SOLID STATE PHYSICS. Crystal structure. (d) (e) (f)

Transcription

1 SOLID STAT PHYSICS y defto, sold state s that partcular aggregato form of matter characterzed by strog teracto forces betwee costtuet partcles (atoms, os, or molecules. As a result, a sold state materal has a depedet geometrc form ( cotrast to lquds, whch tae the form of the cotaer ad a varat volume ( cotrast to gases/vapors gve temperature ad pressure codtos. As temperature creases, a sold state materal ca evolve to aother aggregato form (lqud or gas. Sold state physcs studes the structural, mechacal, thermodyamc, electrcal, magetc, ad optcal propertes of (poly-crystalle ad o-crystalle solds (for example, amorphous materals, such as glass. Crystal structure The propertes of crystalle solds are determed by the symmetry of the crystalle lattce, because both electroc ad phooc systems, whch determe, respectvely, the electrc/ magetc ad thermal respose of solds, are very sestve to the regular atomc order of materals ad to ay (local or o-local perturbato of t. The crystalle structure ca be revealed by the macroscopc form of atural or artfcally-grow crystals (see the pctures below, or ca be ferred from the resultg debrs after cleavg a crystalle materal. (a (b (c (d (e (f Crystals of (a baryt, (b salt, (c hexagoal beryl, (d trgoal quartz, (e mooclc gypsum, ad apatte (f

2 Crystal Structure No-crystalle materals have o log-rage order, but at least ther optcal propertes are smlar to that of crystalle materals because the wavelegth of the cdet photos (of the order of μm s much larger tha the lattce costat of crystals ad so, photos see a effectve homogeeous medum. Other propertes of o-crystalle materals are derved based o cocepts proper to crystalle solds ad, therefore, the crystal structure s extremely mportat uderstadg the propertes of sold state materals. The macroscopc, perfect crystal s formed by addg detcal buldg blocs (ut cells cosstg of atoms or groups of atoms. A ut cell s the smallest compoet of the crystal that, whe staced together wth pure traslatoal repetto, reproduces the whole crystal. The perodcty of the crystalle structure that results ths way s cofrmed by X- ray dffracto expermets. The fgures below llustrate crystals whch the bass cossts of (a oe atom ad (b two atoms. (a (b The group of atoms or molecules that forms, by fte repetto, the macroscopc crystal s called bass. The bass s postoed a set of mathematcal/abstract pots that form the lattce (also called ravas lattce. So, a crystal s a combato of a bass ad a lattce. Although usually the bass cossts of oly few atoms, t ca also cota complex orgac or orgac molecules (for example, protes of hudreds ad eve thousads of atoms. I two dmesos, all ravas lattce pots R ma a ( m + ca be obtaed as superpostos of tegral multples of two o-collear vectors ad a a (m ad are arbtrary tegers. A bass cosstg of s atoms s the defed by the set of

3 Crystal Structure vectors r j m ja + ja, j,,,s, that descrbe the posto of the ceters of the bass atoms wth respect to oe pot of the ravas lattce. I geeral, m,. j j very pot of a ravas lattce s equvalet to every other pot,.e. the arragemet of atoms the crystal s the same whe vewed from dfferet lattce pots. The ravas lattce defed by ( s varat uder the operato of dscrete traslato Tpq pa + qa alog teger multples p ad q of vectors ad a, respectvely, because a T ( pq ( Rm Tpq + Rm R p+ m, q+ s aga a ravas lattce pot. I fact, sce the traslato operato s addtve,.e. T pq Tuv Tp+ u, q+ v TuvT pq, assocatve,.e., ad has a verse, such that T T ad T ( T T ( T T T, commutatve,.e. T pq uv m pq p, q pq uv m pqt uv T pq T p, q I wth I the detty trasformato, t follows that the traslatos form a abela (commutatve group. ecause codto ( s satsfed for all ravas lattce pots, ad a are called prmtve traslato vectors, ad the ut cell determed by them s called prmtve ut cell. The modulus of these vectors, a a ad a, are the lattce costats alog the a a respectve axes, ad the area of the ut cell two dmesos s S a. It s mportat a to otce that the set of vectors ad a s ot uque (see the fgures below, but all prmtve ut cells have the same area. a The prmtve ut cell covers the whole lattce oce, wthout overlap ad wthout leavg vods, f traslated by all lattce vectors. A equvalet defto of the prmtve ut cell s a cell wth oe lattce pot per cell (each lattce pot the fgures above belog to

4 Crystal Structure 4 four adjacet prmtve ut cells, so that each prmtve ut cell cotas 4 (/4 lattce pot. No-prmtve (or covetoal ut cells are larger tha the prmtve ut cells, but are sometmes useful sce they ca exhbt more clearly the symmetry of the ravas lattce. esdes dscrete traslatos, the ravas lattce s varat also to the pot group operatos, whch are appled aroud a pot of the lattce that remas uchaged. These operatos are: Rotatos by a agle π / about a specfc axs, deoted by C, ad ts multples, j C ( C j. Geometrc cosderatos mpose that,,, 4 ad 6, ad that repeatg the rotato tmes oe obtas C, where s the detty operato, whch acts as r r. Moreover, C π does ot represet a symmetry elemet. C D θ θ A The allowed values of ca be determed assumg that we apply a rotato wth a agle θ aroud a axs that passes frst through a pot A ad the through a adjacet lattce pot. The pots A ad are separated by the lattce costat a. If C ad D are the resultg pots, they should also be separated by a teger multple of a. rom the requremet that CD a + as( θ π / a a cosθ ma, or cosθ ( m /, wth m teger, t follows that m ca oly tae the values,,,, ad, the correspodg π /θ tag the values specfed above. As for traslatos, the rotatos also form a abela group. xamples of two-dmesoal fgures wth dfferet rotato symmetres: C C C 4 C 6

5 Crystal Structure 5 Iverso I, whch s defed by the operato r r f appled aroud the org. Reflecto σ j, whch ca be appled aroud the horzotal plae (j h, the vertcal plae (j v, or the dagoal plae (j d. Improper rotato S, whch cossts of the rotato the plae ormal to the rotato axs. Note that S I. C followed by reflecto Whe we combe the pot group symmetry wth the traslatoal symmetry, we obta the space-group symmetry. It s mportat to otce that the bass ca troduce addtoal symmetry elemets, such as helcodal symmetry axes ad gldg reflecto plaes. The fgure bellow represets several symmetry operatos: (a traslatos, (b rotato, (c verso, ad reflecto wth respect to a (d vertcal, ad (e horzotal plae. (a (b (c (d (e Crystal lattces are classfed accordg to ther symmetry propertes at pot group operatos. The fve ravas lattce types two dmesos are show the fgure below. These are: square lattce, for whch a a, ad γ 9, where γ s the agle betwee a ad a, rectagular lattce, for whch a, ad γ 9, a cetered rectagular lattce, whch s a rectagular lattce wth a addtoal lattce pot the ceter of the rectagle, hexagoal lattce, for whch a, ad γ 6 (or for a dfferet choce of the org, a oblque rectagular lattce (called also oblque lattce, for whch a a, ad γ 9, 6 (or.

6 Crystal Structure 6 Wth the excepto of the cetered rectagular lattce, all ut cells the fgure above are prmtve ut cells. The prmtve cell for the cetered rectagular lattce s a rhombus (see fgure at rght ad therefore ths ravas lattce s also called rhombc lattce, case whch ts prmtve ut cell has a, ad γ 9, 6 (or. a ach lattce type has a dfferet set of symmetry operatos. or all ravas lattce types two dmesos, the rotato axes ad/or reflecto plaes occur at lattce pots. There are also other locatos the ut cell wth comparable or lower degrees of symmetry wth respect to rotato ad reflecto. These locatos are dcated the fgure below.

7 Crystal Structure 7 I order to corporate the formato about the pot group symmetry the prmtve cell, the Wger-Setz cell s usually employed. Ths partcular prmtve ut cell s costructed by frst drawg les to coect a gve lattce pot to all earby lattce pots, ad the drawg ew les (or plaes, three-dmesoal lattces at the md pot ad ormal to the frst les. The Wger-Setz cell s the smallest area (volume eclosed by the latter les (plaes. A example of the costructo of a Wger-Setz cell for a twodmesoal oblque lattce s llustrated the fgure below. or a two-dmesoal square lattce the Wger-Setz cell s also a square. The Wger-Setz cell s always cetered o a lattce pot ad corporates the volume of space whch s closest to that lattce pot rather tha to ay other pot. θ r The faces of the Wger-Setz cell satsfy the relato r cosθ R /, where R s the dstace to the earest eghbor ad θ s the agle betwee r ad R. Ths relato ca be rewrtte as ( r R R or, sce the equato s equvalet to the replacemet of R wth R, r R + R, ad fally, ( r + R r. I other words, the faces of the Wger- Setz cell are determed by the tersecto betwee equal-radus spheres cetered at the earest-eghbor pots of the ravas lattce. I a smlar maer, three dmesos, all ravas lattce pots R mp ma + a + pa ( ca be obtaed as superpostos of tegral multples of three o-coplaar prmtve traslato vectors, ad a (m,, ad p are arbtrary tegers, ad the pot group a a

8 Crystal Structure 8 operatos are defed detcally. The volume of the prmtve ut cell, whch ths case s a parallelepped, s Ω ( a a. a There are 4 three-dmesoal ravas lattces, whch belog to 7 crystal systems, as ca be see from the fgure below, where the prmtve traslato vectors are deoted by a, b, c (wth respectve legths a, b, ad c, ad α, β, γ are the agles betwee b ad c, c ad a, ad a ad b, respectvely. These crystal systems, whch are dfferet pot groups edowed wth a sphercal symmetrc bass, are: cubc, for whch a b c, α β γ 9. It cossts of three o-equvalet spacegroup lattces: smple cubc, body-cetered cubc, ad face-cetered cubc. Ths s the crystal system wth the hghest symmetry ad s characterzed by the presece of four C axes (the dagoals of the cube tetragoal, for whch a b c, α β γ 9. It ecompasses the smple ad bodycetered ravas lattces ad cotas oe C 4 symmetry axs. orthorhombc, for whch a b c, α β γ 9. It corporates the smple, bodycetered, face-cetered, ad sde-cetered lattces ad has more tha oe C symmetry axs or more tha oe reflecto plae (actually, three such axes/plaes, perpedcular to each other. hexagoal, for whch a b c, α β 9, γ. It s characterzed by the exstece of a sgle C6 symmetry axs. The covetoal hexagoal ut cell (see the fgure at rght s composed of three prmtve cells. trgoal, for whch a b c, α β γ 9. It cotas a sgle C axs. mooclc, for whch a b c, α γ 9 β. It cludes the smple ad sdecetered lattces, ad has oe C symmetry axs ad/or oe reflecto plae perpedcular to ths axs. trclc, for whch a b c, α β γ 9. Ths s the crystal system wth the lowest symmetry. It s ot symmetrc wth respect to ay rotato axs or reflecto plae.

9 Crystal Structure 9 The relatos betwee these lattces ca be summarzed the fgure at the rght. The dfferet crystal systems have dfferet umbers of ut cell types because other possble ut cell types caot represet ew ravas lattces. or example, both the body-cetered ad the face-cetered mooclc lattces ca be reduced to the sde-cetered lattce by approprately choosg the prmtve traslato vectors. xamples of two sets of prmtve traslato vectors for a body-cetered cubc (bcc lattce are represeted the fgure below at left ad ceter, whle the fgure at rght dsplays a set of prmtve traslato vectors for a face-cetered cubc (fcc lattce.

10 Crystal Structure The prmtve traslato vectors for the left fgure above ca be expressed as a ( a / ( x + y, a ( a / ( x + y +, a ( a / ( x y +, (4 z z z whle those for the rght fgure are a ( a / ( x +, a ( a / ( y +, a ( a / ( z + (5 y z x ad the agles betwee these vectors are 6. A smple lattce has lattce pots oly at the corers, a body-cetered lattce has oe addtoal pot at the ceter of the cell, a face-cetered lattce has sx addtoal pots, oe o each sde, ad a sde-cetered lattce has two addtoal pots, o two opposte sdes. The smple lattces are also prmtve lattces ad have oe lattce pot per cell, sce the eght stes at the corers are shared by eght adjacet ut cells, so that 8 (/8. The o-smple lattces are o-prmtve. The volume of the prmtve ut cell these lattces s obtaed by dvdg the volume of the covetoal ut cell by the umber of lattce pots. I partcular, the body-cetered lattces have two pots per ut cell: the eght at the corers whch cotrbute wth 8 (/8, ad the oe the ceter, whch belogs etrely to the ut cell. The face-cetered lattces have 4 lattce pots per cell: those the corers cotrbute wth 8 (/8, ad those o the faces cotrbute wth 6 (/, sce they are shared by two adjacet cells. ally, the sde-cetered lattces have two lattce pots per cell: the pots at the corer cotrbute wth 8 (/8, ad those o the faces wth (/. The characterstcs of the cubc lattces wth sde a are summarzed the table below. If each lattce pot s expaded to a sphere wth a radus equal to half of the dstace betwee earest eghbors, such that adjacet spheres touch each other, the a pacg fracto ca be defed as the fracto betwee the volume of the spheres cotaed the covetoal ut

11 Crystal Structure cell ad the volume of the ut cell. Note that the volume betwee the spheres oe ca always sert smaller spheres, whch ca stad for other atom types. Smple ody-cetered ace-cetered Volume of a a a covetoal cell Lattce pots per 4 cell Volume of prmtve a a / a /4 cell Number of earest 6 8 eghbors Nearest-eghbor a a/ a/ dstace Number of secod 6 6 eghbors Secod-eghbor a a a dstace Pacg fracto π/6.54 π/8.68 π/6.74 The 4 ravas lattces corporate all possble crystalle structures; they result by tag to cosderato the space-group symmetry,.e. the symmetry at traslatos ad the pot group symmetry of the lattce (the symmetry wth respect to rotato, reflexo or verso. Whe the bass cossts of oly oe atom, the ravas lattce s detcal wth the crystalle structure. ut whe the bass s complex ad cossts of several atoms, say s, the crystalle structure ca be see as formed by the terpeetrato of s ravas lattces. The ravas lattces have always a verso ceter oe of the lattce pots, whereas such a verso ceter ca lac crystals wth complex bases. y coutg the pot groups of the possble dfferet crystals (whch have bases wth dfferet symmetres, oe eds wth crystalle classes that ca be accommodated by the 7 crystal systems. Also, there are space groups that result from the combato of the crystalle structures wth the traslatoal symmetry.

12 Crystal Structure Idex system for lattce pots, drectos ad plaes Whe the org of the prmtve traslato vectors s a lattce pot, aother lattce pot wth a posto R mp ma + a + pa s smply specfed by the set of umbers [[m,,p]]. A egatve teger m, or p s deoted by a sg placed o top of t. or example, [[ m p ]] stays for the lattce pot specfed by the tegers m, ad p, wth m, ad p postve umbers. I partcular, for the three-dmesoal prmtve ravas lattces the coordates of the lattce pot at the org are [[,,]], the other lattce pots dfferg oly through dscrete traslatos alog the three coordate axs. The umber of o-equvalet lattce pots a ravas lattce s gve by the umber of lattce pots per ut cell. I partcular, for the body-cetered lattce, the posto of the lattce pot at the ceter of the cube s deoted by [[/,/,/]], the three addtoal lattce pots face-cetered lattces havg coordates [[,/,/]], [[/,,/]], [[/,/, ]]. I a smlar maer, depedg o the set of opposte stes they ca occupy, the addtoal ste a face-cetered lattce has the coordates [[,/,/]], [[/,,/]] or [[/,/,]]. A drecto, by defto, passes through two lattce pots. To specfy a drecto a crystalle lattce, oe uses the symbol [mp], where m, ad p are three tegers determed by the followg rule: sce oe ca specfy a drecto by the coordates [[, p ]] ad [[ m, p ]] of two pots through whch t passes, the dces m, ad p m,, are defed as the smallest teger umbers that satsfy the proportoalty relatos m m m, p p, p p m p m, (6 p m or m : p : p ( m m : ( : ( p. (7 If oe of the tegers s egatve, the sg s placed o top of the teger. or example, [ m p ] stays for the drecto specfed by the tegers m, ad p. If the drecto s ot cosdered as a oreted axs but as a smple le, the drecto specfed by the tegers m,, ad p s the same as that specfed by m,, ad p (otherwse, the chage of all sgs meas a chage of drecto of the same le. If there are several equvalet drectos (equvalet, from the pot of vew of crystal symmetry, they are deoted as mp. A

13 Crystal Structure partcular stuato s ecoutered the hexagoal lattce, whch lattce drectos are labeled by four umbers (ths stuato s ot further dscussed ths course. xamples: The axs s the [] drecto. The a a examples are llustrated the fgure below. axs s the [ ] drecto. Other a a [] [] a [] [] I three-dmesoal lattces, the oretato of a crystal plae s determed by three o-collear pots the plae. If each pot s stuated o a dfferet crystal axs, the plae s specfed by the coordates of the pots terms of the lattce costats,, ad a. a a Aother way to specfy the oretato of a plae, whch s more useful for structure aalyss, volves the determato of three dces, called Mller dces, accordg to the rule: d frst the tercepts of the plae o the axes terms of lattce costats a, a, ad a, rrespectve of the ature (prmtve or o-prmtve of the ut cell. Tae the recprocal of these umbers. If fractoal, reduce these umbers to the smallest three tegers, say m,, p, wth the same rato. The result, symbolzed by (mp (or ( m p f the secod dex, for example, s egatve, s the Mller dex system of the plae. It s obvous that the Mller dex for a tercept at fty s zero. The faces of a cubc crystal, for example, are deoted by (, (, (, (, (, ad (. Moreover, the plae ( s parallel to (, but cuts the a axs at a /. If, from the pot of vew of crystal symmetry, there s a set of oparallel equvalet plaes, they are symbolzed as {mp}. or example, the set of faces of a cubc crystal s {}. Aga, for the hexagoal lattce there are four Mller dces stead of three. xamples of Mller dces are gve the fgures below.

14 Crystal Structure 4 ( ( ( Note that the Mller dces determe ot oly oe plae but a famly of parallel plaes, sce there s a fte umber of plaes wth the same dces, all of whch cut the coordate axes at s / m, s /, ad s / p, wth s teger. The plae that cuts the axes at / m, /, ad / p s the closest to the org from the famly of parallel plaes. Note also that the plaes wth Mller dces (sm,s,sp are parallel wth the plae (mp, but the dstace betwee them s s tmes smaller. or example, the set of plaes ( s parallel to but twce as close as the ( set of plaes. I cubc crystals, the plae (mp s perpedcular to the drecto [mp] wth the same dces, but ths result caot be exteded to other crystal systems. A example s gve the fgure below.

15 Crystal Structure 5 Smple crystal structures Oe of the most smple crystal structures ad, at the same tme, of geeral terest, s that of NaCl (sodum chlorde. It s llustrated below. The lattce s face-cetered cubc, wth a bass cosstg of oe Cl o (blue at [[]] ad a Na + o (gree at [[/,/,/]]. As ca be see from the fgure below, a ut cube cossts of four NaCl uts, wth Na + os at postos [[/,/,/]], [[,,/]], [[,/,]], ad [[/,,]] ad Cl os at [[]], [[/,/,]], [[/,,/]], ad [[,/,/]]. ach atom has as earest eghbors sx atoms of opposte d. xample of crystals wth ths structure ad ther lattce costats are gve below. Crystal a(å Crystal a (Å Crystal a (Å L 4. Kr 6.6 MgO 4. Lr 5.5 Agr 5.77 MO 4.4 NaCl 5.64 Ag 4.9 MgS 5. NaI 6.47 CaSe 5.9 PbS 5.9 KCl 6.9 ao 5.5 SrTe 6.47 Aother commo structure s that of CsCl (other crystals wth the same structure are gve the table below. The lattce s ths case smple cubc, wth a bass cosstg of oe Cs + o (red at [[]], ad oe Cl o (gree at [[/,/,/]]. The umber of earest eghbors (of opposte d s eght. Crystal a (Å Crystal a (Å Crystal a (Å AlN.88 CsCl 4. TlCl.8 CuZ (β-brass.94 Csr 4.9 Tlr.97 AgMg.8 CsI 4.57 TlI 4. The crystal structure of damod (ad also of S ad Ge semcoductors s represeted below.

