L N O Q F G. XVII Excitons From a many electron state to an electron-hole pair

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Jonas Gardner
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1 XVII Excitons 17.1 Fom a many lcton stat to an lcton-ol pai In all pvious discussions w av bn considd t valnc band and conduction on lcton stats as ignfunctions of an ffctiv singl paticl Hamiltonian. Tis appoac nglcts two impotant pnomna t xcang intaction of lctons and t lcton colation t lat of wic also oftn calld many body ffcts. In tis capt w will consid an impotant class of pnomna. W stat by a bif summay of t singl paticl stats tat w av considd ali. T f cai stats a solutions of L N M p! + V( ) + V H m0 4m0 c ( s ) p+ d y( ) = y ( ). (1) Solutions of Eq. (1) wic satisfy t loc s tom can b wittn O Q P + ik y n k( ) = Nu n k( ), () w N is a nomalization constant and u nk () is a lattic piodic solution of L NM p!! k! k! + V( ) + V V un m0 4m0 c ( s ) p+ + p+ m0 m0 4mc 0 ( s ) k ( ) H KQP. = nkunk() (3) T solution of Eq. (3) at Γ point ( k = 0 )isgivnby H( k = ) u = u. (4) 0 n0 n0 n0 An appoximat solution of (3) was obtaind as a sis xpansion ov t zon cnt stats u n0 wic a assumd known via ti ignvalus n0 wic a t band dg ngis and in tms of som ot smimpiical paamts dscibing t coupling of bands ( t Lutting paamts and t dipol matix lmnt btwn t conduction and valnc band). Fo t F G IO JP 150

2 conduction band t solution of (3) is t in lowst od appoximation givn by ik uc0 (). (5) Fo t valnc band wav function w av to account vn in lowst od fo t k vcto dpndnt mixing of HH and LH bands. Tis mixing (also including t intaction wit t distant bands) sults in t Lutting Kon loc stats + 3/ ik + ik uvk () = N Â cm k u j( ) m () j 0 (6) m =-3/ j w t sum is takn ov t HH and LH stats and t k dpndnt mixing cofficints a ignvctos of t Hamiltonian (38). As discussd pviously t total wav vcto of t full valnc band is zo. Accodingly wn an lcton wit wav vcto k is movd fom t valnc band t total wav vcto of t valnc band is k. Moov sinc at t top of t band t valnc band ffctiv mass is ngativ, moving on lcton lavs a ngativ mass dficit i.. a positiv mass and also a positiv unit cag fo t valnc band. Tfo w can associat to t many lcton valnc band missing an lcton in an ignstat + ik y v k() = u v k() a paticl wit positiv mass and cag +. W nglct t band mixing sinc it dos not bing anyting nw gading t dfinition of t ol stat. In summay w assum in t following tat t ignstats of t Hamiltonian (1) a givn and nxt w will consid manypaticl stats tat can b constuctd fom ts singl-lcton stats. T a two many-lcton stats tat a of spcial intst gading fo instanc t potoabsoption pocss. Ts stats involv (1) t ound stat of t smiconducto valnc band filld conduction band mpty and () an xcitd stat wit on lcton xcitd fom t valnc band to t conduction band. Tis stat can b dscibd as Slat dtminants accounting fo t antisymmty of Fmionic many-lcton systm. T fundamntal issu gading t following discussions is t following. Consid two diffnt ik stats of t catgoy (). In t fist stat t lcton is in a stat uc0( ) ik andanlctonstat uv0( )is not occupid (w nglct t band mixing 151

3 to kp notation compact). It can b sown tat tis many lcton stat will b mixd by t lcton lcton intaction wit anot stat of typ () cosponding to diffnt valus of t wav vctos k, k say. Tis mans tat t colatd many-lcton wav function will b a supposition of stats () cosponding to diffnt valus of k, k. It can b sown tat t matix lmnts dfining t mixing of ts stats a t sam as tos wic w obtain by assuming tat t many-lcton stat is placd by an lcton ol pai and t matix lmnts a calculatd by assuming tat ts paticls a intacting via Coulomb intaction. W sall omit t poof. T ad sould owv kp in mind tat t xpansion ov t non-intacting lcton and ols stats is noting but a compact notation fo coupling t undlying many-lcton stats. Duing t final stag wn w calculat t potoabsoption amplitud w av tun to t many-lcton pictu to obtain a consistnt singl paticl matix lmnt. 17. Expansion of xciton wav function in tms of f paticl stats Wn t Coulomb intaction is nglctd, t wav function of an lcton in t conduction band c and a ol in t valnc band v moving in a lattic witout intacting is givn by a poduct of solutions of Eq. (1) fo conduction and valnc band: Ψ ( ) = u u, ik ik ck vk. (7) W now tun t Coulomb intaction on btwn t lcton and t ol. In pincipl t w a now looking fo a solution of a total Hamiltonian wic includs t Hamiltonian (1) fo t lcton and t ol and in addition t Coulomb intaction btwn tm. T Coulomb intaction is tatd in t sam way as t inclusion of t confinmnt potntial in t cas of QW. T total Hamiltonian and is spaatd into on cosponding to t ignstats (5) and (6) and into Coulomb Hamiltonian mixing ts ignstats. Sinc t solutions (7) wn summd ov t wav vctos and conduction and valnc bands dfins a complt function spac w can wit t xciton wav function 15

