g., for a dimer, / E 1 S 170

66 CALCULATION OF 1-D BAND STRUCTURE N atoms k N e 0 ika Crystal Schrodiger Equatio: H ( k ) E ( k ) ( k ) Dirac braket otatio: Electroic eergies: ˆ * H Hˆ d Ek ( ) Ĥ for ormalized atomic orbitals ad igorig overlap itegrals: m 1 if m ( ) m 0 if m i m ka e m m, 171 N

67 Ĥ k e ika 1. for o site (m = ): ( k) Hˆ ( k) ˆ H N. for resoace (m ), cosider oly the earest eighbors () 1 +1 ˆ ( 1) e H e e ika i ka ika 1 Ek ( ) ( k) Hˆ ( k) ika ika N N( e e ) ( k) ( k) N cos ka 17

68 1 +1 Ek ( ) e Hˆ e e e ika ik 1) a ika ik 1) a ( ( 1 1 cos ka Badwidth i 1D is 4β (this result igores overlap (S itegrals)) i Z dimesios: i W 4 Z 173

69 SLOPE OF THE BANDS DENSITY OF STATES 174

70 Example: Krogma s salt the DOS couts levels the itegral of the DOS up to E F is the total umber of occupied MOs 175

71 SLOPE OF THE BANDS CARRIER VELOCITY the mea velocity of a electro described d by eergy E ad wave vector k is 1 E vk ( ) k highest velocity Geeral result. Electros move forever with costat velocity (i ideal crystals). Zero velocity for electros i isolated atomic levels (zero badwidth) zero velocity less overlap lower tuelig probability lower velocity 176

72 CURVATURE OF THE BANDS CARRIER MASS the effective mass of a charge carrier ear a bad miimum i or maximum is iversely proportioal to the curvature of the bad: 1 1 E * m egative effective mass! hole! k Parabolic approximatio ear miimum/maximum: Ek ( ) E k mm 0 * positive effective mass E k m * 177

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84 ) ( ) ( 1 1 a ik e k ) ( ) ( 1 a ik e k H e e basis atom k E a ik a ik ) ( ˆ ) ( ) )( ( 1) ( 1 H e e H e e a ik a ik ) ( ˆ ) ( ) ( ) ( ) ( H e e a ik a ik a ik a ik ) ( ˆ ) ( 3 1 1) ( ka e e a ik a ik cos 189

85 Similarly: basis atom k E ) )( ( H e e H e e a ik a ik a ik a ik ) ( ˆ ) ( ) ( ˆ ) ( 1 1 1) ( H e e H e e ik ik a ik a ik ) ( ˆ ) ( ) ( ) ( 3 1 1) ( 1 1 ka e e a ik a ik cos E 4 E1 d bl h ll 4 doublig the uit cell size doubles the eergy per uit cell 190

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90 PEIERLS (pay-earls) DISTORTION 1D equally spaced chais with oe electro per io are ustable such a system ca lower its eergy by distortig so as to remove a electroic degeeracy. This is the solid state aalogue of the Jah-Teller effect. - symmetry breakig lifts a degeeracy 195

91 JAHN-TELLER EFFECT a o liear molecule with a degeerate electroic groud state will udergo a symmetry lowerig deformatio that breaks the degeeracy, stabilizig oe orbital ad destabilizig aother. e.g. tetragoal distortio lowers the eergy of a d 9 complex octahedral tetragoal 196

92 PEIERLS DISTORTION OF H ATOM CHAIN E F E F symmetric pairig distortio opes a bad gap at the Fermi level. 197

93 distortio stabilizes the system - effect largest for ½ filled bad - bad gap forms -H molecules charge desity waves

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95 POLYACETYLENE aother 1D Peierls distortio localizatio of pi electros Oxidatio with haloge (p dopig): [CH] + 3x/ I > [CH] x+ + x I 3 Reductio with alkali metal ( dopig): [CH] + x Na > [CH] x + x Na + pure PA = isulator doped PA = coductor Ala J. Heeger, Ala G. MacDiarmid ad Hideki Shirakawa Nobel Prizeii Chemistry 000, orgaic semicoductors D ad 3D Peierls distortios also occur, sometimes formig bad gaps

96 D BAND STRUCTURE maik a x y ( k) ) e m, ik m, 01

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98 How to calculate E(k)? maik a x y ( k ) e, m, ik Crystal Schrodiger Equatio: H ( k) E( k) ( k) m Ek ( ) Hˆ E( k) (cos k a cos k a) x y 03

99 4 M ( a, a ) 4 (0,0) E ( k ) (cos k a cos k a ) W 4Z x y 04

100 E 4 4 DOS E ( k ) (cos k a cos k a ) x y 05

101 Y 06

102 EFFECT OF LATTICE SPACING cosider the p x orbitals. For a square lattice: a E ( k ) cos k a a x a b b 0 0 b cos k y b b E a 4 a 4 b 07 Y X M Y

103 For rectagular lattice with icreased lattice costat i the y-directio directio, β b is smaller tha before: b E( k) cos k a a x b cos k y b E a 4 a X, Y ot equivalet k-poits less overlap i y-directio tha for square lattice 4b 08 Y X M Y

104 For rectagular lattice with very large lattice costat i y-directio directio, β b is almost zero: b E( k) cos k a cos k E a x b y b a 4 a X, Y ot equivalet k-poits o overlap i y-directio, so idetical to 1D bad structure! Y X M 09 Y

105 3D BAND STRUCTURE Brilloui i Zoe of Diamod ad Zicblede Structure (FCC Lattice) Sig Covetio Zoe Edge or surface : Roma alphabet Iterior of Zoe: Greek alphabet Ceter of Zoe or origi: Notatio: <=>[100] directio X<=>BZ edge alog [100] directio <=>[111] directio L<=>BZ edge L<=>BZ edge alog [111] directio 10

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107 Electroic Bad Structure of Si E g <111> <100> <110> 1