16 Crystal Structure 6 Crystal a (Å C (damod.57 S 5.4 Ge 5.66 α-s (grey 6.49 It s a face-cetered cubc (fcc lattce wth a bass cosstg of two detcal atoms, wth coordates [[]] ad [[/4,/4,/4]]. Alteratvely, damod ca be vewed as beg formed from two terpeetratg fcc lattces, dsplaced by /4 of the volume dagoal. Sce the covetoal ut cell of the fcc lattce cotas 4 lattce pots, t follows that the covetoal ut cell of damod has 4 8 atoms. No prmtve cell exsts that cotas oly oe atom. I damod, each atom has 4 earest eghbors ad ext earest eghbors. It s usually ecoutered materals where the covalet bodg prevals. Note that, although a fcc lattce, the pacg fracto of the damod structure s oly.4. A closely related crystal structure to that of the damod s the cubc zc sulfde (zc blede structure. It dffers from damod that the two atoms of the bass are dfferet ( ths case, Z ad S. The covetoal ut cell cotas four molecules, the Z atoms (dar blue the fgure below beg placed at the postos [[]], [[,/,/]], [[/,,/]] ad [[/,/,]], whereas the S atoms (gree occupy the postos [[/4,/4,/4]], [[/4,/4,/4]], [[/4,/4,/4]], ad [[/4,/4,/4]]. ach atom s surrouded by four equally dstat atoms of the opposte d, placed the corers of a regular tetrahedro. Crystal a (Å Crystal a (Å Crystal a (Å SC 4.5 AlP 5.45 IAs 6.4 ZS 5.4 AlAs 5.66 ISb 6.48 ZSe 5.67 GaAs 5.65 SC 4.5 MS (red 5.6 GaSb 6. CuCl 5.4 CdS 5.8 GaP 5.45 Cur 5.69 CdTe 6.48 AgI 6.47 HgSe 6.8

17 Crystal Structure 7 Ule the damod structure, where there s a ceter of verso at the mdpot of every le betwee earest-eghbor atoms, such verso ceters are abset the zc blede structure. Ths s a example of addtoal symmetry operatos related to the bass of the crystal structure. The hexagoal close-paced (hcp crystal structure ca be obtaed from the hexagoal ravas lattce f the bass cossts of two atoms (blue ad red the fgure below, left ad f the atoms oe plae, whch touch each other, also touch the atoms adjacet plaes. The pacg fracto ths case s.74 (as fcc lattces, ad s maxmum. Ths crystal structure s foud the sold state of may elemets, as ca be see from the table below. The hcp structure ca be vewed as vertcal arragemet of two-dmesoal hexagoal structures, such as the sphercal atoms the secod layer are placed the depressos left the ceter of every other tragle formed by the ceters of the sphercal atoms the frst layer. The thrd layer of atoms s the placed exactly above the frst, the fourth above the secod, ad so o. Ths d of arragemet s called AA I a deal hcp structure, the heght betwee the frst ad the thrd layers (the heght alog the c axs the fgure below s c 8/ a.6a. ecause the symmetry of the hcp lattce s depedet of the rato c/a, real hcp structures ths rato ca tae values close to, but ot exactly detcal to the deal.6 value (see the table below. Crystal a (Å c/a Crystal a (Å c/a He.57.6 Mg..6 e.9.58 T Nd.66.6 Zr..59 Z Y Cd Gd α-co.6.6 Lu.5.58

18 Crystal Structure 8 If the c/a rato dffers cosderably from the deal.6 value, the hexagoal structure s o loger closepaced. Ths s the case of graphte, for example, whch s a o-closed-paced hexagoal structure of carbo atoms (see the fgure at rght, wth a.4å ad c.4 Å, whch mples that c/a.9. The fact that the hcp structure has the same pacg fracto as the fcc structure s easly explaed the fgure below. Suppose that we place the frst two plae of atoms as the hcp structure. If the atoms the thrd plae are postoed over the ceters of the tragles formed by the ceters of the atoms the frst plae that have o atoms from the secod plae above them, the resultg structure s fact a fcc. Ths vertcal arragemet s called ACAC The hcp ad fcc structures dffer oly by the vertcal arragemet (AA or ACAC of hexagoal plaes of atoms. A structure closely related to hcp s wurtzte, geerally ecoutered bary compoud semcoductors such as ZS (wurtzte, ZO, N, CdS, CdSe, GaN, AlN, but sometmes also terary compouds such as Al.5 Ga.5 N. I bary compouds (see the fgure at rght, each elemet has a hcp structure, ad the crystal s formed by terpeetratg two such structures, so that a atom oe hcp lattce s equallydstaced from the atoms the other hcp lattce. The crystal structure of the elemets the perodc table s dcated the fgure below. Note that several elemets ca suffer trastos from oe crystalle structure to aother depedg o the exteral codtos: temperature, pressure, etc. I the table below dhcp stads for double hexagoal closed-paced (the heght of the cell alog the drecto ormal to the hexagoal plaes s twce that the hcp structure

19 Crystal Structure 9 Lattce costats of some elemets that crystallze the fcc crystal structure: Crystal a (Å Crystal a (Å Crystal a (Å Crystal a (Å Crystal a (Å Ar 5.6 Au 4.8 Cu.6 N.5 Pt.9 Ag 4.9 Ca 5.58 Kr 5.7 Pb 4.95 Sr 6.8 Al 4.5 β-co.55 Ne 4.4 Pd.89 Xe 6. Lattce costats of some elemets that crystallze the bcc crystal structure: Crystal a (Å Crystal a (Å Crystal a (Å Crystal a (Å a 5.6 e 4.8 Mo.6 Rb.5 Cr 4.9 K 5.58 Na 5.7 Ta 4.95 Cs 4.5 L.55 Nb 4.4 V.9 W 6.8

20 Recprocal lattce The cocept of recprocal lattce s drectly coected wth the perodcty of crystalle materals ad of ther physcal propertes (such as charge desty, electrc feld dstrbuto, etc.. Sce the crystal s varat uder ay traslato wth a ravas lattce vector R mp ma + a + pa ( for ay tegers m, or p, ay fucto ϕ wth the same perodcty as the crystalle lattce must satsfy the relato ϕ r ϕ( r + R, ( ( mp where x, x, s a arbtrary posto vector wth coordates,, ad x measured r ( x x x wth respect to the (geerally o-orthogoal system of coordates determed by, a, a ad a. Ths meas that ϕ ( x, x, x ϕ( x + ma, x + a, x + pa ( or, for a fucto that ca be expaded a ourer seres ϕ ( x, x, x ϕ exp[ ( Gx + G x + Gx] (4 G, G, G t follows that, for ay m,, ad p, exp( mg a, exp( G a, exp( pg a. (5 Thus, G, wth,,, ca oly tae dscrete values G πs / a, (6 ad (4 ca be rewrtte as

21 Recprocal lattce ϕ ( r ϕ exp( G r (7 s, s, s where G s + (8 b + sb sb s a vector a coordate system defed by the vectors b,,,, such that b a πδ. (9 j j Smlar to the ravas lattces that are costructed startg wth the prmtve vectors a, oe ca defe a recprocal lattce terms of the prmtve vectors b, such that G (8 are pots the recprocal lattce. A recprocal lattce ca oly be defed wth respect to a gve drect lattce. As demostrated the followg, the G vectors have dmesos (ad meag of wavevectors related to plae waves wth the perodcty of the drect lattce. If the vectors a are chose ad the volume of the prmtve cell the drect space s Ω ( a a a, the vectors b ca be chose as b π / Ω( a, b π / Ω( a, b π / Ω( a. ( ( a ( a ( a It follows the that the volume of the prmtve cell of the recprocal lattce s gve by π Ω rec b ( b b ( / Ω. ( xamples of drect ad correspodg recprocal lattces two dmesos are gve the fgures below.

22 Recprocal lattce or a d ( x, a d ( x +, the vectors of the recprocal lattce are determed from y y codto (9, ad are foud to be b ( π / d( x, b ( π / d( x +. y y Whe x ad y are ot orthogoal, but x y ε (see the fgure above, for a dx cy ad a d x + cy, we obta (please chec! b c + dε d + cε π x π cd( ε cd( ε c dε d cε y, b π x + π y. cd( ε cd( ε I three dmesos, the recprocal lattces for the ravas lattces the cubc system are summarzed the table below Real space Recprocal space Lattce Lattce costat Lattce Lattce costat SC a SC π / a CC a CC 4π / a CC a CC 4π / a The recprocal lattce of a cubc lattce s also cubc sce, ths case, f x, y, z are orthogoal vectors of ut legth, ax, ay, a az ad Ω a, from ( t follows that b a a (π / a, b (π / a, b (π / z,.e. the recprocal lattce s smple cubc wth a x lattce costat π / a. a y a Aalogously, the recprocal lattce to the bcc lattce wth (see the frst course ( a / ( x + y, a ( a / ( x + y +, a ( a / ( x y +, ad Ω a / has z z z prmtve vectors b (π / a( x +, b (π / a( y +, b (π / a( z +,.e. s a fcc y z x lattce wth a volume (of the prmtve ut cell recprocal state of Ω (π / a, rec whereas the recprocal lattce of the fcc lattce, wth a ( a / ( x +, a ( a / ( y +, y z

23 Recprocal lattce 4 a ( a / ( z +, ad Ω a / 4 s a bcc lattce wth Ω 4(π / a ad prmtve vectors b x (π / a( x + y, b (π / a( x + y +, b (π / a( x y +. I both cases the z z rec z cubc structure of the recprocal lattce has a lattce costat of 4π / a. Observato: The recprocal lattce of a recprocal lattce s the drect lattce. ecause the product of a prmtve ravas lattce vector ad of a prmtve vector of the recprocal cell s a teger multple of π,.e. that G R π ( mh + pl, ( mp hl + for all tegers m,, p ad h,, l, t follows that exp( G R for ay vector R the ravas lattce ad ay vector G the recprocal lattce. Ths mples that the fucto exp( G r has the same perodcty as the crystal because exp[ G ( r + R] exp( G r exp( G R exp( G r. As a cosequece, cell exp( G r dv ( s depedet of the choce of the cell ad a traslato wth a arbtrary vector d should ot chage the value of the tegral. More precsely, f cell exp[ G ( r + d] dv exp( G r dv (4 cell the [exp( G d ] cell exp( G r dv, from whch t follows that cell exp( G r dv Ωδ (5 G, ad that the set of fuctos exp( G r form a complete, orthoormal bass for ay perodc fucto whch has the same perodcty as the crystal,.e. whch ca be wrtte as ϕ ( r ϕ exp( G r. (6 G G

24 Recprocal lattce 5 If the formula above s regarded as a ourer trasformato of the perodc fucto ϕ, the coeffcets precsely, sce ϕ G ca be retreved by performg a verse ourer trasformato. More cell ϕ( rexp( G' r dv ϕg G cell ϕg G cell exp[ ( G G' r] dv exp( G rexp( G' r dv ϕgωδ G GG' (7 t follows that Ω ϕ ϕ( rexp( G r dv. (8 cell G Relatos betwee the drect ad recprocal lattces Oe geometrcal property that ca be easly show s that the recprocal lattce vector G mp mb + b + pb (9 s perpedcular to the plae (actually, to the set of parallel plaes wth Mller dces (mp the ravas lattce. The closest plae to the org from the set of plaes (mp cuts the a coordate axes at a / m, a /, ad a /, respectvely. p To show that (mp s perpedcular to G t s suffcet to demostrate that G s perpedcular to two o-collear vectors the (mp plae, whch ca be chose as mp mp u a a / m, v a p a / m, ( / / ad satsfy, deed, the relatos u G v G ( mp mp because of (9. The, t follows that the ormal to the (mp plae that passes through the org ca be expressed as G G. ( mp mp / mp

25 Recprocal lattce 6 a /p a a / a a a /m A cosequece of ths result s that the dstace betwee two cosecutve plaes wth the same Mller dces (mp s versely proportoal to the modulus of. Sce we ca always draw a plae from the (mp famly through the org, the dstace betwee two successve plaes s equal to the dstace betwee the org ad the closest plae to org from the (mp famly. Ths dstace s obtaed by calculatg the projecto o the ormal to the (mp,.e. o ( t s foud that G mp G G, of ay of the vectors a / m, a /, or / p. Usg mp mp / mp a d mp a a a π. ( m p G mp So, d mp m b + b + p b + m( b π b + p( b b + pm( b. (4 b

26 Recprocal lattce 7 As already poted out the dscusso about Mller dces, the dstace betwee ay two plaes the famly (sm,s,sp, s s tmes smaller tha betwee ay two plaes the famly (mp. The two famles/sets of plaes are parallel. I partcular, for the smple, body-cetered ad face-cetered cubc ravas lattces wth the prmtve traslato vectors gve the Crystal Structure secto of the course, the dstace betwee two cosecutve plaes wth the same Mller dces s, respectvely, d sc mp d bcc mp d fcc mp a, (5a m + + p a, (5b ( + p + ( p + m + ( m + a (5c ( + p m + ( p + m + ( m + p Due to the form of (7, the vectors G of the recprocal lattce ca be uderstood as wavevectors of plae waves wth the perodcty of the lattce ad wavelegths π / G, smlar to wavevectors optcs that are perpedcular to wavefrots ad have dmesos related to the wavelegth λ as π / λ. The frst rllou zoe Aalogous to the Wger-Setz cell drect lattces, oe ca defe a prmtve ut cell the recprocal lattce that has the same symmetry as ths lattce. Ths prmtve ut cell s ow as the frst rllou zoe. The costructo of the frst rllou zoe s smlar to that of the Wger-Setz cell,.e. we draw les to coect a gve lattce pot the recprocal lattce to all earby lattce pots, ad the draw ew les (or plaes, three-dmesoal lattces at the md pot ad ormal to the frst set of les. These les (plaes are called ragg plaes sce (as we wll see later all vectors that fsh o these surfaces satsfy the ragg codto. The frst rllou zoe s the the area (volume recprocal space that ca be reached from the org, wthout crossg ay ragg plaes. Hgher-order rllou zoes, say the th rllou zoe, are the defed as the area (volume recprocal space that ca be reached from the org by crossg exactly ragg plaes. The costructo of the frst (lght blue, secod (lght brow ad thrd (dar blue rllou zoes for a two-dmesoal lattce s llustrated the fgure below. The ragg plaes eclosg the th rllou zoe correspod to the th order X-ray dffracto.

27 Recprocal lattce 8 Although hgher order rllou zoes are fragmeted, the fragmets, f traslated, loo le the frst rllou zoe. Ths process s called reduced zoe scheme. All rllou zoes, rrespectve of the order, have the same volume. The hgher-order rllou zoes for a two-dmesoal square lattce are llustrated the fgure below. As for Wger-Setz cells, the faces of the frst rllou zoe satsfy the relato G G /, where G s the dstace to the earest eghbor the recprocal space. Ths relato ca be rewrtte as G G or, sce the equato s equvalet to the replacemet of G wth G, we obta ( + G,.e. the frst rllou zoe s the tersecto of spheres wth the same radus cetered at earest eghbor pots the recprocal lattce.

28 Recprocal lattce 9 I partcular, sce the recprocal lattce of the bcc lattce s a fcc lattce, the frst rllou zoe of the bcc lattce (see the polyhedro the fgure a below s the Wger-Setz cell of the fcc. The reverse s also true: the frst rllou zoe of a fcc lattce (the trucated octahedro/rhombododecahedro fgure b below s the Wger-Setz cell of the bcc lattce. or certa ravas lattce, partcular bcc, fcc ad hexagoal, the pots of hghest symmetry the recprocal lattce are labeled wth certa letters. The ceter of the rllou zoe s all cases deoted by Γ. Other symmetry pots are deoted as follows (see also fgures: sc lattce: bcc lattce: fcc lattce: M ceter of a edge R corer pot X ceter of a face H corer pot jog four edges N ceter of a face P corer pot jog three edges K mddle of a edge jog two hexagoal faces L ceter of a hexagoal face U mddle of a edge jog a hexagoal ad a square face W corer pot X ceter of a square face

29 Recprocal lattce hexagoal lattce: A ceter of a hexagoal face H corer pot K mddle of a edge jog two rectagular faces L mddle of a edge jog a hexagoal ad a rectagular face M ceter of a rectagular face Dsperso relatos of electros ad phoos for dfferet crystal drectos use ths labelg (see the fgures below, the labels dcatg the drecto but also the symmetry of the crystal, sce dfferet labels are used for dfferet symmetres.

30 X-ray dffracto o crystalle structures The drect observato of the perodcty of atoms a crystalle materal reles o the X-ray or partcle (electro or eutro dffracto/scatterg o these spatally perodc structures, sce the wavelegth of the cdet beam s these cases comparable to the typcal teratomc dstace of a few Å. Optcal dffracto s ot sutable for ths purpose sce the wavelegth of photos s much too log (about μm comparso to the lattce costat (a few Agstroms. I a dffracto expermet, both the X-ray or partcle source ad the detector are placed vacuum ad suffcetly far away from the sample such that, for moochromatc radato, the cdet ad outgog X-ray or partcle beams ca be approxmated by plae waves. The X-rays ca be used ether trasmsso or reflecto cofguratos. The dffracto pcture offers formato regardg the symmetry of the crystal alog a certa axs. I partcular, the postos of the spots gve formato about the lattce ad the testy aalyss reveal the composto of the bass. The X-rays peetrate deeply the materal, so that may layers cotrbute to the reflected testy ad the dffracted pea testes are very sharp ( agular dstrbuto. To obta sharp testy peas of the scattered radato, the X-rays should be specularly reflected by the atoms oe plae.

31 X-ray dffracto λ hc / or X-rays, the wavelegth s determed from the relato hν hc / λ or, whch equals a few Å f s of the order of few ev. I fact, λ(å.4/(ev. X-rays are scattered mostly by the electroc shells of atoms a sold, sce the ucle are too heavy to respod. lectros ca also have de rogle wavelegths smlar to the lattce costats of crystals. I ths case ( h / λ / m, ad for a electro eergy of 6 ev, the correspodg wavelegth λ h / m s about 5 Å. Actually, f the etc eergy of the electros s acqured a accelerato voltage potetal U, such that eu, oe has λ(å.8/[u(v] /. or eutro dffracto we have to cosder a smlar relato, except that the electro mass m has to be replaced by the eutro mass M. The, λ(å.8/[(ev] /. Whe a wave teracts wth the crystal, the plae wave s scattered by the atoms the crystal, each atom actg le a pot source (Huyges prcple. ecause a crystal structure cossts of a lattce ad a bass, the X-ray dffracto s a covoluto of dffracto by the lattce pots ad dffracto by the bass. Geerally, the latter term modulates the dffracto by the lattce pots. I partcular, f each lattce pot acts as a coheret pot source, each lattce plae acts as a mrror. The X-rays scattered by all atoms the crystalle lattce terfere ad the problem s to determe the ravas lattce (cludg the lattce costats ad the bass from the terferece patters. The wave that s dffracted a certa drecto s a sum of the waves scattered by all atoms. Hgher dffracto testes wll be observed alog the drectos of costructve terferece, whch are determed by the crystal structure tself. G The dffracto of X-rays by crystals s elastc, the X-rays havg the same frequecy (ad wavelegth before ad after the reflecto. The path dfferece betwee two cosecutve plaes separated by d s A d sθ. rst-order costructve terferece occurs f

32 X-ray dffracto d sθ λ, ( codto ow as ragg s law. The ragg law s a cosequece of the perodcty of the crystal structure ad holds oly f λ d. Ths s the reaso why the optcal radato s ot sutable to detect the crystalle structure, but oly X-rays ad electro or eutro beams ca perform ths tas. Hgher order dffracto processes are also possble. The ragg relato determes, through the agle θ, the drectos of maxmum testy. These drectos are detfed as hgh-testy pots o the detecto scree, the posto of whch reveal the crystal structure. or example, f the sample has a cubc crystal structure oreted such that the drecto [] (the dagoal of the cube s parallel to the cdet beam, the symmetry of the pots o the detector scree wll reveal a C symmetry axs. O the cotrary, f the dffracto patter has a C6 symmetry axs, the crystal s hexagoal, f t has a C4 symmetry axs t s a tetragoal crystal, whereas t s cubc f t shows both a ad a C symmetry axs. C4 The ragg formula says othg about the testy ad wdth of the X-ray dffracto peas, assumes a sgle atom every lattce pot, ad eglects both dffereces scatterg from dfferet atoms ad the dstrbuto of charge aroud atoms. A closer loo at the teracto betwee the X-rays ad the crystal of volume V reveals that the ampltude of the scattered radato (whch s proportoal to the ampltude of the oscllato of the electrc ad magetc felds of the total dffracted ray s determed by the local electro cocetrato ( r exp( G r, whch s a measure of the stregth of the G G teracto, ad has the same perodcty as the crystalle lattce. The dffracto testy I. or elastc X-ray scatterg, the phase of the outgog beam, wth wavevector ', dffers from that of the comg beam that propagates wth a wavevector through exp[ ( ' r], so that ( rexp[ ( ' r] dv ( rexp( Δ r dv G exp[ ( G Δ r] dv ( G where Δ ' s the scatterg vector, whch expresses the chage wavevector. The result the above tegral depeds o the volume of the crystal. If the crystal has legth L ad N prmtve cells the drecto (,, of a orthogoal coordate system (f the

33 X-ray dffracto 4 crystal system s ot orthogoal, a trasformato of coordates to the x x, y x, z x axes should be performed, the tegral alog the drecto s gve by L / L / π exp ( s a Δξ x dx a s[ π ( s π ( s Δξ N Δξ ] L sc[ π ( s Δξ N ] ( where s, Δ ξ are the compoets of G ad Δ o the axs ad costat o the same drecto. The fucto ad teds to the Drac delta fucto for large x. Therefore, large-volume crystals scatterg occurs oly f a L / N s the lattce sc ( x s x / x has a maxmum value for x, Δ G, (4 case whch V G. (I fte-volume crystals there s a sort of ucertaty the agular rage of Δ aroud G for whch the scatterg ampltude taes sgfcat values: as the volume decreases, the agular rage creases. The above codto suggests that X-ray dffracto expermets reveal the recprocal lattce of a crystal, opposto to mcroscopy, whch exposes the drect lattce (f performed wth hgh-eough resoluto. ragg plae ' The dffracto codto + G + G. I partcular, the form Δ ' G ca be rewrtte as ' + G or h ' h + hg of the dffracto codto represets the mometum coservato law of the X-ray photo the scatterg process; the crystal receves the mometum hg. or elastc scatterg ' ad thus G + G,

34 X-ray dffracto 5 or G G /, equato that defes the faces of the frst rllou zoe (the ragg plaes. The geometrc terpretato of ths relato (see the fgure above s that costructve terferece/dffracto s the strogest o the faces of the frst rllou zoe. I other words, the frst rllou zoe exhbts all the wavevectors that ca be ragg-reflected by the crystal. The dffracto codto s equvalet to ragg s law, whch ca be wrtte for a certa set of plaes separated by the dstace G G d d mp as ( / λsθ π / d mp π, or, wth G m b + b + p (for the drecto of G wth respect to the set of plaes, b see the fgure llustratg the ragg law. The Laue codto The dffracto codto Δ G ca be expressed stll aother way: f we multply both terms of ths relato wth the prmtve traslato vectors of the drect lattce, we obta the Laue codtos a Δ πs, a Δ πs, a Δ πs, (5 where s are tegers. The Laue equatos have a smple geometrcal terpretato: Δ les smultaeously o a coe about,, ad a,.e. les at the commo le of tersecto of a a three coes. Ths codto s qute dffcult to satsfy practce. Moreover, aalogy to optcal dffracto expermets, the Laue codto ca be vewed as a codto of costructve terferece betwee waves dffracted by two atoms separated by a prmtve traslato vector or, by exteso, betwee waves dffracted by all atoms the crystal. At ragg reflecto, the radato scattered by all atoms arrves phase at the detector, ad testy peas are obtaed. The wald sphere The drecto of terferece peas ca be easly determed also va a smple geometrcal costructo suggested by wald. Namely, oe costructs a sphere (a crcle two dmesos see the red crcle the fgure above aroud a pot O the recprocal lattce chose such that the cdet wavevector wth O as org, eds o a arbtrary lattce pot A. The org of the wald sphere (or crcle s ot ecessary a lattce pot.

35 X-ray dffracto 6 O A The radus of the sphere (crcle s the waveumber of the cdet (ad outgog radato '. A maxmum testy s foud aroud a drecto ' f ad oly f the wald sphere (crcle passes through aother pot of the recprocal lattce. The drecto ' s determed by the org O of the wald sphere ad ths lattce pot o the surface (crcumferece, whch s separated from the tp of (from A by a recprocal lattce vector. It s possble that for certa cdece agles ad wavelegths of the X-rays o such preferetal drecto ' exsts. Therefore, to obta peas the scattered testy t s geeral ecessary to vary ether the wavelegth or the cdece agle of the comg X-rays such that a suffcet umber of recprocal lattce pots fd themselves o the wald sphere (crcle, order to determe uambguously the crystal structure. I the frst method, called Laue method, the radus of the wald sphere (crcle s vared cotuously (see, for example, the gree crcle

36 X-ray dffracto 7 the fgure above, whle the secod method, called the rotatg crystal method or Debye- Scherrer-Hull method, the wald sphere (crcle s rotated aroud the orgal lattce pot wth respect to whch the wald sphere (crcle was costructed. The result s represeted wth the dar blue crcle the fgure above. I aother dffracto method (the Debye-Scherrer method polycrystalle samples are used, whch are ether fxed or rotate aroud a axs. I ths case, the cdet beam s scattered by oly those crystalltes (radomly oreted wth plaes that satsfy the ragg codto. ecause the sample cotas crystalltes wth all oretatos, the dffracto patter o the scree s o loger formed from dscrete pots, but from cocetrc crcles. The fluece of the bass o the scattered ampltude If the Laue/dffracto codto Δ G s satsfed, a explct accout of the bass fluece mples that the assumpto of pot/sphercal sources at the lattce pots have to be modfed. I ths case, we have foud that V N ( rexp( G r dv NS, (6 cell G G where Ω ( rexp( G r dv, N s the total umber of lattce pots, ad cell G S G cell ( r exp( G r dv (7 s called the structure factor. It s defed as a tegral over a sgle cell, wth r at oe corer. If there s oly oe lattce pot the bass ad the electro dstrbuto ( r δ ( r, S G. If there are s atoms the bass at postos r j, j,,..,s, the total electro desty ca be expressed as a superposto of electro cocetrato fuctos j at each atom j the bass, so that the structure factor s expressed as tegrals over the s atoms of a cell: S G j s s j ( r r j exp( G r j exp( G r dv j j s j ( ρexp( G ρ dv j ( ρ exp( G ρexp( G r j s j f j exp( G r j dv (8

37 X-ray dffracto 8 where ρ r ad f ( ρ exp( G ρ dv s the atomc form factor, whch depeds r j j j oly o the type of elemet that the atom belogs to. The tegral has to be tae over the electro cocetrato assocated wth a sgle atom. The atomc form factor s a measure of the scatterg power of the jth atom the ut cell. If the charge dstrbuto has a sphercal symmetry, oe ca use sphercal coordates chose such that the polar drecto s alog G. I ths case, dv πρ sϕdρdϕ, G ρ G ρ cosϕ Gρ cosϕ, where ϕ s the agle betwee ρ ad G, ad the atomc form factor becomes f j π ( ρ ρ dρ exp( Gρ cosϕsϕdϕ 4π ( ρ ρ (s Gρ / Gρ dρ. (9 j π j The atomc form factor decreases rapdly wth the dstace ad, the lmt ρ, whe s G ρ / Gρ, f j j 4π ( ρ ρ dρ Z, ( where Z s the umber of electros a atom. Also, whe G Δ (for a dffracted ray collear wth the cdet ray, the phase dfferece vashes ad aga f j ( G Z. f ca be vewed as the rato of the radato ampltude scattered by the electro dstrbuto a atom to that scattered by oe electro localzed at the same pot as the atom. The overall electro dstrbuto a sold, as obtaed from X-ray dffracto expermets, s almost the same as for free atoms,.e. atoms whch the outermost (valece electros are ot redstrbuted formg the sold. X-ray dffracto expermets are thus ot very sestve to small redstrbutos of electros. xample: cosder a bcc lattce as a sc lattce wth a bass cosstg of two atoms at [[]] ad [[/,/,/]]. The prmtve lattce vectors for the ravas ad the recprocal lattces are ths case ax, ay, a az, ad b (π / a, b (π / a, b (π / z, a a x y a respectvely. The dffracto pea of the sc lattce that s labeled by (mp correspods to G m b + b + pb (π / a( mx + y + p ad for ths dffracto pea z

38 X-ray dffracto 9 S mp f + j f f j exp( G r exp[ (π / a( mx + y + pz ( a / ( x + y + z] + f j exp[ π ( m + + p] f exp[ (π / a( mx + y + pz ] ( The bcc dffracto testy s gve by I mp mp + S f + f + Re[ f f exp[ π ( m + p]]. ( If f f f, 4 f, f m + + p eve I mp f [ + exp[ π ( m + + p]] (, f m + + p odd So, for the bcc structure wth the same type of atoms, the (mp dffracto peas of the sc lattce dsappear wheever m + + p s a odd teger. I partcular, t dsappears for a ( reflecto (see the fgure below sce the phase dfferece betwee successve plaes s π, ad the reflected ampltudes from two adjacet plaes are out-of-phase/destructve terferece occurs. π π Observato: for a sc lattce wth oe atom the bass, the dffracto testy would have bee the same, rrespectve of the party (eve or odd of m + + p. Ths example llustrates the effect of the bass o the dffracto testy.

39 Crystal bdg The stablty of sold state materals s assured by the exstg teractos (attractve ad repulsve betwee the atoms the crystal. The crystal tself s deftely more stable tha the collecto of the costtuet atoms. Ths meas that there exst attractve teratomc forces ad that the eergy of the crystal s lower tha the eergy of the free atoms. O the other had, repulsve forces must exst at small dstace order to prevet the collapse of the materal. Oe measure of the stregth of the teratomc forces s the so-called cohesve eergy of the crystal, defed as the dfferece betwee the eergy of free atoms ad the crystal eergy. Smlarly, the cohesve eergy per atom s defed as the rato betwee the cohesve eergy of the crystal ad the umber of atoms. Typcal values of the cohesve eergy per atom rage from to ev/atom, wth the excepto of ert gases, where the cohesve eergy s about. ev/atom. I partcular, the cohesve eergy determes the meltg temperature of sold state materals. Crystals wth <.5 ev have wea crystal bdgs, whle the others are characterzed by strog crystal bdgs. U U As show the fgure above, the potetal/bdg eergy U, whch descrbes the teracto betwee two atoms, approach (or fty for a teratomc dstace R (or to, ad has a mmum at a certa dstace R R. It s composed of a attractve eergy part, domat at R > R, ad a repulsve eergy part that prevals at R < R. The, the most stable state of the system, whch occurs at the lowest possble eergy, s characterzed by the cohesve eergy, the correspodg teratomc dstace, R, beg ow as the U

40 Crystal bdg equlbrum teratomc dstace. The last parameter has typcal values of Å, whch mples that the stablty of the crystal s determed by short-rage forces. The teratomc force, defed as ( R U / R, ( s egatve (attractve for R > R, ad postve (repulsve for R < R. The attractve ad repulsve forces, whch have dfferet orgs, cacel each other at the equlbrum teratomc dstace. The geeral form of the potetal eergy s A U ( r, wth > m. ( m r r The repulsve force betwee atoms the sold has the same org all crystals: Paul excluso prcple, whch forbds two electros to occupy the same orbtal (the same quatum state. The repulsve force s characterzed (see the formula above by the power-law expresso U A / r, wth > 6 or, sometmes, by the expoetal expresso U λ exp( r / ρ, where λ ad ρ are emprcal costats that ca be determed from the lattce parameters ad the compressblty of the materal. Whch expresso s better suted to descrbe the repulsve force depeds o whch oe better fts wth expermetal values. The repulsve potetal s short-raged ad thus t s effectve oly for earest eghbors. The attractve forces create bods betwee atoms/molecules the sold, whch guaratee the crystal stablty ad are of dfferet types depedg o the crystal. Oly the outer (valece electros partcpate the bodg. There are several types of bodg, depedg o the mechasm resposble for crystal coheso: oc, covalet ad metallc, whch gve rse to strog crystal bdgs, ad hydroge bodg ad va der Waals teracto, whch determe wea crystal bdgs. Crystal bdg ert/oble gases. Va der Waals-Lodo teracto The crystals of ert gases have low coheso eergy ad meltg temperature, ad hgh ozato eerges. They are the smplest crystals, wth a electro dstrbuto close to that of free atoms. rom a electrcal pot of vew they are solators, ad from a optcal pot of vew, are trasparet the vsble doma. The wea bdg betwee the costtuet atoms

41 Crystal bdg favors compact crystalle structures, partcular fcc ravas lattces wth oe atom the bass (the oly exceptos are He ad He 4, whch crystallze the hcp crystal structure. Idvdual atoms of Ne, Ar, Kr, or Xe have completely occuped exteral shells, wth a sphercally symmetrc electroc charge dstrbuto. I crystals, the presece of other atoms duces a redstrbuto of the electrc charge ad a perturbato of the sphercal charge symmetry that ca be descrbed wth the model of fluctuatg dpoles. Coulomb attracto ca occur betwee two eutral spheres, as log as ther teral charges polarze the spheres. I a classcal formalsm (vald sce electrostatc forces have a log rage, ths model assumes that the movemet of the electro atom duces a stataeous dpole momet p whch geerates a electrc feld r r r O p ( p r ( r r 5 4πε r r ( at the posto of atom separated from atom through a dstace. Ths electrc feld duces a fluctuatg dpole atom (the dstace betwee the atoms as well as the magtude ad drecto of p fluctuate tme, wth a momet r r p α(, (4 r where α s the atomc polarzablty. The eergy of the dpole-dpole teracto betwee the two fluctuatg dpoles s U attr ( r p p ( p r ( p p ( r 5 4πε r r r, (5

42 Crystal bdg 4 ad ts mmum value s attaed whe p (5 wth ts expresso (4, we get p p r, case whch, replacg the value of U attr,m ( r 4πε 4αp r 6 C r 6. (6 Ths va der Waals (or Lodo teracto s the domat attractve teracto oble gases. The hgher-order cotrbutos of the dpole-quadrupole ad quadrupole-quadrupole 8 teractos are characterzed by the respectve potetals C / r ad C / r, ad do ot cotrbute sgfcatly to the coheso eergy of the oble gases crystals. The same 6 C / r depedece of the eergy s recovered a quatum treatmet, the secod-order perturbato theory. Assumg a power-law expresso for the repulsve forces wth, the teracto potetal s gve by the Leard-Joes formula U ( r 4 6 σ σ γ, (7 r r where the parameters γ ad σ are determed from X-ray ad coheso eergy expermets. The teracto eergy of atom (atom, geeral wth all other atoms the crystal s the U 6 σ σ U ( r j 4γ (8 j j rj rj ad the eergy of the crystal composed of N atoms s U ( N / U. or a perodc cryst arragemet of atoms the lattce, wth earest-eghbors at a dstace R, r j p j R ad U cryst 6 σ σ Nγ S S6 (9 R R where

43 Crystal bdg 5 S 6 j pj 6, S j pj ( are rapd coverget seres, that ca be calculated after the crystalle structure s determed by X-ray measuremets. Ther values are, respectvely,. ad for the fcc structure, wth almost the same values for hcp structures. The crystal eergy s mmum value for the R value whch s the soluto of 5 6 U cryst / R Nγ [S ( σ / R 6S ( σ / R ],.e. for R / 6 σ ( S / S6. ( The rato R / σ. 9 for a fcc ravas lattce, the correspodg coheso eergy per atom (at zero temperature ad pressure beg S6 S6 S6 U U cryst ( R / N γ S S6 γ 8. 6γ. ( S S S Quatum correctos reduce the bdg eergy above by 8%, %, 6%, ad 4% for Ne, Ar, Kr, ad Xe, respectvely. The quatum correctos are more mportat for ert gas crystals wth smaller equlbrum teratomc dstace (smaller lattce costats. The above model determes also the compressblty modulus of oble gases wth volume V (ad volume per atom v V / N R /, defed at low temperatures as p γ V 75 V σ U cryst U γ S6 V v 4 S T cost V v σ S R R R R 5/. ( Ne Ar Kr Xe R (Å U (ev T melt (K γ (ev...4. σ (Å Β ( 9 Pa

44 Crystal bdg 6 Ioc bdg The oc bdg s foud oc crystals formed from postve ad egatve os, for example Na + ad Cl NaCl. I ths bodg type, electros are trasferred from the low electroegatve atom, whch becomes a postve o, to the hgh electroegatve atom, whch s trasformed to a egatve o (see the fgure below. The electroegatvty s the average of the frst ozato eergy ad the electro affty. It measures the ablty of a atom or molecule to attract electros the cotext of a chemcal bod. I NaCl the ozato eergy (actually the frst ozato eergy, whch s the eergy requred to move a electro from a eutral solated atom to form a o wth oe postve charge: Na + Na + + e of Na s 5.4 ev ad the electro affty (the eergy a absorbed whe a electro s added to a eutral solated atom to form a o wth oe egatve charge: Cl + e Cl + a of Cl s.56 ev. The electro affty s egatve f eergy s released the process. or most elemets the electro affty s egatve, but t taes postve values atoms wth a complete shell. The et eergy cost of the oc bodg (.e. the dfferece betwee the eergy of the os ad that of the two atoms s the a 5.4 ev.56 ev.58 ev per par of os, wthout tag to accout the Coulomb eergy betwee the os. I geeral, the electroegatvty creases wth the group umber the perodc elemet table, from the frst to the seveth group (elemets the eght group have complete shells. Depedg o the dfferece electroegatvty betwee two atoms, the bodg betwee them s Ioc (for large dfferece. xample: Na-Cl. Polar covalet bodg (for moderate dfferece. xample: H-O. Covalet bodg (for small dfferece. xamples: C-O, O-O

45 Crystal bdg 7 I oc crystals the bodg s acheved by the log-rage electrostatc force ad so, a classcal treatmet s meagful. The electroc cofgurato of the os s smlar to that of ert/oble gases,.e. the electroc charge has a sphercal symmetry, whch s oly slghtly perturbed crystal. The perturbatos are localzed the regos whch the os are closer. I partcular, NaCl the electroc cofguratos of the Na + ad Cl os are smlar to that of oble gases Ne (s s p 6 ad Ar 8 (s s p 6 s p 6, respectvely (see below. Ne Ar I oc crystals, the coheso eergy U s o loger equal to the dfferece betwee the attractve ad the repulsve potetals that act upo a o at the equlbrum posto, deoted ths case by U m (ad whch stll determes the echlbrum teratomc dstace, but has a correcto term equal to a, such that the dfferece betwee the eergy of free atoms ad of the os the crystal (whch defes the coheso eergy s U U m + a. I other words, Na + + Cl Na 4Cl 4 + U crystal + a ad U m + a s the eergy released per molecule whe the eutral costtuets form a oc crystal.. The Coulomb force betwee oe postve Na o ad oe egatve Cl o, separated by a dstace R s gve by Coulomb e (4 4πε R wth R.8 Å the earest-eghbor dstace NaCl, so that the respectve attractve potetal eergy,

46 Crystal bdg 8 e, (5 4πε R U Coul equals 5. ev per par. It follows the that the et eergy ga the oc bodg, s 5. ev.58 ev.54 ev per par of os. The electrostatc eergy ga per NaCl molecule a fcc crystal s obtaed by addg dfferet cotrbutos: that of the (opposte type 6 earest-eghbors of a certa o, U that of the secod earest-eghbors (of the same o type, U that of the 8 thrd earest eghbors of opposte type, U The result s e 6 4 πε R, e, 4πε R e 8, ad so o. 4πε R U o e 6 + πε R 8 e ( ±... 4πε R j pj 4 (6 The seres above coverge evetually to e e.748 M, where M 4πε R 4πε R U o s the Madelug costat, whch taes specfc values for each crystal structure. or other crystal structures: CsCl, zc blede, ad wurtzte, we have, respectvely, M.76,.68 ad.64. (If the seres s slowly coverget or eve dverget, the terms the sum are rearraged such that the terms correspodg to each cell cacel each other the cell remas eutral charge. The total attractve eergy a NaCl crystal wth N o pars s gve by U U N /, where the factor the umerator accouts for the fact that there are attr o two types of os: Na ad Cl, ad the factor the deomator s troduced order to cout every o par oly oce. or NaCl, U attr 86 J/mol (expermets gve 776 J/mol. The dscrepacy (of about % betwee the expermetal ad theoretcal values s explaed by the exstece of the (o-classcal repulsve forces. Smlarly, f we add up the repulsve potetal felt by a atom from all others (the expoetal form s used ow, we obta

47 Crystal bdg 9 U rep λ exp( r j j / ρ zλ exp( R / ρ (7 where we cosder that ρ << R ad z s the umber of earest eghbors. The teracto eergy of the whole crystal, whch cossts of N o pars/molecules s Me ( R N zλ exp( R / ρ, (8 4πε R U cryst ad ts mmum value per molecule, U U cryst, m N e M ρ, (9 4πε R R occurs for the equlbrum teratomc dstace R foud from the codto du cryst / dr ( zλ / ρexp( R / ρ + Me /(4πε R. The frst term (the Madelug term (9, whch expresses the electrostatc cotrbuto of the teractos, s domat sce ρ << R. Wth the same defto as above, the compressblty modulus taes ow the form 9 fr d U e M 4 dr 6πε fr R R R ρ ( where f v / R s the rato betwee the volume per partcle v ad the thrd power of the earest-eghbor dstace. f respectvely. 8 /, ad 6 / for CsCl, NaCl, ad zc blede, L LCl Na NaCl K KCl Rb RbCl R (Å U (ev ( Pa λ ( ev ρ (Å

48 Crystal bdg Covalet bodg The covalet bodg forms molecules composed of detcal partcles, for example hydroge. I ths case two atoms form a (homopolar bod by sharg a par of electros (oe from each atom, wth opposte sps. Most atoms ca form more tha oe covalet bod. or example, C has four outer electros (of sp type ad thus ca form 4 covalet bods. The covalet bod s hghly drectoal ad dfferet bods repel each other. Therefore, the correspodg crystal has geerally a low pacg rato. or example, C ad S ca have damod structure, wth atoms joed to 4 earest eghbors at tetrahedral agles; ths structure has a pacg rato of oly.4 compared to.74 for close-paced structures. The electros covalet bods are strogly localzed alog the bod, so that the crystals are semcoductors or solators, wth ot very good electrcal coductvty. To descrbe the covalet bodg hydroge, we troduce the ormalzed atomc A, orbtals s for the j (j, electro that ca belog to ether atom A or as ψ (, so that s j the ormalzed wavefucto of the total system ca be ether symmetrc (labeled wth + or atsymmetrc (labeled wth A A Ψ + / ( + S ][ ψ ( ψ ( + ψ ( ψ (] (a [ A s s s s A A Ψ / ( S ][ ψ ( ψ ( ψ ( ψ (] (b [ A s s s s where S A * A ( ψ s ( ψ s ( dr s the overlap tegral. Note that the symmetrc A A wavefucto for oc crystals ca be expressed as [ ψ ( ψ ( + ψ ( ψ (]. Ψ + Ψ + - Ψ - + Ψ + Ψ -

49 Crystal bdg The symmetrc wavefucto (also called sglet correspods to two atparallel sp, wth quatum umber S of the operator (wth S the total sp, whle the atsymmetrc wavefucto (also called trplet correspods to parallel sps,.e. S (wth the sp projecto quatum umber m s,, ad ; there are three atsymmetrc wavefuctos!. The form of the wavefuctos above s determed from the codto that the total wave fucto for fermos (cludg sp must be atsymmetrc upo partcle exchage. The eergy egevalues are represeted above as a fucto of the dstace betwee the atoms. A boud state ca exst the sglet state, wth +.4 ev f the covalet bodg forms betwee H atoms,.e. the strogest bdg occurs f the sps of the two electros are atparallel. S - atom A + atom To characterze the crystalle structure of damod oe must geeralze the prevous formula order to corporate the p atomc orbtals. Ideed, the last occuped orbtals of these materals are: C(s p, S(s p, ad Ge(4s 4p. Whe both s ad p-type orbtals are volved, they hybrdze (see fgures below. [The s atomc orbtals have quatum umbers,, (prcpal quatum umber, l (orbtal quatum umber, ad m (magetc quatum umber; the projecto of l. The p orbtals have,,, l, ad m,,.] The s ad p atomc orbtals hybrdze whe the eergy dfferece betwee them s much smaller tha the bdg eergy.

50 Crystal bdg I partcular, whe oe s orbtal wth wavefucto orbtal, wth wavefucto above wth wavefuctos Ψ px Ψ s ad oe p orbtal, say the, hybrdze, the result s two lear sp orbtals (see fgure p x Ψ / ( Ψs + Ψ p x / s Ψ p x, Ψ ( Ψ. ( O the cotrary, sp hybrd orbtals form betwee oe s orbtal ad two p orbtals, the resultg plaar structure (see fgure above havg orbtals arraged plae wth a agle of betwee them. The electros the hybrd orbtals are strogly localzed ad form σ bods; they do ot partcpate electrcal coducto. Oe p orbtal remas perpedcular to the plae, where t forms a π bod wth other out-of-plae p orbtals from eghborg atoms; ths s the case of graphte or graphee (bdmesoal crystal. The electros the π orbtals are delocalzed ad partcpate electrcal coducto. The three hybrd orbtals are gve by Ψ Ψ / ( Ψs + Ψ p x / [ Ψs (/ Ψ p x + / Ψpy, ( / s p x py ], Ψ [ Ψ (/ Ψ / Ψ ]. Smlarly, the electroc cofguratos that forms from oe s orbtal ad three p orbtals s called sp. Ths electroc cofgurato s characterstc for damod. The agular part of the s ad p orbtals are ( polar coordates / Ψs (4π, (4 / / / p (/ 4π sθ cosϕ, p (/ 4π sθ sϕ, (/ 4π cosθ Ψ x Ψ y Ψp z

51 Crystal bdg so that the four hybrd atomc orbtals that are lear combatos of atomc orbtals form a tetrahedro (see fgure below ad are gve by Ψ (/ ( Ψ + Ψ + Ψ + Ψ, (5a s px py pz Ψ (/ ( Ψ + Ψ Ψ Ψ, (5b s px py pz Ψ (/ ( Ψ Ψ + Ψ Ψ, (5c s px py pz Ψ (/ ( Ψ Ψ Ψ + Ψ. (5d 4 s px py pz C, S ad Ge form crystals whch the covalet bdg s domat, the va der Waals cotrbuto to the coheso eergy, also ecoutered crystals from a sgle elemet, beg eglgble. However, crystals wth a bass composed of two atoms A ad, wth ad, respectvely, 8 valece electros, the covalet bdg s accompaed by a oc cotrbuto. The resultg bod s called polar covalet bod. The oc cotrbuto ( fracto s.8 SC,.6 GaSb,. GaAs, ad.44 IP. Smlarly, oc crystals the covalet bdg ca also cotrbute to the coheso eergy, the fracto of the oc cotrbuto beg oly.86 AgCl,.94 NaCl, ad.96 Rb. Whe covalet bodg forms betwee dfferet atoms, the hybrd orbtals cosdered above are modfed, as ca be see from the fgures below. lectroc cofgurato the CH 4 molecule.

52 Crystal bdg 4 odg betwee the s orbtals of the H atom ad the p x ad p y orbtals of O atom H O (a wthout, ad (b wth hybrdzato Covalet crystals are characterzed by: hgh meltg temperatures (the coheso eergy per atom s about ev hardess (but also frable ther coductvty depeds strogly o temperature ad mpurty atoms hgh value of the delectrc costat geerally trasparet IR, but strogly absorbet vsble ad ear-ir. Note: Crystal bods form betwee valece electros,.e. the electros o the outer shells, whch partcpate chemcal reactos/determe the physcal propertes of the materal. I cotrast, core electros are those o er shells. Hydroge bdg of crystals ecause eutral hydroge has oly oe electro, t should form a covalet bod wth oly oe other atom. However, just as oppostely charged os are attracted to oe aother ad ca form oc bods, the partal charges that exst at dfferet atoms polar covalet bods ca teract wth other partally charged atoms/molecules. Partcularly strog polar covalet bods are foud, for example, whe a hydroge atom bods to extremely electroegatve os such as O water/ce (see the fgure below, left, (see the fgure below, rght, N or Cl. The partal charges the fgure below are deoted by δ. The hydroge bod forms betwee the hydroge atom wth a strog partal postve charge ad electroegatve os wth strog

53 Crystal bdg 5 partal egatve charges eghborg molecules. The bdg eergy s of the order of. ev. or example, the coheso eergy per molecule ce s. ev. hydroge bod hydroge bod The hydroge bod s weaer tha, although smlar to, oc bod sce t forms betwee partal charges rather tha full (complete charges. I hydroge bods the hydroge atom s the hydroge bod door ad the electroegatve o s the hydroge bod acceptor. As the polar covalet bdg, the hydroge bod ca be vewed as a mxture of oc ad covalet bodg, the oc bodg beg domat. or example, the typcal hydroge bod that ls two H O molecules ce, the bdg ca be cosdered as a superposto of three bdg types: O (covalet H (oc O O (oc H (oc O O (oc H (covalet O I hydroge, the proto radus s wth fve orders of magtude smaller tha the radus of ay other o, ad so t allows the exstece of oly two earest eghbors of the proto (more tha two atoms would get each other s way,.e. the hydroge bod s drectoal. Despte t s wea, the hydroge bod s extremely mportat lvg orgasms, whch are maly composed of water, sce water as well as protes ad uclec acds posses a great capacty to form hydroge bods. I partcular, the hydroge bdg occurs as tramolecular bdg betwee the orgac complemetary bases thyme ad adee, ad cytose ad guae DNA. It ca also be ecoutered betwee costtuets of crystals such as KH PO 4, KD PO 4 (KDP, Ca(OH, or Mg(OH.

54 Crystal bdg 6 Metallc bodg The metallc bodg ca be uderstood as the bodg betwee postvely charged metallc ucle/os ad delocalzed coducto electros, see as a sea of free electros. It prevals elemets whch the valece electros are ot tghtly boud wth the ucleus ( metals, for example. However, the metallc bod we caot spea about os, sce there s o partcular electro that s lost to aother o. Ule other bodg types, the metallc bodg s collectve ature, so that o sgle metallc bod exsts. It s ether tra- or termolecular sce o molecule ca be dstgushed metals. Metallc bodg ca be uderstood as a omolecular, extremely delocalzed commual form of covalet bodg. The delocalzato s most proouced for s ad p electros, wth l ad l, respectvely, beg much weaer for d ad f electros, whch have quatum umbers l ad l, respectvely. I metals, a atom acheves a more stable cofgurato by sharg all ts valece electros wth all other atoms the crystal. However, besdes delocalzato, metallc bodg also requres the avalablty of a far larger umber of delocalzed eergy states tha of delocalzed electros. These states are referred to as electro defcecy; they assure the etc eergy for delocalzato. The metallc bodg s ecoutered, for example, alale metals such as L, K, Na, wth electroc cofguratos that resemble those of oble gases wth a addtoal s electro o the outer shell. Havg few electros o ther outer shells, alal metals have oly partly flled eergy levels, ad therefore are electro defcet. I formg the crystal, the wavefuctos of the outer s electros overlap wth those of ther earest eghbors, ad the electros become delocalzed. Ther dyamcs resembles that of free electros, so that alale metals the lattce s occuped by the postvely charged os wth the oble gas

55 Crystal bdg 7 structure (they occupy fact oly as much as % from the volume of the crystal, whle the valece electros occupy the remag volume. Ule covalet crystals, where the electroc charge s dstrbuted a strogly ouform maer (the bods are spatally oreted, the electroc desty metallc crystals s hghly uform. Ths explas the hgh elastcty ad malleablty of these materals. The total Coulomb potetal, whch cludes electro-electro, electro-o ad oo teractos, s U Coul U U < (the frst ad thrd terms o the rhs are e e + U e + postve, the mddle oe s egatve. Therefore, the attractve potetal s of electrostatc ature, beg balaced by the repulsve teracto due to the Paul excluso prcple. It should be metoed that t s ot ecessary for metals to have metallc bodg. or example, may trasto metals show covalet propertes (ot all electros partcpate covalet bods, ad are good electrcal coductors. Note: I trasto metals (e, Co, N, Cu, Z, Ag, Au, M, etc. the d orbtals are oly partally occuped ad the outermost s orbtals are fully occuped. xample: 4s full, d complete. Crystals wth metallc bodg are usually characterzed by hgh electrcal ad thermcal coductvty, wth wea temperature depedece hgh elastcty hgh optcal reflectvty a large frequecy badwdth broad rage of meltg temperatures: low meltg temperatures for alale metals (L, Na, K, Rb, Cs, termedate for oble metals (Cu, Ag, Au, ad hgh values for metals such as T, Zr, Mo, W. The correspodg coheso eerges vary betwee ev ad 5 ev.

56 Lattce oscllatos. Phoos Let us cosder a crystalle materal cosstg of a large umber N o of heavy postvelycharged os (composed of the ucleus ad the valece electros o the er atomc orbtals wth masses M ad stuated at postos R, α,.., N, surrouded by ad teracto α α wth N electros o the outer atomc orbtals wth masses m ad at postos deoted by r, el o,.., N el. The total Hamltoa of the system s the H T el + T o + V el + V o + V el o h h + α + U el ( r r j + U o ( Rα Rβ + V ( r Rα ( m α M, α, α α α β, j j β The terms of the rght-had-sde deote, order, the etc eergy of the electros, the etc eergy of the os, the (Coulomb teracto eergy of electro pars, the teracto eergy of o pars, ad the teracto eergy betwee electros ad os. Sce m << M α, the electro veloctes are much hgher tha the o veloctes, so that the electros see a froze dstrbuto of os, whle the os ca oly sese the average (ot stataeous! spatal dstrbuto of electros. I other words, for a gve o cofgurato the electros are a quas-equlbrum state that s slowly varyg tme due to o s moto, whereas the os evolve slowly a potetal dstrbuto geerated by the average cofgurato of the electros. Ths adabatc approxmato, ow also as the or- Oppehemer approxmato, allows a factorzato of the total wavefucto of the system, Ψ ( r, R wth r r, r,..., }, R R, R,..., } to a electroc part, ψ ( r; R, { r N el { R N o whch the o s postos are cosdered as parameters, ad to a oc part, φ (R : Ψ ( r, R ψ ( r; R φ( R. ( These electroc ad oc parts satsfy the followg equatos: h + U el ( r rj + m,, α j j V ( r R α ψ ( r; R el ( R ψ ( r; R (

57 Lattce oscllatos. Phoos α h + α M α, α α β β U o ( Rα Rβ + el ( R φ ( R φ( R, (4 where s the eergy of the whole system ad el s the eergy of the (subsystem of electros. Let us assume further that a crystalle lattce wth s atoms the bass, the os move aroud ther equlbrum postos, so that R R u << R. The, the R α α α α α teracto eergy betwee pars of os ca be expaded a Taylor seres aroud the equlbrum postos. Tag to accout that U o ( Rα Rβ U o ( Rα Rβ U ( o Rα Rβ uα + uβ (5 Rα Rβ sce the force that acts upo a o at equlbrum (whch s proportoal to ths dervatve vashes, we fd that μν U o ( α Rβ U o ( R R + Aαβ α β α β α β,, α, β, μ, ν α β α β R u u, (6 μ α ν β where A μν αβ U o R ( R μ α a R R ν β β Rα, β Rα, β (7 ad the dces μ x,y,z (ad ν deote the projectos of the posto vectors o a Cartesa coordate system, the frst spatal dervatve of U o vashes due to the requremet that the force (whch s proportoal to ths dervatve that acts upo a o at equlbrum vashes, ad hgher-order terms the Taylor expaso are eglected. The last approxmato s called harmoc. ecause the frst term the Taylor expaso of U o s costat, t ca (together wth el be cluded the referece eergy of the system, so that the harmoc approxmato the lattce dyamcs s descrbed by the Hamltoa

58 Lattce oscllatos. Phoos H o μ ( pα + α, μ M α A μν αβ α, β, μ, ν u u μ ν α β. (8 The dyamcs of the lattce oscllatos ca the be expressed the caocal form as μ H μν ν p& α Aαβ uβ, u β, ν o μ α μ μ H pα u& α, (9 p M o μ α α ad the equato of moto for the dsplacemet of o α the μ drecto, M α μ μν ν u& α + Aαβ uβ, ( β, ν descrbes fact a set of coupled harmoc oscllators. The couplg stregth wth eghborg os s characterzed by the coeffcets μν A αβ. A harmoc potetal eergy correspods to forces that are lear the dsplacemets. I the harmoc approxmato oe ca vew the lattce vbrato as a teracto of coected elastc sprgs (classcal harmoc oscllators, as the fgure below. The lattce oscllatos are thus smlar to elastc waves that propagate through such a cha of coected sprgs. If a atom s dsplaced from ts equlbrum ste by a small amout, t wll ted to retur to ts equlbrum posto due to the force actg o t. Ths results lattce vbratos. Due to teractos betwee atoms, varous atoms move smultaeously, so we have to cosder the moto of the etre lattce.

59 Lattce oscllatos. Phoos 4 Note that coupled harmoc oscllators, the force that acts o a o α from other os β, gve by u α u β μ α M μ αu&& α β, ν μν αβ. Ths meas that the expresso of the force should be A u ν β, s proportoal to the relatve dsplacemet, μ α μν ν μν ν ν Aαβ u β Aαβ ( u β uα. ( β, ν β, ν Ths s possble oly f the followg equato s satsfed: β μν μν μν A αβ Aαα + Aαβ. ( β α The physcal relevat solutos for the system of harmoc oscllators are of plaewave type,.e. are oscllatory tme, wth the same frequecy for all os. These are the ormal oscllatos. ecause of the perodcty of the crystalle lattce, the ampltude of os dsplacemets dfferet ut cells are the same, so that oly the phase of the oscllatos vary from oe ut cell to the other. So, for a crystal wth s atoms the bass ad N elemetary cells, α { χ, }, χ,,s,,,n, we loo for solutos of the form ( N o sn u μ χ μ, λ χλ χ λ t ( t u ( e ( exp[ ( R ω ( ] ( where u, λ ( s the ampltude of the ormal oscllatos of type λ (several logtudal ad trasverse oscllatos that propagate alog the drecto /, ad ( s the polarzato vector of χ-th atom the ut cell (ot ormalzed to uty!. Itroducg ths soluto (, we fd that the polarzato vectors of the atoms satsfy the followg system of s equatos (ν,,, ad γ,,s e χλ λ M χω ( e μ χλ ν e γλ, (4 μν ν μν A χ, m γ exp[ ( Rm γ R χ ] e γλ χγ ( m, γ, ν γ, ν

60 Lattce oscllatos. Phoos 5 where μν χγ μν ( A exp[ ( R R ]. I geeral, we have a system of ds m χ, m γ mγ χ equatos, where d s the umber of dmesos. It ca easly be demostrated that the matrx of the coeffcets whch mples the orthogoalty of the polarzato vectors μν χγ ( s hermtc, μ * μ [ e χλ ( ] eχλ' ( sδ λλ ' χ, μ, (5 from whch t follows that the system of s equatos has o-trval solutos oly whe μν Det ( M ω ( δ δ. (6 χγ χ λ χγ μν Ths codto represets a characterstc equato for ω λ, whch has s solutos/braches for a gve, called the ormal oscllato frequeces of the lattce, wth correspodg polarzato vectors e μ χλ (. The depedece of the oscllato frequeces ω λ o s called the dsperso relato of the ormal oscllato of the λ-th brach. ad thus μν rom the defto of (, f are real, t follows that χγ μν μν μν * A χ, mγ χγ ( [ χγ ( ] μ * ( [ μ * e χλ eχλ ( ], ω λ ( [ ωλ ( ] (7 or ω ( ω ( sce the oscllato frequeces are real ad postve. λ λ If, three dmesos, we cosder a lattce wth oe atom the bass,.e. wth s, the we have three oscllatos braches (there are three degrees of freedom for each atom, λ,,, ad the lmt of log wavelegths, μ μν ν M ω λ ( eλ ( ( eλ ( (8 ν μν μν μν μν ( A, m A, + A, m m m, (9

61 Lattce oscllatos. Phoos 6 the last equalty followg from (. ecause the left-had-sde of (9 vashes ad the polarzato vectors are fte, t follows that ω ( ad the dsperso law for the three braches the log-wavelegth lmt ca be wrtte as λ ω ( v ac, (. ( λ λ The parameters v ac,λ are called acoustc veloctes sce a smlar relato as that above holds for acoustc waves propagatg a cotuum, elastc ad sotropc medum. Moreover, the oscllatos that tae place the drecto of the propagato vector are called logtudal ad those ormal to are trasverse: we have oe logtudal ad two trasverse acoustcal oscllatos. The fgure below llustrates the dsperso relato of a crystal, whch we ca detfy oly acoustcal braches, whch meas that the there s oly oe atom the bass of ths crystal. or complex lattces, wth s >, there are aga three oscllato braches wth a acoustc-le dsperso relato the lmt, as above, correspodg to the stuato whch all atoms the lattce have the same dsplacemets (oscllate phase, ad thus the complex structure of the lattce s ot mafest. These are the acoustc oscllato braches. However, ths case we have also s oscllato braches wth o aalog the dyamc of cotuum meda ad for whch the dsperso relato the log-wavelegth lmt has the form

62 Lattce oscllatos. Phoos 7 ω λ ( ωλ β λ. ( These are the optcal oscllato braches, characterzed by the cut-off oscllato frequeces ω λ ; the parameters β λ are geerally postve. Ths type of oscllato braches s called optcal because, whe the ut cell cossts of os wth dfferet type of charges (postve ad egatve, these oscllatos form a stataeous dpolar momet that teracts strogly wth the electromagetc radato. I ths case os wth dfferet sg oscllate at-phase,.e. ther dsplacemets are opposte drectos. As for the acoustc oscllatos, three dmesos we have oe logtudal ad two trasverse optcal oscllato braches for each s value. Oscllatos of a fte atomc cha wth oe atom the bass or exemplfcato, let us cosder frst a smple oe-dmesoal fte lattce (a atomc cha cosstg of detcal atoms (more precsely, os wth mass M, separated by the lattce costat a, as the fgure below. We expect acoustcal oscllato brach. ds (for d ad s,.e. a sgle, a + or (thermal vbratos of the crystalle lattce, whch the os move slghtly aroud ther equlbrum postos R a, ther actual postos satsfy the relatos R R u << R, where the dsplacemets ca occur ether alog the cha or trasverse to the cha of atoms. It should be oted that oe-dmesoal lattce vbratos are ot ecoutered oly atomc chas. or example, a smple cubc crystal wth oe atom the prmtve cell, whe a wave propagates alog the drectos of the cube edge, face dagoal, or body dagoal, etre plaes of atoms move phase wth dsplacemets ether parallel or perpedcular to the drecto of the wavevector (see the fgures below. R

63 Lattce oscllatos. Phoos 8 We ca descrbe the dsplacemets of the plae from ts equlbrum posto wth a sgle coordate, u. The problem becomes ths way oe-dmesoal. or each wavevector there s oe soluto wth logtudal polarzato ad two wth trasverse polarzato. The parameter A s dfferet for logtudal ad trasverse waves. We have already see that, the harmoc approxmato, the dyamcs of the system s equvalet to that of coupled harmoc oscllators, the harmoc potetal actg o a o descrbg a force that s lear the dsplacemet. or smplcty, we assume further that oly the teracto betwee earest eghbors s sgfcat, case whch the force exerted o th -th atom the lattce s lear the o s (relatve dsplacemets ad hece gve by u A( u + u + A( u u ( where A s the teratomc force (or, equvaletly, the elastc costat betwee earesteghbor os. Applyg Newto s secod law to the moto of the -th atom wth mass M, / dt M ( d u, we obta d u M A( u ( ( + u + A u u A u u+ u. ( dt A smlar equato should be wrtte for each atom the lattce. The solutos of the equato above have the form

64 Lattce oscllatos. Phoos 9 u ( t t u exp[ ( R ωt] u exp[ ( a ω ]. (4 Such solutos represet travelg waves, whch all the atoms oscllate wth the same frequecy ω ad the same ampltude u ad have wavevector. Solutos of ths form are oly possble because of the traslatoal symmetry of the lattce. Note that (4 there s o eed for the dces χ, λ or μ because here s oly oe atom the bass (χ, we have oe oscllato (λ ad oe dmeso (μ. I addto, all atoms oscllate wth the same ampltude, ad o polarzato vectors eed be troduced (t ca be cluded. Isertg (4 to the equato of moto ( we obta u Mω A[ exp( a exp( a] (5 or exp( a + exp( a M ω A A[ cos( a] 4As ( a /. (6 ad the dsperso relato, represeted the fgure below, s ω ( A / M s( a /. (7 ω (A/M / st rllou zoe Please observe that equato ( follows also drectly, by partcularzg ( for oedmesoal moto ad assumg that oly the earest-eghbor atoms teract wth each

65 Lattce oscllatos. Phoos aother, so that A, m for m, ±. I ths case, A, A, + A, A, A, such that the sum rule ( s satsfed. Smlarly, equato (5 could have bee drectly wrtte from Mω A exp( a + A A exp( a]. [,, +, + ecause the dsperso relato s perodc: ω ( ω( + π / a, wth the perodcty gve by the recprocal lattce vector, all dstct frequecy values ca be foud the terval π / a < π / a, (8 whch correspods to the frst rllou zoe. The maxmum (cut-off frequecy ω max A / M s obtaed for the mmum wavelegth of λ π /( π / a a. The exstece m of a mmum wavelegth ca be uderstood as resultg from the codto that waves wth wavelegths smaller tha a caot propagate the lattce, beg reflected at the boudares of the frst rllou zoe. The sgfcace of the perodcty of the dsperso relato s evdet from the fgure below: chagg by oe recprocal lattce vector gves exactly the same movemet of atoms. I the log-wavelegth lmt a / <<, we have s( a / a /, ad ω ( A / M a v (9 ac so that the oscllatos are acoustc ad characterzed by the acoustc velocty v ac a A / M.

66 Lattce oscllatos. Phoos Moreover, sce the oscllato frequecy does ot deped learly wth, we ca defe separately the phase velocty,.e. the velocty of the phase of the wavefrot, ad the group velocty,.e. the propagato velocty of the wavepacet ad of the wave eergy. Ther modulus are gve, respectvely, by v v A s( a / s( a / ω a vac, ( M a / a / ph d A ω a cos( a / vac cos( a /. ( d M gr v ac v v ph v ac /π v gr I the log-wavelegth rage,, v v v, whle at the edges of the frst rlou zoe, for ph gr ac π/a ±π / a, v v / π ad. ph ac v gr te lattces or fte oe-dmesoal lattces cosstg of N detcal atoms, the requremet of symmetry (of equvalece of physcal propertes whe the equato of moto refers to dfferet atoms mposes the cyclc boudary codto u u + N. Ths so-called or- Karma codto expresses the depedece of the propertes o the surface,.e. we have a fte sold, wth o surfaces; a fte cha wth o ed. rom the cyclc boudary codto t follows that exp( Na, or π m, ( Na wth m a teger. There are N allowed m values for the frst rllou zoe: N / m < N /, (4 whch correspod to the N degrees of freedom of the system. ecause N s usually a large umber, the dscrete ature of the waveumber s dsregarded ad t s cosdered as a cotuous varable. elow: example for N.

67 Lattce oscllatos. Phoos ω Desty of oscllatos a smple fte oe-dmesoal lattce How may oscllatos (wth dfferet values exst the the frequecy terval ( ω, ω + dω? Ths umber s referred to as the desty of oscllatos per ut frequecy ad s deoted by dn osc / dω D( ω. ecause for the fte oe-dmesoal lattce the waveumber vares oly dscrete steps of possble ths waveumber rage, so that Δ π / Na, there s oly oe oscllato dn osc Na (5 d Δ π D ad, tag to accout the double degeeracy due to the symmetry of ω ( (two values correspod to the same N/πω max ω, we obta ω max ω dn osc dn osc d N D( ω dω d dω π N π A/ M s ( a / A/ M N π cos( a / A/ M ω / ω max N π ω max ω (6

68 Lattce oscllatos. Phoos Oscllatos of a fte atomc cha wth two atoms the bass Let us assume that we have a fte oe-dmesoal lattce wth lattce costat a, cosstg of equally spaced os wth dfferet masses ad M > (see the fgure M M below. The bass has therefore two atoms, placed at equlbrum postos R, a, ad, + R ( / a. I ths case d, s, so there should be two oscllato braches, oe acoustcal ad oe optcal a uu -, - v - u -, u u, uv, u + u +, vu + +, u +, + Smlarly to the atomc cha wth oe atom the bass, whe there are two atoms per ut cell we have two equatos of moto of the geeral form M ( d u / dt, oe for every type of atom. To dstgush betwee the dsplacemets of the two atoms, we deote wth u, ( t t u e exp[ ( a ω ] the dsplacemet of atoms wth mass (the yellow oes M the fgure above ad wth u, ( t t u e exp[ ( a( + / ω ] that of atoms wth mass M (the gree oes. (Note that, for smplcty, we omt for ow the dex λ, whch has two values: ac ad opt, correspodg to the acoustcal ad optcal oscllatos. So, we have M M or d u, dt d u A( u, u, + A( u, u, A(u, u, u,, (7a, + + dt A( u, u, + A( u, u, A(u, u, u,, (7b M ω e Ae + Ae[exp( a / + exp( a / ] Ae + Ae cos( a /, (8a M ω e Ae + Ae[exp( a / + exp( a / ] Ae + Ae cos( a /. (8b

69 Lattce oscllatos. Phoos 4 Ths system of coupled equatos has soluto oly whe ts determat vashes,.e. whe A M ω Acos( a / A exp cos( a / A M ω, (9 or 4 4 ωγ a ω ω ω + s (4 4 wth M A + M M M γ. ω, 4 M M ( M + M rom (4 t follows that there are two solutos (two types of lattce oscllatos for every value of ; these are the optcal ad acoustc braches. The two solutos are ω a ω ( γ s, ω a ω ( + γ s. (4 As for the oe-dmesoal lattce wth oe atom the bass, ω + π / a ω (, so, (, that all relevat values are foud the frst rllou zoe. Aga, the dsperso relato could have bee obtaed drectly from (4. More precsely, whe oly the earest eghbors teract we would have obtaed Mω e ( A,;, exp( a / e ( + A,;,e ( + A,; +, exp( a / e (, M ω e ( A,;, exp( a / e ( + A,;,e ( + A,; +, exp( a / e (, so that the dsperso relato s recovered f A, ;, A,; +, A, ;, A,;, A ad + A, ;, A,;, A (the sum rule ( s aga satsfed!. rom the solutos (4 oe ca detfy the oscllato braches: the acoustc oe correspods to the frst soluto, for whch π ω ω ac ( ω (, ω ac ± γ > ωac (, (4 a

70 Lattce oscllatos. Phoos 5 ad the optcal brach s cosstet wth the secod soluto, for whch ω π ω ±. (4 a opt ( ω ( ω, ω opt + γ < ωopt ( t -π/a π/a t ecause for the acoustc brach ω ( for, from the system of coupled ac equatos (8 t follows that e / e, whch mples that the dsplacemet of the two ( ac types of os s the same/occurs the same drecto ad the ut cell moves as a whole; t oscllates phase (see the fgure above, bottom, rght. O the other had, the logwavelegth lmt of the optcal brach, ( e / e opt M / M,.e. the os are dsplaced opposte drecto ad we have out-of-phase oscllatos. The oscllatos occur such that the ceter of mass of each o par s fxed,.e. M e + M e (see the fgure above, top, rght. I the log-wavelegth lmt, whe s( a / a /, the dsperso relato of the acoustc brach ca be approxmated as ω ac ω γ a ω γa, (44 4.e.

71 Lattce oscllatos. Phoos 6 ω ac ( vac, v A ω a a ac γ 4 ( M + M, (45 whle the dsperso relato for the optcal brach becomes ω γ a γ a ω opt + ω (46 8 or ω opt ω β, β ωγ a /. (47 ( Note that, to calculate the desty of states for the fte oe-dmesoal lattce wth two atoms the bass, we ca follow the same treatmet as for the atomc cha wth oe atom the bass, tag to accout that we must calculate separately the desty of states for the two oscllato types, whch have dfferet dsperso relatos. Desty of states a fte three-dmesoal crystal The desty of states/oscllatos a three-dmesoal crystal s obtaed, as the oedmesoal crystal, by mposg the approprate boudary codtos for. ecause there s a large umber of atoms a crystal, whch teract strogly wth ther eghbors, the cotrbuto of the atoms at the surface of the crystal to ay physcal pheomea s eglgble. Therefore, we ca employ aga the or-karma cyclc codto u χ μ μ u χ u( + Nμ χ (48 μ χ μ λ χ t wth u u e exp( R ω, where N s the umber of atoms the x drecto, μ,,, wth x x, x, y x z μ. As the oe-dmesoal case, the wavevector compoet alog the μ drecto the frst rllou zoe has a dscrete spectrum, μ μ π m μ, (49 N a μ μ

72 Lattce oscllatos. Phoos 7 wth N μ N μ mμ < (5 where a s the lattce costat alog x, ad m are tegers,.e. t vares steps of μ μ Δ μ π / L μ, wth L μ N μ aμ the legth of the crystal alog the xμ drecto. The dscrete values a two-dmesoal lattce are represeted by pots the fgure below. μ y x allowed values It follows the that a state/oscllato occupes a volume the space gve by ( π Δ ΔΔ Δ, (5 V where V L LL NΩ s the volume of the crystal wth sn snn N atoms that form a lattce wth a prmtve cell of volume Ω aaa ad s atoms the bass. The desty of states/oscllatos the space s the defed as dn osc d ( V π. (5 Δ The desty of oscllatos the frequecy space, defed as dn osc dn osc d D( ω, (5 dω d dω

73 Lattce oscllatos. Phoos 8 represets the umber of oscllatos (wth dfferet values that exst the frequecy terval ( ω, ω + dω. I ths formula d s the volume space betwee the surfaces ω ( ad ω( + dω( (see the fgure below. The desty of oscllatos s dscrete (as for oedmesoal crystals but, for suffcetly large crystals, the sum over the dscrete states ca be replaced by a tegral. We ca calculate t observg that d d ds ω d ds d ω ω ω( cost ω( cost ω ds dω, (54 from whch t follows that V ds D( ω (π. (55 ω ω( cost ω We ca express also the desty of states as D ( ω dω [ V /(π ] d, or V D( ω dω d, (56 (π st Z whch represets a partcular case of approxmatg a sum over the frst rllou zoe by a tegral, approxmato that for a arbtrary fucto f ( s V V f ( f ( Δ f ( d. (57 (π (π st Z or f ( ( ω(, we have

74 Lattce oscllatos. Phoos 9 ( ω( ( ω D( ω dω (58 f the desty of states s ormalzed each brach χ, χ,,s such that ωmax D( ω dω N (59 wth ω max the maxmum value of the oscllato frequecy the brach. Note that for the three-dmesoal crystals, we have ot defed the desty of states ormalzed at ut volume (as for the atomc cha wth oe atom the bass, but have ept the crystal volume throughout the calculatos! Quatzed oscllatos/phoos a oe-dmesoal fte lattce wth oe atom the bass We have see that the oe-dmesoal lattce wth oe atom the bass, the os act as coupled harmoc oscllators. Here we show that ths system of coupled oscllators ca be reduced to a equvalet system of depedet harmoc oscllators by the troducto of ormal coordates. The, we assocate a ormal oscllato to each ormal coordate. The Hamltoa of the fte oe-dmesoal lattce wth oe atom the bass ca be wrtte as H o N p N + A' uu'. (6 M, ' I the quatum treatmet of the system of coupled harmoc oscllators, the posto ad mometum coordates become (cojugate operators, such that ˆ ˆ ˆ ˆ ˆ ˆ, [ u ˆ, ˆ '], [ p ˆ, ˆ ' ]. (6 [ u, p' ] u p' p' u hδ ' u We ca troduce ormal posto ad mometum operators, p ˆ / / Q N uˆ exp( a, P N pˆ exp( a, (6 ˆ terms of whch the orgal operators ca be expressed as

75 Lattce oscllatos. Phoos ˆ u N / Q exp( a ˆ, p ˆ N / Pˆ exp( a. (6 The ormal operators satsfy the commutato relatos ˆ ˆ [ ˆ ˆ [ Q, P ' ] N u, p' ]exp( ' ' a a N h exp[ ( ' a] hδ ', ' [ Q ˆ, ˆ Q ' ], [ Pˆ, Pˆ ' ],, (64 sce N exp[ ( ' a] N exp( πm / N[exp( πm ] exp[ π m / N] Nδ Nδ exp( πm / N q ' (65 for, ' the frst rllou zoe, for whch ' πm / Na wth m a teger. rom the commutato relatos t follows that the ormal posto ad mometum operators for the same waveumber are also cojugate. ecause there are N ormal oscllatos ad û, pˆ are hermtc operators, Q ˆ + Q ˆ, P ˆ + ˆ P, ad, for the frst rllou zoe (where t taes N dscrete values we have ormal operators: N ormal posto operators Qˆ, ad N ormal mometum operators Pˆ. I the ormal posto ad mometum operators, the etc eergy term the Hamltoa becomes N N pˆ M MN Pˆ Pˆ, ' N exp[ ( + ' a] ' M Pˆ Pˆ (66 wth the frst rllou zoe. I a smlar maer, A, ' ' uˆ uˆ ' Q ˆ Q ˆ N, ' Q ˆ Q ˆ N, ' ' A exp[ ( ' + a] A ', ' ' exp( aexp( ' ' a ' ' exp[ '( ' a] M ω ( Qˆ Q ˆ (67 f we tae to accout that The Hamltoa of the system of os, A' exp[ '( ' a] Mω ( ' ad ω ( ω(. '

76 Lattce oscllatos. Phoos + o Q Q M M P P H ˆ ˆ ( ˆ ˆ ˆ ω, (68 descrbes a system of harmoc oscllators that are ot, however, totally depedet sce the terms correspodg to ad are stll coupled. To avod ths stuato, we troduce ahlato ad creato operators for each, ad, ad express the ormal posto ad mometum operators as â + â ˆ ˆ ( ( ˆ + + a a M Q ω h, ˆ ˆ ( ( ˆ + a a M P h h ω. (69 The creato ad ahlato operators satsfy the commutato relatos ' ' ' ' ˆ ˆ ˆ ˆ ] ˆ, ˆ [ a a a a a a δ + + +, ] ˆ, ˆ [ ' a a,. (7 ] ˆ, ˆ [ ' + + a a or the frst rllou zoe, the Hamltoa operator of the lattce s ow a sum of N depedet harmoc oscllators: ˆ ( ˆ ˆ ( ˆ ˆ ˆ ˆ ( ( ˆ ˆ o N a a a a a a H H ω ω ω h h h, (7 wth, the hermtc umber operator, for whch ad, ( the thrd term of the equato above, we substtute frst for. ] ˆ, ˆ [ ' H H a a N ˆ ˆ ˆ + ] ˆ, ˆ [ ' N N ] ˆ, ˆ [ H N or each mode, the egestates (oc states ad egevalues of the Hamltoa ad the umber operators are + a! ˆ (, (7a + ( ω h,,,, (7b where s the fudametal state of the oscllator, ad

77 Lattce oscllatos. Phoos + a ˆ, a ˆ + +, N ˆ (7 wth for ˆ. The egestate of the lattce Hamltoa s the (there are N dscrete values a N N,,...,... (74 ad the eergy of the collecto of harmoc oscllators s h ω( + + ph. (75 where h ω( / s the zero (fudametal eergy, ad ph h ω( s the eergy of the quatzed oscllatos of the lattce a excted state. The state of each quatum oscllator ca be see as that of each wth a assocated eergy exctato quata, h ω(. Ths exctato quata s assocated to a quas-partcle amed phoo, aalogy wth the photo, whch s the quata of the electromagetc feld. I the oe-dmesoal lattce cosdered here, the phoo s called acoustc phoo sce there are oly acoustc oscllato braches a oe-dmesoal lattce wth oe atom the bass, ad the lattce has phoos at the wavevector, phoos at the wavevector, ad so o. for, Smlarly, a oe-dmesoal fte lattce wth two atoms the bass ad N values + Hˆ o h ωλ ( aˆ, λ aˆ, λ +, (76a, λ, +, +, + N, + N, ( aˆ, ( aˆ,...( ˆ, ( ˆ, a a N N,,,,...,, λ λ N λ (76b (!(!...(!, λ, λ N, λ wth,,...,, ad h ω λ (, λ +, (77, λ,

78 Lattce oscllatos. Phoos where the phoos assocated wth the λ brach are the acoustc phoos, ad those assocated wth λ are the optcal phoos. I a three-dmesoal lattce wth s atoms the bass, there are three degrees of freedom for every atom, ad hece there are acoustc phoos ad A typcal phoo dsperso spectrum for s s llustrated below. s optcal phoos. optc braches acoustc braches Phoos are quata of the collectve/thermal lattce oscllatos. The crystalle lattce ca be vewed ether as a collecto of coupled harmoc oscllators or as a gas of free/oteractg phoos, whch obey the laws of quatum statstcs. I partcular, phoos are bosos ad obey the ose-ste statstcs. However, sce they are ot real partcles, ther umber s ot depedet of temperature ad volume, so that the electro-chemcal potetal of the phoo gas must be zero. The, the thermal equlbrum umber of phoos wth frequecy ω ( s gve by the Plac dstrbuto λ ph, λ exp[ hω ( / λ. (78 T ] So, the umber of phoos s small at low temperatures h ω λ >> T, for whch ph exp[ hω / T ], but becomes hgh at large temperatures h << T, for whch λ ω λ ph /[ + ( h ωλ / T ] T / hω λ.

79 Phooc heat capacty The thermal propertes of solds, ad partcular the heat capacty, are determed by both phoos ad electros. We refer ow to the phoo, or lattce, cotrbuto to the heat capacty. The heat capacty s defed as the heat Δ Q requred to rase the temperature by Δ T,.e. C ΔQ / ΔT. If the process s carred out at costat volume, Δ Q must be replaced by ΔU, whch represets the crease the teral eergy U of the system. The, the heat capacty at costat volume s C V U. ( T V The phoo cotrbuto at the heat capacty s obtaed from the lattce eergy term h ω +, λ, λ hω, λ + (, λ, λ exp( hω, λ / T of the teral eergy U eq +, wth the eergy the equlbrum cofgurato of eq the system. Actually, expressg the teral eergy of the system terms of the free eergy ad the etropy S as U + TS, ( apart from the heat capacty at costat volume C ( U / T T( S / T we ca defe V V V also a heat capacty at costat pressure, C T( S / T. These two parameters are related P P through C C T ( V / T /( V / P P V P T, ad oe ca be determed from the other. These parameters are the same oly the harmoc approxmato of lattce oscllatos. I a classcal statstcal theory, based o the classcal partto fucto, the mea eergy of a oe-dmesoal oscllator (resultg equally from ts etc ad potetal eergy parts s T, value that becomes T for a three-dmesoal oscllator. The, for sn three-dmesoal oscllators volume s sn T, ad the phooc heat capacty at costat

80 Phooc heat capacty C ph sn, (4 T V.e. C ph per atom, or C ph N A R 6cal/molK per mole, wth the Avogadro umber. Ths s the Dulog-Pett law, ad t predcts a temperature-depedet heat capacty. Ths predcto agrees wth expermetal data at hgh temperatures, but ot at low temperatures, where expermets dcate that T as T. To expla the low-temperature behavor of the heat capacty, oe should dsregard the classcal statstcal theory, whch s o loger vald whe the separato betwee the eergy C ph levels of the oscllator s comparable to or hgher tha statstcal mechacs. The specfc heat of the lattce s the defed as N A T, ad use stead the quatum C ph d dt hω,, λ T λ exp( hω, λ / T [exp( hω / T ], λ. (5 Ths expresso does ot volve the zero eergy of the lattce ad s called for ths reaso the phooc heat capacty. To specfcally calculate the phooc heat capacty we eed to ow the phoo dsperso relato. Ths relato s qute complcated for three-dmesoal crystals ad therefore approxmatos are geerally made. The ste model I ths model each atom or molecule s cosdered as a partcle that oscllates the average feld of ts eghbors. Therefore the system wth sn degrees of freedom all partcles have the same oscllato frequecy ω. The phoo heat capacty s ths case gve by C ph ( T sn ( Θ / T, (6 where Θ s the ste temperature, defed by h ω Θ ad ξ exp( ξ ( ξ (7 [exp( ξ ] s the ste fucto.

81 Phooc heat capacty ξ exp( ξ At hgh temperatures, for ξ Θ / T <<, ( ξ exp( ξ, so that ( + ξ C ( T, (8 ph sn as the Dulog-Pett law, but at low temperatures, for ξ >>, where ( ξ ξ exp(, the heat capacty has a expoetal temperature depedece of the form ξ C ph Θ ( T sn exp( Θ / T. (9 T Although lmc ph T ( T, the low-temperature depedece of the heat capacty s ot proportoal to T (see the fgure above, rght. The dscrepacy s due to the approprate treatmet of the acoustc phoo cotrbuto to the heat capacty. Ule for optcal phoos, for whch the frequecy s almost costat as a fucto of, the frequecy of acoustc phoos has a much wder terval of varato ad the oscllatos dfferet lattce cells must be cosdered as correlated (the atoms oscllate phase!. Therefore, sce the ste model descrbes a satsfactory maer the optcal phoo cotrbuto to capacty s expressed as C ph, the heat C opt ac ( T C ( T C ( T ( ph ph + ph

82 Phooc heat capacty 4 where / ( ( ( T N s T C opt ph Θ ( ad the cotrbuto of the acoustc phoos s estmated from the Debye model. The Debye model I the Debye model the frequecy of acoustc phoos a geeral, asotropc crystal s wrtte as v ac, ( (,, ϕ θ ω ω λ λ λ, λ,, ( wth θ, ϕ the polar agles, ad ther cotrbuto to the heat capacty s gve by,,,,,,,, ( ] / [exp( / exp( ] / [exp( / exp( λ λ λ λ λ λ λ λ ω ω ω ω ω ω ω ω λ λ d D T T T T T T C ac ph h h h h h h ( where Ω Ω cost ac cost cost v d V d d d V ds V D (,, (, (,, ( ( / ( ( ( ω λ ω λ ω λ ω ω ω λ ϕ θ ω π ω π ω π ω (4 s the desty of states. The frequecy tegral the expresso of s performed betwee ad ac C ph, ( max, ϕ θ ω λ. If,, ( 4 cost ac v ac v d Ω λ ω λ ω ϕ θ π (5

83 Phooc heat capacty 5 s a agular average of the acoustc velocty (the equalty holds as detty the sotropc crystal, the we ca troduce also a agle-depedet maxmum oscllato frequecy (the Debye frequecy ω max ωd whch, the Debye model, s also depedet o the polarzato λ. Ths maxmum oscllato frequecy follows from the ormalzato codto of the N acoustc oscllato braches: ωd D d ω V V D D( d Ωω, d 4 cost vac, (, ω ω λ ω λ ω ω λ π λ ω λ θ ϕ π π v, ac N, (6 ad so ω 6 N / V. (7 D v ac π I ths case C ac ph D V T hω exp( hω / T d ω π v h T [exp( hω / T ] 9 ac T N Θ D J 4 ω ( Θ D / T 4 (8 where ΘD defed through h ω Θ s the Debye temperature, ad D D ξ x exp( x J ( ξ dx (9 [exp( x ] s the Debye-Grüese tegral, whch has o aalytcal soluto. The Debye temperature s proportoal to the acoustc velocty, ad so s hgher for hgh Youg modulus values ad for lower crystal destes. It s usually determed by measurg the temperature depedece of the resstace aroud the Debye temperature. At hgh temperatures, for T >> Θ, the argumet the tegral s very small, sce x <<, ad after expadg t seres oe obtas D J 4

84 Phooc heat capacty 6 C ac ph 9 9 T N Θ D T N Θ ΘD / T D 4 x exp( x ( + x ΘD / T x dx dx 9 N T N Θ D ΘD / T x exp( x dx ( whle at low temperatures that T << Θ D the upper lmt of the tegral ca be exteded to, so C ac ph 4 4 T x exp( x π T 9 N dx N T Θ D [exp( x ] 5 Θ D ( Ths temperature depedece ca be uderstood from a qualtatve argumet: at low temperatures oly the phoo modes wth eergy h ω < T are excted. These modes are, the space, sde a sphere (the thermal sphere, so that the umber of modes s proportoal to ω T. If each mode has a average exctato eergy of T, the total eergy of exctato s proportoal to The total phooc heat capacty, 4 T ad hece the heat capacty s proportoal to C C + C expermets for both hgh ad low temperatures (see the fgure below. ph ac ph opt ph T., s ow agreemet wth

85 Phooc heat capacty 7 The Debye temperatures for some elemetal crystals are gve the table below lemet Θ D (K lemet Θ D (K lemet Θ D (K L 44 e 44 Cu 4 Na 58 Mg 4 Ag 5 K 9 Ca C Rb 56 Sr 77 S 645 Cs 8 a Ge 74

86 lectro dyamcs the crystalle lattce We have see that, the adabatc approxmato, the equato satsfed by the (subsystem of valece electros wth mass m ad postos,,, the feld of os at postos, α,, s r el N α R o N ; ( ( ; ( ( (,, R r R R r R r r r ψ ψ α α el j j j el V U m + + h, ( where,, ad the teracto eergy betwee electros,, s a b-partcle term of Coulomb type. Ths equato s qute dffcult to solve ad, therefore approxmate methods are employed. I oe of them, called the self-cosstet feld method (or Hartree-oc method, the teracto eergy betwee electros },...,, { el r N r r r },...,, { o R N R R R /(4 ( j j el e U r r r r πε j j j U el, ( r r s replaced by a effectve feld of the remag electros, whch ca be expressed as a sum of oe-partcle terms,. More exactly, f N el ef U, ( s r ψ s the oe-partcle wavefucto for the th electro, wth the sp value, the atsymmetrc (wth respect to the permutato operator of two electros wavefucto of the system of electros ca be expressed as a Slater determat, s, (, (, (, (, (, (, (, (, (! ( el el el el el el el el el N N N N N N N N N el s s s s s s s s s N r r r r r r r r r r ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ L M M L L, ( ad the eergy of the system s gve by + j j j j el j j j j j el j el U U H,, ( ( ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ r r r r, ( wth

87 lectro dyamcs h h o H + V ( r R α + V (4 m α m the oe-partcle Hamltoa. The secod term the rght-had-sde of el s the Hartree term, whle the thrd term s the Hartree-oc or exchage term. If the total eergy of the electro system could be wrtte as a sum of oe-electro eerges: ψ ψ, (5 el the the system of teractg electros would be equvalet to a system of depedet electros that satsfy the Schrödger equato of moto h o ef ef + V + U ψ H + U ψ ψ m (, (6 wth ef ψ U ψ ψ ψ j U el ( r r j ψ ψ j ψ ψ j U el ( r r j ψ jψ. (7, j, j The effectve feld ef U ca be determed f the oe-electro wavefuctos ψ are ow, but the latter ca oly be foud f the effectve feld s gve. It s a self-cosstet problem, whch ca oly be solved teratvely. More precsely, a set of tral fuctos s frst chose, ( ( such that H ψ ψ, wth the help of whch the effectve potetal ef ( s calculated. ( If ths effectve potetal s troduced the oe-electro equato, oe obtas ψ as a U soluto of ef ( ( ( ( H + U ψ ψ, ad so o. The teratve process stops after, let s say p steps, whe ψ ψ ( p ( p. I cocluso, the moto of oe electro a crystal ca be descrbed wth the oe-electro approxmato. We use further ths approxmato ad drop the subscrpt for smplcty.

88 lectro dyamcs lectros a perodc lattce. loch fuctos We have see that a electro a lattce s descrbed by the Schrödger equato h + V ( r ψ ( r ψ ( r m, (8 o where the eergy potetal V V + U the remag electros, ad ef descrbes the teracto wth all postve os ad V r V ( r + R (9 ( for ay traslato vector R of the fte lattce. The fluece of lattce perodcty o the electro wavefucto ca be deduced observg that h h r + R + V ( r + R ( r + R ( r + R V ( r ( r + R m ψ ψ + m ψ, ( sce r + R r. Ths mples that the electro wavefucto after the traslato wth R satsfes the same equato as before, ad hece ψ ( r + R C ψ ( r, ( wth, because the ormalzato codto o the volume V mposes that C V ψ ( r + R dr C ψ ( r dr. Moreover, at two successve traslatos we have C Cm C+ m, so that V C exp( R, ( or ψ ( r + R exp( R ψ ( r, where s, for ow, a wavevector (more precsely, a quaswavevector. The symmetry propertes of the electro wavefucto at traslatos ca be used to fd the geeral form of the egefuctos of the Schrödger equato for electros a

89 lectro dyamcs 4 crystalle lattce. More precsely, sce ψ r exp( R ψ ( r + R exp( r ( {exp[ ( r + R ] ψ ( r + R }, the wavefuctos ca be wrtte as ψ ( r u ( rexp( r, ( where u r exp[ ( r + R ] ψ ( r R s a perodc fucto wth the same perodcty ( + as the lattce, u ( r u ( r + R. lectro wavefuctos of the form gve above are called loch fuctos. Note that for free electros, whch satsfy the Schrödger equato wth vashg potetal eergy, ad whch have wavefuctos ψ ( r V / exp( r, wth V the ormalzato volume, all pots space are equvalet (the probablty to fd the electro at a pot r, ψ ( r /V, s depedet of r, ad the wavefucto s at the same tme a egefucto of the mometum pˆ h wth egevalues p h (besdes beg a egefucto of the Hamltoa, wth egevalues h / m p / m. I cotrast, for loch waves ψ ( r u ( r cost., ad oly pots that dffer through a traslato vector R are equvalet. Therefore h s the quas-mometum of the electro the crystal ad s the quas-wavevector; the followg, we wll stll refer to as wavevector, for smplcty. The fluece of lattce perodcty o the electro eergy ca be ferred observg frst that the electro eergy a crystal s -depedet ad real (the Hamltoa s a hermtc operator: (. The Schrödger equato ca be expressed also as h ( + m + V ( r u ( r ( u ( r (4 sce ψ ( r [ u ( rexp( r] exp( r[ u + u ] exp( r( + u (5 ad hece ψ ( r exp( r( + u Schrödger equatos satsfed by the wavefucto for. ecause the eergy s real, from the (detcal ad ts complex cojugate:

90 lectro dyamcs 5 h ( m + V ( r u ( r ( u ( r, (6 * * h ( + V ( r u ( r ( u ( r m, (7 t follows that the eergy s a eve fucto of : ( (. (8 I partcular, the so-eergetc surfaces of a electro a lattce,.e. the surfaces space for whch ( cost., have a verso symmetry; they are ot, however, spheres as for free electros. If the crystalle lattce we perform a traslato of wth a vector the recprocal space, G, we obta ( ( + G, (9 because, by replacg wth + G ( we obta ψ ( r + R exp( R ψ ( r exp[ ( + G R ] ψ ( r sce exp( G R (the Schrödger equato ad the wavefucto are detcal after a traslato the space. Therefore, the eergy s a perodc fucto the recprocal lattce space ( space, ad taes dstct values oly sde the frst rllou zoe (remember that the whole space ca be dvded rllou zoes wth all possble orders, whch have the same volume ad ca be reduced to the frst rllou zoe by applyg symmetry operatos. I the Schrödger equato for the electro s a parameter. As for phoos, where we have dfferet solutos ω ( for a gve, assocated to dfferet oscllato braches, we ca have also dfferet egefuctos ad egevalues of the electroc Hamltoa (dfferet eergy values ( for a gve wavevector. lectro velocty a crystal We have see that the electros a crystalle lattce are loch waves. The so-called loch electros are quas-partcles sce ther propertes are depedet o the crystalle lattce. If a electro wth wavevector s sde a lattce, oe ca troduce the cocept of average velocty through the quatum mechacal defto

91 lectro dyamcs 6 * h v v ψ pˆ ψ ψ ( r ψ ( r dr, ( m m V wth pˆ ( h / the mometum operator. or loch electros the average velocty ca be expressed as v * h u ( r ( u ( r d r m + V h h + u * ( r u ( r d r m m, ( V whch dffers from the expresso for free electros, v h / m p / m, through the secod term. To calculate the average electro velocty, we dfferetate (4 wth respect to (we apply the gradet space, ad obta h ( + + u ( r. ( m f the potetal eergy does ot deped explctly o. rom ths result ad the ormalzato codto u V * ( r u ( r dr we ca express ( as h u * u d m ( r( + ( r r, ( V h so that, fally, v h. (4 Ths relato s detcal to that obtaed for free electros wth a parabolc dsperso relato, whch h / m ad, thus, h / m ad v h / m h. Therefore, (4 s a more geeral expresso of average velocty tha (, the dfferece beg that (4 the mass of electros does ot appear explctly; a crystalle lattce the free electro mass m ca be replaced by a effectve mass, so that (4 holds, but ot (.

92 lectro dyamcs 7 lectro accelerato a crystalle lattce. ffectve mass Let us ow cosder that the electros the crystal are accelerated by a exteral force. We separate the effects of the exteral forces o the electro ad the Coulomb teractos sde the crystal, by modelg the loch electro as a quas-partcle that reacts oly to the exteral forces. The fluece of the Coulomb teractos o the electro dyamcs s characterzed by a effectve electro mass. ecause, accordg to the hrefest s theorem, the classcal equatos of moto are vald for the average values of the quatum observables, we have d ( d d( h v dt dt dt dp v dt v, (5.e. the quas-mometum p h of a electro the crystal plays the same role as the mometum of a free electro. Here s oly the exteral force; the teral forces the crystal are tae to accout whe calculatg the dsperso relato (, ad thus are cluded the expresso of the average velocty. The accelerato of the electro the crystal ca be defed as dv d d a ( dt h dt h dt h (6 or, wrtte o Cartesa compoets μ,ν x,y,z, a μ dv dt μ ν h μ ν ν ν m eff μν ν. (7 rom the relato above t follows that, geeral, the accelerato of a electro the crystal has a dfferet drecto tha that of the appled force. The relato betwee the accelerato ad the appled force ca be used to troduce, as for the case of free electros, a secod-order tesor parameter, called effectve mass. Its verse, defed as m eff μν h μ ν, (8

93 lectro dyamcs 8 s a symmetrc tesor, sce ( / eff μν (/ meff νμ m. The verse of the effectve mass ca tae both postve ad egatve values, ule for a free electro, where t s a scalar parameter ad always postve. The effectve mass corporates the effect of the crystal lattce o the moto of electros; t s ot a characterstc of the electro (as s ts free mass but of the electro-lattce teracto. The effectve mass s versely proportoal to the curvature of the dsperso relato ad s fte at flexo pots, where the curvature vashes. If s a extremum (maxmum or mmum of the eergy dsperso relato the space, for whch (, a Taylor seres expaso of aroud the harmoc approxmato ( whch oly terms up to the secod order are cosdered ca be wrtte as h ( ( + ( μ μ ( ν ν + μ, ν μ ν μ, ν m eff μν ( μ μ ( ν ν (9 We have deoted (. The above form s postvely defed f the effectve mass taes oly postve values, ad, reducg t to the prcpal axes we obta a ellpsodal soeergetc surface that satsfes the equato h ( ( meff, meff, meff, If two of the semaxes of the ellpsod are equal, for example f m m eff, eff, the commo value of the effectve mass s called trasverse effectve mass m, ad m m s the logtudal effectve mass; the ellpsod s ths case a ellpsod of rotato. If all dagoal compoets of the effectve mass tesor are equal, the so-eergetc surfaces become spheres, m eff, meff, meff, meff, ad the dsperso relato of the electro the crystal s smlar to that of the free electro wth mass m, except that m s replaced wth the effectve mass: t eff, l h (, ( m eff f the referece eergy s chose such that. The accelerato has the same drecto as the exteral force oly alog the prcpal axes of the ellpsod, or for sphercal so-eergetc surfaces.

94 lectro dyamcs 9 ergy bads We have see that the electro eergy ca tae multple values for a gve the frst rllou zoe. The eergy spectrum for the electros a crystal ca be quattatvely determed startg from two models: the approxmato of quas-free electros (the weabdg approxmato or the approxmato of quas-boud electros (the tght-bdg approxmato. These approxmatos correspod to two extreme cases. I the frst oe t s assumed that the state of the electro the crystal ca be modeled as a perturbed state of a free electro, whle the secod approxmato, the state of the electro s a perturbed state of a boud electro a solated atom, the perturbato both cases beg due to the perodc lattce potetal. The wea-bdg approxmato s partcularly sutable for treatg the eergetc spectrum of valece electros metals, whle the tght-bdg approxmato s more sutable for semcoductor ad solatg materals. I both cases the perodcty of the crystalle lattce leads to the formato of allowed ad forbdde eergy bads. I the fgure below t s show how eergy bads, separated by gaps, form from atomc s ad p orbtals as the separato betwee atoms decreases S; a s the lattce costat.

95 lectro dyamcs The tght-bdg approxmato Let us assume that the wavefucto of the loch electro the crystal, ψ (r, ca be expressed as a lear combato of atomc wavefuctos: ψ r C ϕ ( r R ( ( where ϕ r R s the wavefucto of the atom at posto R ad C exp( R sce ( the electro wavefucto must be varat (up to a phase factor at traslato. The atomc wavefuctos satsfy the Schrödger equato for the solated atom, h H aϕ + U a ( r R ϕ aϕ m, ( where H s the Hamltoa ad U s the potetal eergy of a solated atom ad s a a the correspodg eergy, whereas the Schrödger equato for the electro the crystal s a h Hψ ( r + V ( r ψ ( r ψ ( r [ H a + W ( r] ψ ( r m, (4 wth W ( r V ( r ( r R < U a the perturbato eergy due to the crystal. The fact that V r < ( r R, ad hece W ( r <, expresses the stablty of the crystal, the potetal ( U a eergy the lattce beg lower tha a solated atom. The electro eerges the crystal ad the solated atom are llustrated the fgure below. U a (r V(r

96 lectro dyamcs The last equato ca be re-wrtte as ] ( [ exp( + a W H ϕ r R, (5 or ] ( ( [ exp( + a W ϕ ϕ r R, (6 whch becomes, after left-sde multplcato wth ad tegrato over the crystal volume V, ( exp( * m m m R r R ϕ ] ( ( ( ][ ( exp[ + m a m m S A R R R R R R. (7 Here r R r r R r R R d W A m V m m ( ( ( ( * ϕ ϕ (8 s the exchage tegral, whch depeds o the overlap of the atomc wavefuctos ad the crystal perturbato ad defes the exchage teracto eergy, ad r R r R r R R d S m V m m ( ( ( * ϕ ϕ (9 s the overlap tegral, whch depeds oly o the overlappg degree of the atomc wavefuctos. If l m R R R s the traslato vector betwee the th ad mth atoms, t follows fally that + l l l l l l a S A exp( ( exp( ( R R R R. (4 Thus, the eergy of the electro the crystal dffers from the eergy a solated atom through a perodc fucto of. Istead of a dscrete eergy level (as atoms we have ow a eergy bad wth a wdth determed by the maxmum ad mmum values of the perturbato term.

97 lectro dyamcs I the smplfyg assumpto that the atomc wavefuctos are rapdly decreasg wth the dstace, such that the overlap tegral s eglgble eve for eghborg atoms,.e. such that, Rl ( m S( R l, ad hece S( R l exp( Rl, (4, Rl ( m l we obta that a C A exp( Rl, (4 l where the sum exteds ow oly over the earest eghbors, for whch atomc orbtals, for whch A >, ad A( R A < l for s * C A( ϕ ( r R W ( r ϕ ( r R dr < (4 V s a costat (depedet of parameter. or a smple cubc lattce wth perod a, whch a atom at the org of the coordate system has 6 earest eghbors at postos [[]], [[ ]], [[]], [[ ]], [[]], ad [[ ]], exp( Rl exp( a + exp( a + exp( l [cos( a + cos( a + cos( a] a + exp( a + exp( a + exp( a (44 a C A[cos( a + cos( a + cos( a], (45 whch s a eve fucto of the wavevector compoets; all possble eergy values are obtaed for wavevector compoets the frst rllou zoe,.e. for π / a π / a. The expresso above dcates that, the crystal, the eergy level of the solated atom shfts dowwards wth C due to the teracto betwee atoms, whch reders the crystal more stable tha a solated atom, ad trasforms to a eergy bad, whch s perodc the wavevector compoets ad exteds betwee ad, wth m max

98 lectro dyamcs a C 6A, for, (46a m a C + 6A, for ±π / a. (46b max The wdths of the eergy bads, Δ A, crease wth the exchage tegral (wth the overlap of the atomc orbtals ad are wder for the hgher dscrete atomc eergy levels sce the wavefuctos of the outer atomc levels are more exteded space. Dfferet eergy bads form startg from dfferet atomc orbtals, ad a addtoal subscrpt label these eergy bads. I the fgure below (left dfferet eergy bads are represeted the frst rllou zoe. These bads are formed ( the creasg eergy order from atomc orbtals for whch A s postve, egatve, postve, etc. m eff wth wdths These (allowed eergy bads are separated by eergy gaps (forbdde eergy bads. The wdth of the frst eergy bad, for example, s gve by g g m, max, ( a, a, + A ( C C 6( A. (47

99 lectro dyamcs 4 If both ad A correspod to s states, the eergy gap s determed from the extremtes of A the two bads at dfferet values (ceter ad edges of the frst rllou zoe, whereas the eergy gap betwee bads that form from s ad p atomc orbtals s determed by states wth the same : at the edges of the frst rllou zoe, for A A p < (the exchage tegral s postve ad A A s >, ad at the ceter of the frst rllou zoe f A ad A. > < ffectve mass electroc eergy bads Let us calculate the effectve electro mass at the extreme pots of the eergy bad wth the dsperso relato a C A[cos( a + cos( a + cos( ]. Near the ceter of the a frst rllou zoe, whe a << ad cos( a ( a /, a + C 6A + Aa ( + + m Aa, (48 ad hece the eergy depeds quadratcally o the wavevector, + h / m m eff, as for free electros, wth a effectve mass m eff h ( h / Aa. (49 The effectve mass s postve for eergy bads that form from s atomc orbtals, for whch A >. or the partcular case cosdered here, that of a smple cubc lattce, the so-eergetc surfaces cost. the eghborhood of the ceter of the frst rllou zoe are spheres. Note that m eff depeds o the dsperso relato, ad hece o the crystal structure. O the cotrary, at the edges of the frst rllou zoe, troducg the ew varables ' ± ( π / a, such that cos( a cos( m ' a + π cos( ' a, the dsperso relato ca be expressed as ' a C + A[cos( ' a + cos( ' a + cos( ' a] (5 ad, for ' a << ad cos( ' a ( ' a /, a ' C + 6A Aa ( ' + ' + ' max Aa '. (5

100 lectro dyamcs 5 The dsperso relato s aga smlar to that of a free electro,.e. has the form + h ' / ' max m eff, wth a effectve mass m eff h ( Aa h /, (5 whch s egatve f A >. A example of such a stuato s represeted the fgure above (rght. Note that the effectve mass s egatve at the ceter of the frst rllou zoe ad postve at ts edges for eergy bads that form from p atomc orbtals, for whch A <. I geeral, m > the eghborhood of the mmum eergy value the bad ad m < eff ear the maxmum eergy value the bad. Although (5 shows that the so-eergetc surfaces are also sphercal at the edges of the frst rllou zoe, they have complcated forms at termedate eergy values, betwee the ceter ad the edges of the frst rllou zoe. Due to the perodcty of electro eergy the space, the so-eergetc surfaces are the same all cells the recprocal space, so that the so-eergetc surfaces are multple coected. The same geeral results are obtaed the wea-bdg approxmato. eff lectros ad holes If the electro the crystal, wth a electrc charge mass m eff > s placed a electrc feld, ts accelerato e ad a postve (sotropc effectve a d v / dt / m e / (5 eff m eff s smlar form to that of a free electro. O the other had, the eghborhood of the maxmum eergy value the bad m < ad the electro equato of moto s gve by eff a dv dt /( m e /( m. (54 / eff eff The egatve effectve mass the equato above has o aalog for free partcles. It has, as cosequece, that a electrc feld wll decelerate the electro, stead of acceleratg t. To avod such a awward terpretato, t s cosdered that the moto of the electro wth a egatve effectve mass the electrc feld s equvalet wth the moto of a quas-partcle

101 lectro dyamcs 6 wth electrc charge + e ad a postve effectve mass, equal to m. Ths quas-partcle s called hole. The hole s a quas-partcle sce, ule the electro, t has o physcal meag free space; holes exst oly crystals ad reflect the behavor of valece electros. I a geeralzed sese, eve the electro the crystal ca be see as a quas-partcle edowed wth a effectve mass, whch s dfferet from that of the free electro. If the electros the upper part of a occuped eergy bad (the valece bad acqure suffcet eergy from thermal vbratos, for example, to go to a uoccuped state the ext eergy bad, called coducto bad, the remag empty states ca be cosdered as holes. Sce the empty states ca be occuped by other electros, the holes ca be see as movg throughout the valece bad,.e. they ca be regarded as free quaspartcles the valece bad. I other words, a eergy bad occuped wth electros wth the excepto of ts upper part, ca be see as partally occuped wth holes. ecause the electros that partcpate at electrcal ad thermal coducto are those able to move (quas-free the crystal, the cocept of holes allows a major smplfcato the treatmet of the system of electros the valece bad: t s o loger ecessary to deal wth the moto of the etre system of electros, but oly wth the moto of a much smaller umber of holes. I geeral, the holes the valece bad have ot oly a opposte electrc charge, but also a dfferet effectve mass tha the electros the coducto bad, sce ther dsperso relato s dfferet (the valece ad coducto bads orgate from dfferet solated atomc levels. The holes move the drecto of the appled electrc feld, whereas electros move the opposte drecto!. eff Desty of electro states Real crystals have fte szes ad, therefore, the soluto of the Schrödger equato for the electros the crystal depeds o the boudary codtos. As for phoos, we assume that large crystals the surface pheomea do ot fluece sgfcatly the electro dyamcs sde the crystal, ad hece use the cyclc (or-karma boudary codtos. I the orthorhombc symmetry, for example, f the dmesos of the crystal alog the orthogoal Cartesa coordates x,,, ( codtos mpose that x x, x y, x z are deoted by L, the cyclc ψ ( x, x, x ψ ( x ψ ( x + L, x + L, x, x + L, x ψ ( x, x + L + L, x ψ ( x, x, x + L (55

102 lectro dyamcs 7 or, sce ψ ( r u ( rexp( r wth (r the same all lattce cells, u L πm, πm / L (56 wth m teger umbers. Smlarly, a fte crystal wth N atoms alog the drecto, such that L N a, wth a the respectve lattce costats, t follows that πm, (57 N a where, for wavevectors the frst rllou zoe, the tegers m ca tae oly N values the tervals N m < N /. (58 / Thus, for a smple lattce (wth oe atom the bass, the umber of dstct eergy states of electros a allowed eergy bad s N NN N, ad these states ca be occuped by N electros because, accordg to the Paul prcple for fermos, oly two electros wth opposte sps ca occupy a eergy state characterzed by a gve. I large crystals the dstace betwee eergy levels s qute small ad the eergy bad s approxmated as a (quas-cotuous fucto of, case whch ay sum over states the space ca be replaced (as the case of phoos by a tegral over the frst rllou zoe. More precsely, for ay fucto (, V V ( ( Δ ( d (π (π st Z (59 where ( π (π Δ ΔΔ Δ (6 L L L V s the volume space occuped by a dstct electro state.

103 lectro dyamcs 8 I partcular, the desty of states the space, defed as the umber of electro states wth a gve sp oretato, dn, the volume elemet d, s gve by el dn el d ( V π, (6 Δ or N el V (π st d. Z The desty of states per ut volume, defed as the umber of states per ut volume wth a gve sp oretato the eergy terval d s the dn el d D( d, (6 V d d (π V V where s the volume space betwee the so-eergetc surfaces ( ad ( + d. As for phoos, d dsd ds( d / wth ds the ftesmal elemet o the ( cost. surface ad d ormal to ths surface, ad the expresso above smplfes to D( (π ds ( cost, (6 whch becomes m m (m D( (π / eff eff eff / dω ( h Ω π h 4π h (64 for sphercal so-eergetc surfaces, for whch ( + / h m eff. I ths case h / m eff ad ds dω, wth dω the elemet of sold agle.

104 lectro dyamcs 9 Classfcato of sold state materals Oe of the most mportat achevemets of the eergy bad theory s the possblty to classfy sold state materals metals, solators, ad semcoductors. Ths classfcato s ot based o the structure of the eergy bads, whch s qute the same all materals, but o the degree of occupato of these bads. The avalable umber of eergy states s occuped by electros agreemet wth the Paul excluso prcple. More precsely, at low temperatures ( prcple, at T K the states are occuped the order of creasg eergy value, such that oly two electros (wth opposte sps are allowed o a eergy state wth a gve value. The avalable states are occuped by electros up to a eergy level called erm eergy, or erm level. ecause electros are fermos, ther quatum statstcal dstrbuto fucto at temperature T s descrbed by the erm-drac formula f ( + exp[( / T ] (65 wth the oltzma costat. The temperature depedece of the erm-drac dstrbuto fucto s llustrated the fgure below. At T K, the dstrbuto s a step fucto, equal to for eerges smaller tha the erm eergy, ad equal to otherwse. Two stuatos ca exst: at low temperatures, T K, the erm level s sde a eergy bad (see the fgure below, left,.e. electros occupy partally the last eergy bad. The materal s the a metal ad ca easly coduct electrcty sce the electros the vcty of accelerated by a small appled electrc feld, ca occupy avalable empty states wth hgher eergy.,

105 lectro dyamcs at low temperatures, T K, the electros occupy completely a umber of eergy bads, so that the erm level s sde the eergy gap betwee the last occuped bad, called valece bad, ad the followg empty bad, called coducto bad (see the fgure above, rght. I ths case a small electrc feld does ot provde suffcet eergy for the electro to jump to empty states the hgher eergy bad, ad therefore o electrc curret ca flow through the materal. We deal ths stuato wth a delectrc. At hgher temperatures, T K, the delectrc materals, tur, ca be ether solators or semcoductors, depedg o the wdth of the eergy gap betwee the valece ad coducto bads. If g < ev, thermal fluctuatos ca excte electros from the valece to the coducto bad, where they ca cotrbute to electrcal coducto, ad the materal s ths case a semcoductor. As a result, (udoped semcoductors the umber of electros the coducto bad s equal to the umber of holes the valece bad. Whe a appled electrc feld s appled, the curret has two cotrbutos: oe from electros ad the other from holes, whch are drfted opposte drectos. I solators, > ev ad o electrcal coducto exsts at moderate temperatures or electrc felds. ecause the total umber of electros a crystal wth N ut cells ad s atoms the bass, each atom havg Z electros, s NsZ, ad a eergy bad ca accommodate N electros, t follows that the umber of occuped eergy bads s NsZ / N sz / g. The, smple lattces wth s, there s always a partally occuped bad for odd Z values ad these materals should be metals. Ths s the case of moovalet alale metals (L, Na, K, Cs, Rb ad oble metals (Cu, Au, Ag, whch have oly oe valece electro. However, t s ot true that materals wth s ad eve Z are always delectrcs. or example, bvalet elemets such as e, Mg, Ca, Sr,, are metals. The explaato s that, these cases, the completely

106 lectro dyamcs empty coducto bad overlaps the completely occuped valece bad over a small eergy terval (see the fgure below. The electrcal coductvty s poorer, though, f the overlap s slght. Note that geue delectrc materals, at T K there s o overlap betwee the coducto ad valece bads. ecause at the top of partally occuped eergy bads the charge carrers are holes, a mxed coducto (electros ad holes s expected whe the coducto ad valece bads overlap over a arrow eergy rage. The materals that have a mxed coducto at T K are called semmetals; they dffer from metals that metals the coducto s always due to electros oly. The semmetals have more complcated bad structures tha metals ad delectrcs. xamples of semmetals are As, Sb, ; these materals have a electrcal coductvty wth up to four orders of magtude smaller tha metals. I metals, the so-eergetc surface the recprocal space defed at T K through ( (66 s called erm surface. The erm surface separates the states wth a low occupato desty from those wth a hgh occupato desty ad determes the physcal propertes of metals, especally the electrcal propertes, whch oly electros wth a eergy terval of the order of T aroud the erm eergy partcpate. xamples of smple ad complex erm surfaces are gve the fgures below. Note that N, whch has strog magetc propertes, the erm surfaces for electros wth opposte sps are dfferet.

107 lectro dyamcs

108 Statstcs of charge carrers The statstc propertes of charge carrers are determed whe these are statstcal/thermodyamcal equlbrum wth the crystalle lattce. To fd these propertes we eed to ow the dstrbuto fucto ad the desty of states of electros ad holes. or a geeral system of electros characterzed by a dstrbuto fucto (probablty of occupyg avalable electroc states f (, the cocetrato of electros s f ( ( f ( ( f ( ( d V, σ V (π max f ( dsd f ( D( d (π m ( where the factor orgates from the sum over the sp states wth dex σ. The desty of electroc states for sphercal so-eergetc surfaces s gve by / (meff D( ( 4π h /. ( ecause there are dffereces the dstrbuto fucto of charge carrers metals ad semcoductors, the followg we treat these cases separately. Statstcs of electros metals I metals the erm eergy s sde a eergy bad ad oe ca defe the erm surface. The desty of electros metals wth sphercal so-eergetc surfaces s gve by ( wth ad, at a fte temperature T, the electros occupy the eergy states accordg to the erm-drac dstrbuto fucto f ( + exp[( / T ]. ( I ths case, the electro cocetrato becomes

109 Statstcs of charge carrers (meff π h / / exp[( d / T ] + (4 f the upper lmt of the tegral s approxmated wth. Ths approxmato s justfed sce the tegrad decreases rapdly. Wth the chage of varables fally obta x / T, y T, we / / (meff T / ( y (5 π h where x dx α ( y (6 α exp( x y + are the erm-drac tegrals. They are evaluated umercally. Note that the same result s obtaed for a arbtrary. I ths case, (4 appears the umerator of the tegrad ad the lower lmt of the tegral, but dsappears the fal result (equatos (5 ad (6, f the ew varables are chose as ( / T x, ( / T y. The temperature depedece of the erm-drac dstrbuto fucto s represeted the fgure below. At all temperatures, f ( /. At T K, where the erm-drac dstrbuto s a Heavsde fucto,.e.

110 Statstcs of charge carrers, < f ( (7, > the cocetrato of electros metals s / / T / / / (meff T / (meff T ( meff / ( x dx π h π h T, (8 π h wth the erm eergy level at T K, or ( / h π h meff meff, (9 where ( π / s the erm waveumber. rom (9 t follows that the erm eergy at T creases as the electro cocetrato creases, the system of electros beg the fudametal state f the erm sphere s completely occuped ad a excted state f electros occupy states wth >. or fte but low temperatures, y / T >>, ad sphercal so-eergetc surfaces, the erm-drac tegrals ca be approxmated wth α + y π α( α + α ( y +, ( α + 6 y ad so (meff π h / / π T + 8. ( O the other had, the electro cocetrato metals does ot deped o temperature because a crease T affects oly the thermal exctato of electros o hgher eergy levels. It follows thus that the erm level must deped o temperature ad, from

111 Statstcs of charge carrers 4 / ( / π T + 8 ( / π T + 8, ( we obta π T. ( Ths temperature depedece s very wea ad the erm eergy s, the frst approxmato, almost costat as T vares. Statstcs of charge carrers semcoductors I semcoductors the erm level s stuated betwee the valece ad the coducto bad, whch are separated by a eergy gap. At T K, all eergy states below are occuped wth electros accordg to the Paul prcple,.e. accordg to the erm-drac dstrbuto fucto, ad all states above the erm level are empty. ecause the holes semcoductors ca be vewed as states ot occuped by electros, the probablty that a state, wth eergy ( s occuped by a hole (.e. s empty for electros s f (. (4 + exp[( / T ] + exp[( / T ] If / T >> the erm-drac dstrbuto fucto has almost the step-fucto form characterstc for T K. I ths case the dstrbuto fucto s called degeerate ad s ecoutered materals wth large cocetratos of electros, such as metals ad heavly doped semcoductors. O the cotrary, whe / T << or / T >>, the expoetal term the erm-drac dstrbuto fucto s much larger tha uty ad the dstrbuto fucto resembles the classcal Maxwell-oltzma dstrbuto, f cl ( exp( / Texp( / T <<, (5

112 Statstcs of charge carrers 5 whch ca be see as the tal of the erm-drac dstrbuto. Ths case correspods to low electro cocetratos, partcular to udoped (trsc semcoductors, whch the electro cocetrato s wth few orders of magtude smaller tha metals. The Maxwell- oltzma dstrbuto fucto s called odegeerate. Itrsc semcoductors The desty of states for electros a trsc semcoductor s smlar to that metals (for a arbtrary ad s foud to be D m π h (m 4π h / / ( ( c (6 for sphercal so-eergetc surfaces ( + h /, where s the mmum eergy c m c the coducto bad, ad m s the effectve mass of electros the vcty of ths mmum eergy. I a aalogous maer, the desty of states for the holes wth effectve mass m p the valece bad s foud to be D p m (m ( π h 4π h / p p / ( v, (7 where v s the maxmum eergy the valece bad. Ths expresso was foud tag to accout that the hole eergy creases the opposte drecto as that of the electro. oth electro ad hole desty of states must be multpled wth f the sp degeeracy s cluded. However, the dsperso relato s ot always sphercal. or example, the coducto bad of Ge ad S are ellpsodal so-eergetc surfaces wth equvalet mma arraged symmetrcally the frst rllou zoe ( N 4 for Ge ad N 6 for S. Ths case ca be eq reduced to that of sphercal so-eergetc surfaces f, the eghborhood of these mma N eq eq stuated at, the dsperso relato h ( ( ( c (8 m m m

113 Statstcs of charge carrers 6 s trasformed to a sphercal dsperso relato h ' c + ( ' + ' + ' (9 m ' by a chage of varables ' m / m '. The fgures below llustrate the soeergetc surfaces S (left ad Ge (rght. There are 4 complete ellpsods (8 halfellpsods the frst rllou zoe Ge. The, the calculato of the desty of states we must accout for the fact that / / d ddd [( mmm / m ' ] d', from whch t follows that the desty of states becomes D (m ' ( m / / ( ( / c 4π h m ' m m /. ( ecause there are N eq equvalet mma/ellpsods, the desty of states s gve by D (m 4π h / c c ( ( c / ( where the effectve mass of the desty of states the coducto bad s defed as / / m c N eq ( mmm. (

114 Statstcs of charge carrers 7 Cocetrato of charge carrers ad erm eergy I a trsc (udoped semcoductor, the requremet of electrcal eutralty mposes the same cocetrato for both electros ad holes, p. ( I trsc semcoductors the cocetratos of charge carrers are small ( ad p are smaller wth several orders of magtude tha the electro cocetrato metals, so that the avalable eergy states are o loger occuped accordg to the erm-drac dstrbuto fucto, as metals, but accordg to the classcal Maxwell-oltzma dstrbuto. So, the electro a hole cocetratos trsc semcoductors wth sphercal so-eergetc surfaces are gve by max m ( ( d D f ( ( / / c y T m h π, ( ( / / v p y T m p h π (4 wth, T y c c / ( T y v v / (. The expressos are smlar to that foud metals, wth T x c c / (, T x v v / (, respectvely, except that ths case the erm-drac tegrals become exp( exp( ( dx x x y y α α, (5 sce we use the Maxwell-oltzma dstrbuto. ecause exp( / ( ( / y y π, T N T N T T m c c c c c exp exp exp 4 ( / h π π, (6 T N T N T T m p v v v v v p exp exp exp 4 ( / h π π, (7 where / 4 ( h π π T m N c, / 4 ( h π π T m N p v, (8

115 Statstcs of charge carrers 8 are the effectve destes of states the coducto ad valece bads, respectvely. rom the relatos above t follows that, ule the degeerate case metals, the cocetratos of electros ad holes o-degeerate semcoductors deped strogly o temperature. The erm eergy s determed from the eutralty codto,.e. from m / exp T c m / p v exp T. (9 c The result s g / c + v + 4 m T l m p ( v T ad, at T K, the erm level s at the mddle of the eergy gap: + /. As the ( c v temperature creases, the erm level shfts towards the eergy bad wth the smaller effectve mass (see the fgure above; t remas at the cetre of the eergy gap, rrespectve of temperature, oly f m m p. A specal stuato s ecoutered semcoductors wth a small eergy gap wdth ad m / >> or m / << (the frst stuato s g c v p m p m much more commo. I ths case, the erm level ca eter sde the bad wth the smaller effectve mass, ad the semcoductor becomes degeerate at eve moderate temperatures. A example of such a materal s ISb, for whch g.8 ev ad m p / m. It should be emphaszed that, startg from the expresso of the electro cocetrato semcoductors, t follows that the codtos of applcablty of the Maxwell-oltzma statstcs s c 4π h exp / T (πm T <<. ( If the eergy referece s chose such that c, exp( / T << for small electro cocetratos, hgh temperature, ad hgh effectve masses. O the cotrary, f these codtos are ot satsfed ad exp( / T >>,.e. f the electro cocetrato s hgh, the temperature s low, ad the effectve mass s small, the electro gas s degeerate, ad the crtero above s ow as the degeeracy crtero. lectros metals satsfy the degeeracy crtero.

116 Statstcs of charge carrers 9 The trsc cocetrato of carrers ca be defed as g / 4 / g p p N c N v exp ( mm p T exp. ( T πh T / Ths cocetrato does ot deped o the erm level, ad ts temperature depedece s summarzed the formula (see fgure ( T T / g exp, ( T l( /T / ad the bad gap ca be determed from the slope of the fgure at the rght sde. or a large class of semcoductors, the eergy gap wdth depeds o temperature as: g ( T αt. g /T xtrsc semcoductors A extrsc semcoductor s a doped semcoductor,.e. t cotas door ad/or acceptor mpurtes I door mpurtes, the umber of valece electros s hgher tha the host materal, ad the extra electros are ot volved bdg wth the atoms of the host materal; they are stll localzed aroud the door mpurty but ca easly partcpate at the process of charge carrer trasport whe a electrc feld s appled. The umber of free electros that cotrbute to electrcal coducto creases the presece of door mpurtes. O the cotrary, acceptor mpurtes have a smaller umber of valece electros tha the host materal, ad ther stable bdg wth host atoms requres a addtoal electro from the host materal. As a result, the umber of holes creases whe acceptor mpurtes are preset. +

117 Statstcs of charge carrers or S (see the schematc fgures above, whch forms covalet bods wth other four atoms, P s a door mpurty sce t has fve valece electros, ad s a acceptor mpurty because t has oly three valece electros. Door ad/or acceptor mpurtes are purposefully troduced to a host semcoductor to crease ts electrcal coducto. O the other had, the utetoal dopg of a crystal troduces defects to the crystalle lattce ad degrades ts electrcal performaces. Therefore, dopg s a process that should be carefully cotrolled. The door ad acceptor mpurtes have dscrete eergy levels sde the badgap of the host materal, whch ca be occuped by electros ad, respectvely, holes. The charge carrers o the mpurty levels are localzed aroud doors ad acceptors ad are ot free to move aroud the host crystal uless the mpurtes are ozed,.e. the electros o door eergy levels are promoted (by thermal eergy or appled electrc felds to the coducto bad ad the holes o acceptor levels are excted to the valece bad of the host materal. Let us deote by N ad N the cocetrato of door ad acceptor mpurtes, by d a N d ad N a the cocetratos of electros ad holes localzed o the doors ad acceptors, respectvely,.e. the cocetratos of eutral doors ad acceptors, ad by N + d N d N d ad N a N a N a the cocetratos of the ozed mpurtes. The ozato of doors ad acceptors leads to a crease of the umber of free charge carrers wth respect to the trsc semcoductor case. If ad p are the total cocetratos of free electros ad holes, the codto of charge eutralty s, (4a + + N a p + N d or a d d a + N + N p + N + N. (4b Ule the case of free charge carrers, the dstrbuto fucto of the electros localzed o door mpurtes ad that of the holes localzed o acceptor mpurtes s ot the erm- Drac fucto, whch s vald whe two electros wth opposte sps ca occupy a eergy level, accordg to the Paul prcple. The reaso s that oly a sgle electro ca occupy a eergy level o a mpurty atom. If aother electro s brought o ths level, ts eergy vares sgfcatly due to the strog electrostatc teracto betwee electros.

118 Statstcs of charge carrers Accordg to the erm-drac dstrbuto fucto, the rato betwee the probablty that the state s occuped ad the probablty that the state s empty s, f ( f ( exp T. (5 O the cotrary, sce oly oe electro ca exst o a mpurty level, the same rato s ow f d ( f ( d exp T (6 for electros o doors (the level s occuped twce as fast, or f d ( + (/ exp[( / T ], (7 ad f a ( + (/ exp[( / T ] (8 for holes o acceptor mpurtes. I geeral, the factor (/ frot of the expoetal term the deomator should be replaced by ( / g, wth g the degeeracy of the eergy level. Tag to accout that the desty of states of the dscrete door/acceptor eergy levels are D ( N δ (, d,a (9 the cocetratos of the electros localzed o the doors ad of holes localzed o acceptors are gve by N d f d ( D ( d d + N d d exp T, (4

119 Statstcs of charge carrers N a f a ( D ( d a + N a exp T a, (4 ad the eutralty codto becomes N a N d + d + exp T N d + N c exp T N a + + exp T a c + N v v exp T (4 from whch oe ca determe the posto of the erm eergy level. The erm level extrsc semcoductors s dfferet tha trsc semcoductors! xtrsc semcoductors wth oly oe mpurty type Let us cosder that the semcoductor s odegeerate ad has oly door mpurtes wth cocetrato N d + N d + N d. I ths case N,, ad (4b ca be expressed as a N a + N p + d N d, (4 wth ad p determed as above, wth the help of the Maxwell-oltzma statstcs. The electros the coducto bad are geerated ether through the ozato of door mpurtes (through the trasto from the door level to a eergy level the coducto bad, process that requres the eergy gd (equal to the ozato eergy of the door mpurty, or through the ozato of the atoms the crystal (the trasto of a electro from the valece the coducto bad, process that requres a eergy equal to g c. ecause <<, the cotrbuto of the two processes dffers as a fucto of the gd g temperature. More precsely, at low temperatures the domat process s the ozato of mpurtes, whereas at hgh temperatures the electro trastos betwee the valece ad coducto bads preval. We have extrsc coducto at low temperatures, ad trsc coducto at hgh temperatures. d

120 Statstcs of charge carrers I xtrsc coducto regme At low temperatures, at whch the coducto electros orgate from the ozato of door mpurtes, for a gve door cocetrato + N d N d, or N d >> p the eutralty codto becomes N c c N d exp N d. (44 T d + exp T Ths s a secod order equato for exp( / T, whch ca be easly solved by troducg the varables x exp[( / T], y ( N / N exp[( / T]. I terms of these d d c c d varables (44 ca be wrtte as x + x y, from whch t follows that N d c d d + T l + 8 exp. (45 4 N c T or extremely low temperatures, for whch 8 ( N d / N c exp[( c d / T] >>, (45 ca be approxmated as + T l ( N / N exp[( / T], or d d c c d c + d T N d + l N c, (46 whch reduces to c + d (47 at T K. The temperature depedece of the erm level ca be determed tag to accout that N c / ( πm T /(4π h T /. At temperatures of oly few K, whe N c < N d, shfts towards the coducto bad but, as the temperatures creases utl N c N d, the erm level taes aga the value at T K. Thus, ths temperature terval reaches a maxmum value at a temperature

121 Statstcs of charge carrers 4 4 / N πh, (48 e m / d Tmax / determed from the codto d / dt, or l( N / /. I (48 e s ot the d N c electrc charge, but the bass of the atural logarthm! The maxmum value of the erm eergy s foud to be,max + π h. (49 4 / c d / ( Tmax + N 7 / d em The erm level ca eve reach the mmum value of the coducto bad,.e., max c for a crtcal cocetrato mpurty N d, cr 4 em h π / / gd. (5 At ths crtcal cocetrato the semcoductor becomes degeerate. The temperature depedece of the erm level s represeted the fgure below, left. c gd / d g / l(/t /4 T s T v T /T A further crease temperature, whch correspods to decrease the erm level value towards saturato temperature Ts d. The temperature terval N c > N d, leads to a, utl ths value s reached for a so-called < T < Ts s called the wea ozato rego (see rego the fgure above. I ths temperature terval, from (46 t follows that

122 Statstcs of charge carrers 5 N c c N c N d d c N N c d gd exp exp exp, (5 T T T.e. / N d, ad, sce N / c T, the temperature depedece of the electro cocetrato 4 s T / exp( / T. The ozato eergy of the door mpurtes,, ca thus be gd gd determed from the slope of the / 4 l( / T f (/ T plot (see the fgure above, rght. At stll hgher temperatures, for whch 8 ( N / N exp[( / T] <<, (45 ca be approxmated as + T l{( N / N exp[( / T]}, or d d c d c c d c d + T l( N / N, (5 c d c the logarthm beg egatve sce N c >> N d temperature creases ad becomes lower tha d. Thus, the erm eergy decreases as the, level reached at the saturato temperature gd T s. (5 l[ N ( T / N ] c s d I ths temperature terval the electro cocetrato s gve by (see (5 exp N d, (54 c N c T result that shows that the door mpurtes are totally ozed, ad the electro cocetrato s depedet of temperature for T > T s. The regme s a exhaustg regme for door mpurtes ad the fgure above s dcated as rego. II Itrsc coducto regme or hgh-eough temperatures the hole cocetrato starts to crease ad becomes comparable wth the electro cocetrato. I partcular, f p >> N d the eutralty codto (4 ca be wrtte as p +. I ths regme of hgh temperatures the doors are N d completely ozed, the charge carrers orgatg from the ozato of the host semco-

123 Statstcs of charge carrers 6 ductor materal. or a odegeerate semcoductor, whch troduced the eutralty codto leads to p / d N, (55 the soluto of ths equato beg d d N N, p / d d N N. (56 ecause at hgh temperatures the host materal s the ma source of charge carrers, the expressos for electro ad hole cocetratos trsc semcoductors apply, ad the erm eergy level, determed from ] / exp[( T N c c, wth from (56, s T N N N N N T N N N T g d v c c d c d c d c exp 4 l 4 l. (57 The expresso above ca be studed two extreme stuatos:, case whch / 4 << N d N d,, (58 N d p / ad ( agreemet wth our prevous results for the totally ozed/exhausted mpurtes + c d c N N T l, (59, case whch / 4 >> N d p (6

124 Statstcs of charge carrers 7 ad (as for the trsc semcoductor c + v T N + l N v c c + v + 4 m p T l m. (6 The temperature depedece of the erm level ths rego of trsc coducto s dcated the fgure above (see rego. At hgh-eough temperatures the crease of the electro cocetrato the coducto bad orgates from electro trastos from the valece bad. The trasto temperature from the exhaustg regme of mpurtes to the rego of trsc coducto ca be determed from (58 ad (6,.e. from foud to be N d, ad s g T. (6 l[ N ( T N ( T / N ] c v d Summarzg, the temperature depedece of the electro cocetrato shows three dstct regos (see the fgures below. The logarthmc depedece of the cocetrato o the verse of the temperature ca be approxmated wth a straght le regos ad (see fgure below, left f we eglect the fluece of the factors / T ad / 4 T, respectvely, comparso wth the expoetals terms, ad the parameters ad ca be determed from the correspodg slopes. O the cotrary, rego the electro cocetrato s approxmately costat, sce the door mpurtes are exhausted. g gd l freeze-out /T /T s /T extrsc T s T trsc T

125 lectroc specfc heat lectroc specfc heat metals I metals, the electroc specfc heat per ut volume, calculated at costat volume, s defed as del Cel ( dt where the eergy per ut volume of the system of o-teractg electros s gve by el ( f ( ( ( f ( ( f ( D( d, ( V, σ V wth D / / ( (meff / 4π h (see the course o electro statstcs metals. I the ormalzed coordates / T x, / T y ad for sphercal so-eergetc surfaces the eergy per ut volume becomes el / / (m eff π h exp[( d / (m T ] + eff T π h / T / ( y / ( y T, ( ( y / where x dx α ( y (4 α exp( x y + are the erm-drac tegrals. The last equalty ( follows because (see the course o the statstcs of electros metals / (meff T / ( y. (5 π h So, tag to accout that

126 lectroc specfc heat dα ( y / dy α α ( y, (6 the electroc heat capacty ca be expressed as C el 5 (m eff T π h / / (m ( y + eff T π h / T d / dy dy dt 5 / / ( y + T ( y dy dt. (7 At low temperatures, from α + y π α( α + ( y + α + 6 y α, π T ad y / T >> we fd that / / ( y ( y π y + 5 y π T + 5 T 5π T + 5 T (8a T dy dt d dt T T π T + (8b ad so C el π T cl π T Cel, (9 T cl where the erm temperature s a parameter defed as T / ad C (/ s el the classcal electroc specfc heat. cl C el s obtaed usg the same geeral expresso (7 as above, but wth the erm-drac dstrbuto fucto replaced by the Maxwell-oltzma dstrbuto, case whch α α ( y x exp[ ( x y] dx, (a y / ( y /, (b / ( / d / dt / T /, (c Tdy / dt /. (d

127 lectroc specfc heat The equalty (c follows from (5 ad the requremet that d / dt, cosderg that the odegeerate case d ( y / dt ( d / dy( dy / dt ( dy / dt ( y. The, (d s α α cl obtaed from (c ad (8b, so that, fally, C (/. el α The rato T /T ca be see as the fracto of excted electros at temperature T, the other electros beg froze due to the Paul prcple. The value of ths rato at room temperature s typcally. The lear relato betwee the electroc specfc heat metals ad temperature s geerally expressed as C el γt, ( where γ π / s ow as the Sommerfeld costat. Although ths costat has bee derved usg the approxmato of sphercal so-eergetc surfaces, ts value remas the same for geeral surfaces. Tag to accout also the phooc cotrbuto to the specfc heat (see the lecture o phooc heat capacty, at low temperatures the specfc heat s gve by (see the fgure below C V Cel + C ph γ T + at, ( where 4 π a 5 o Θ D (

128 lectroc specfc heat 4 wth o the o cocetrato. The electroc term domates at very low temperatures, for whch γ > at,.e. for T 5 < Θ π 6 D o Θ T D. (4 I partcular, the Debye temperature ca be determed from the slope of the curve C V / T γ + at f ( T at very low temperatures, whle γ s determed from the value of ths depedece at T. The values of γ for several metals are gve the table below. Metal γ 4 (J/mol K Metal γ 4 (J/mol K Metal γ 4 (J/mol K L 7 Ag 6.6 Z 6.5 Na 7 Au 7. Al.5 K e. e 49.8 Cu 6.9 Mg.5 Co 47. Ca 7. a 7 N 7. Carrer specfc heat trsc semcoductors I a odegeerate trsc semcoductor wth sphercal so-eergetc surfaces, the eerges of the system of electros ad holes are gve by, respectvely (see ( ad (b el ( y T / T, h p T. (5 ( y / The total eergy of charge carrers s however equal to carr T + g + p T (6 sce the free electros the coducto bad have a addtoal potetal eergy of g. ecause a trsc semcoductor (6 ca be wrtte as p, wth the trsc carrer cocetrato,

129 lectroc specfc heat 5 T +, (7 carr ( g ad the carrer specfc heat s d d, (8 dt dt carr Ccarr ( T + g + whe the wea temperature depedece of the badgap s eglected. ecause g / g N c N v exp T exp, (9 T T t follows that d dt g + T T ( ad 5 g g g Ccarr ( T T T Ths expresso s vald f g T, sce otherwse the degeeracy of the system of electros ad holes must be tae to accout. rom ( t follows that at low temperatures the cotrbuto of charge carrers to the specfc heat a trsc semcoductor ca be eglected, due to the expoetal temperature depedece of. I a extrsc semcoductor the specfc heat of charge carrers ca be calculated a smlar maer. More precsely, (6 oe must troduce the correct cocetratos of free carrers all coducto regmes, ad must accout for ther specfc dstrbuto fucto ad temperature depedece. The carrer specfc heat of free electros ad holes s foud, the, to deped o both the cocetrato of door ad acceptor os ad of ther eergy levels. At low temperatures ths cotrbuto to the specfc heat s, aga, eglgble.

130 Ketcs of charge carrers solds oltzma etc equato Whe a electrc or a magetc feld s appled o a crystal, the dsplacemet of charge carrers duces trasport (or etc pheomea. The dstrbuto fucto of charge carrers wth eergy ( equlbrum s descrbed by the erm-drac fucto f (. ( + exp[( / T ] O the other had, the presece of exteral felds, the system of charge carrers s o loger equlbrum ad the correspodg dstrbuto fucto f (, r, t depeds, geeral, o spatal coordates ad tme. I a semclasscal treatmet, the umber of partcles that follow a certa trajectory s coserved the absece of scatterg processes, so that / dt. However, scatterg/collso processes of electros o phoos, mpurtes or defects the crystalle lattce are uavodable, so that the total dervatve of the dstrbuto fucto does ot vash ay more, but s equal to the varato of the dstrbuto fucto due to collsos. More precsely, df df dt f f f f + r& + &, ( t r t coll or f t f t coll v r f h f ( where v s the electro velocty the crystal ad d p / dt hd / dt s the exteral force. I a statoary state, whe the dstrbuto fucto s depedet of tme, f / t, ad, f we cosder the effect of the Loretz force e ( + v oly, we obta the etc oltzma equato

131 Ketcs of charge carrers solds e v r f ( + v h f f t coll. (4 To fd the dstrbuto fucto f (, r, t from ths equato t s ecessary to ow the collso term the rght-had-sde. Ths s a dffcult problem, whch ca be smplfed by troducg the relaxato tme dstrbuto fucto whe the exteral felds are swtched off: τ (, whch descrbes the retur to equlbrum of the f f f, (5 t τ ( coll or f f f f t exp[ t / τ ( ]. (6 ( The relaxato tme s thus the terval after whch the chage the equlbrum dstrbuto fucto decreases e tmes after the exteral felds are tured off. The troducto of the relaxato tme parameter s possble whe the collso processes are elastc,.e. whe the eergy of charge carrers s ot modfed at scatterg, ad act depedetly (there s o terferece of electro states. Moreover, the equalty τ >> h / T must be satsfed, where h / T τ c s the collso tme. Ths equalty expresses the fact that the collso tme ca be eglected,.e. the collsos are stataeous. I addto, the exteral felds must ot modfy the eergy spectrum of electros the crystal; ths codto prohbts tese magetc felds, for example, whch lead to the quatzato of electro eergy levels. The quatum ature of electros s apparet oly the collso term, through the electro quatum states that satsfy the Paul prcple. A detaled balace betwee the umber of electros the state characterzed by the wavevector ad those the state ' leads to the collso term f t coll ' P( ', f ( '[ f ( ] P(, ' f ( [ f ( ' ] ' (7 where P(, ' s the electro trasto probablty per ut tme from state to the state '. I the equlbrum state

132 Ketcs of charge carrers solds P( ', f ( '[ f ( ] P(, ' f ( [ f ( ']. (8 We cosder dstrbuto fuctos that ca be approxmated as perturbatos of, f.e. that ca be expressed as f f ( + f (, wth ( df f ( χ( << f (, (9 d where χ( a, as yet, uow vectoral fucto (t has specfc forms for dfferet scatterg processes. Uder these codtos we ca express the collso term as f t coll P(, ' f ' ( [ P(, ' f T ' ( f ( '] [ f ( '[ f ( '[ f ( ' ][ χ( f ( ] f ( ] χ( f ( [ f ( [ ' '] f ( '] f ( '] ( f oly the lear terms f are retaed ad the detty df T f ( [ f ( ] ( d s employed. The, f f ( ' χ( P(, ' τ ( f( t coll ' f ( χ( ' ', ( or / τ ( P (, '( ' / for elastc collsos, whe '. Here, are χ χ ' the projectos of, ' o the vector χ. The calculato of the relaxato tme ca be performed for dfferet scatterg mechasms, the temperature depedece of ths parameter beg geerally expressed as ' χ χ r τ ( A( T, (

133 Ketcs of charge carrers solds 4 where type: A(T s a eergy-depedet coeffcet, ad r s characterstc for each collso r / for collsos wth acoustc phoos metals, r / for the same mechasm semcoductors, r / for scatterg o optcal phoos polar semcoductors at hgh temperatures, r for scatterg o eutral mpurtes metals, ad r / for collsos wth ozed mpurtes semcoductors. If several scatterg mechasms coexst,. (4 τ ( τ ( lectrcal coductvty The electrcal coductvty a crystal s characterzed by the tesor σˆ that appears the defto of the desty of electrc curret: j σˆ, (5 or o compoets μ, ν x, y, z j μ σ μν ν. (6 ν I a sotropc sold the electrcal coductvty s a scalar parameter ad j σ. O the other had, for f ( f ( + f(, the desty of electrc curret per crystal volume ca be expressed as e e j evf ( vf( vf ( d V σ V 4, π (7 sce the equlbrum desty fucto f ( does ot brg ay cotrbuto f the sum above s performed over all postve ad egatve values (, ad hece f (, s a eve fucto of, whereas v h s a odd fucto of. The perturbato term of the equlbrum dstrbuto fucto, f (, s determed from the etc oltzma equato.

134 Ketcs of charge carrers solds 5 More precsely, f oly a electrc feld of testy s appled ad there s o temperature gradet the sample,.e. f f r, the etc oltzma equato ca be wrtte as e h f e h f( + τ ( f. (8 The secod term ca be eglected at small electrc felds, whe oly lear effects are cosdered, case whch the electrcal coductvty s depedet of the electrc feld, ad the equato above ca be solved to obta df df f ( eτ ( h eτ ( v. (9 d d rom (5, (7 ad (9 t follows that the tesor of the electrcal coductvty ca be expressed as e df v μ v ν d. ( σ μν τ ( 4π d I sotropc crystals, for a electrc feld alog the x drecto, the coductvty s scalar: e df v σ σ xx τ ( x d. ( 4π d or sphercal so-eergetc surfaces, h /, m eff v h / m, ad sphercal μ μ eff coordates wth θ the polar agle ad ϕ the azmuthal agle, x sθ cosϕ, y sθ sϕ, z cosθ, d d sθdθdϕ, ad π π max e h df 4 e h df 4 σ σ xx τ ( d s θdθ cos ϕdϕ τ ( d. 4π meff d π meff m d ( Tag to accout that 4 d / / 5 ( meff meff d / h ad that the relaxato tme depeds o eergy, such that ts statstcal average ca be defed as

135 Ketcs of charge carrers solds 6 df / τ ( d d τ (, ( df / d d the electrcal coductvty becomes e σ τ (, (4 m eff where we have used the fact that for metals the electro cocetrato ca be wrtte as (meff π h (meff π h / / / exp[( exp[( d / / / T ] + T ] + df + d / (meff d π h / df d / (5 d Smlar relatos are obtaed alog the prcpal axes for a crystal wth ellptcal soeergetc surfaces. Alteratvely, the desty of electrc curret ca be expressed as j e v eμ (6 where the moblty μ of charge carrers/electros s troduced through v μ. The mus sg dcates that the moto of electros s opposte to the drecto of the appled electrc feld. The relato betwee the electrc coductvty ad the moblty s thus σ eμ,.e. e τ ( μ. (7 m eff I asotropc crystals the moblty, as the electrc coductvty, s a tesor.