4 Y Â Â ( ) = F ( k, k ) u u x, cv, cv, k, k ik ik ck vk (8) w F cv, ( k, k ) a wigs of t f cai wav functions. In Eq. (8) ik u dnots a ol-stat wit momntum k. vk Expansion (8) as to b summd ov all conduction and valnc bands and intgatd ov all wav vcto valus. Tfo it as littl pactical significanc unlss fut simplifications can b mad. W assum tat: T Coulomb intaction is not abl to caus band mixing. T intgation ov wav vcto can b limitd to vicinity of t G point. T atomic loc stats a slowly vaying functions of t wav vctos, justifying t us of G point atomic loc stats as a common facto. Ts appoximations nabl witing Eq. (8) w i i x, c v k k 0 0 c0 v0 (9) k, k Y ( ) ª u u Â F( k, k ) = u u F(, ) F(, ) = ÂF( k, k ) k, k ik ik (10) is t xciton nvlop wav function. In t sam way as fo QW nvlop wav function t loc stats av bn factoizd. H t is no mixing btwn t valnc band stats but lat w will gnaliz ts sults to includ band mixing. Tis lads to spaat solutions dscibing avy-ol and ligt-ol xcitons. In Eq. (9) w av assumd tat only fo a small subspac of t wav vcto valus clos to band dg t wigt facto Φ( k, k) is nonzo. Tis mans tat t xciton wav function (9) must spad ov a lag aa in spac. 153

5 W will now consid analytic solution of xciton wav function (10) in bulk smiconducto. Tis is a spcial cas in wic t atomic loc stats and t xciton nvlop function can b fully spaatd. In all ot cass t xciton wav function ca b obtaind by using xpansion (9) and solving t matix ignvalu poblm numically. Anot xcption to tis is xciton in a vy naow QW. Tis will b discussd lat. It can b sown tat fo wav function Eq. (9) to b a lowst od solution to total Hamiltonian t nvlop function must b solution of L N M!! m m 4p - O Q P F = EF (11) x x w and a t coodinats of lctons and ols and m and m ti band dg ffctiv masss. Eq. (11) is fomally idntical to t wav quation of ydogn atom (including t nucla motion). T spaation of t lcton-ol motion fom t motion of lcton and ol in t piodic lattic basd on t sam agumnts as t us of nvlop wav function to account fo local band dg vaiation in fo instanc QW tostuctus Spaation of t cnt of mass motion and t lativ motion: W assum isotopic masss in t following. Dfin t CM-coodinat R = m m + + m m (1) and t coodinat of lativ motion = -. (13) T nvlop function can now b factoizd into CM and lativ motion pats F(, ) Æ F( R, ) = ( R) f( ) and t wav quation is givn by g L!! O R M 4 g ( R) f( ) m p Eg ( R) f( ) (14) NM QP = 154

6 w M = m + m and m = (14) can b wittn mm. Dfining E = ER + E wav quation m + m L NM! Rg( R)! f( ) M g 4 QP = ER + E ( R) m f() p Eq. (15) is fulfilld fo all valus of R and only if L NM and L NM Rg( R) - M g( R)! O QP = E f() - - m f( ) 4p! R O QP = E O (15) (16a) (16b) T solution of Eq. (16a) is a plan wav dscibing t popagation of t cnt of mass g( R) = i KR (17) w t CM wav vcto is dtmind fom K ME = R fi E R =!! K M. (18) T wav function dscibing t lativ motion is a solution of wic can b wittn L N M! - - m 4p O Q P Eq. (16b) f() = E f(). (19) Tis is t ydognlik quation fo Wanni (o Mott) xcitons. E 4 m 1 1 =- =-E 4 ( p )! n n (0) 155

7 w 4 m E = ( p ) 4!. (1) t xciton o adius is givn by m ax = 0 a 0 () m 0 w t o adius a 0 = Å. T wav function of t gound stat is givn by f af = 1s 3 1 a pa - /. Exampl GaAs ε = 1. 9ε 0 m = m 0, m = 045. m 0 µ = m 0 w obtain a = 118 Å ( Wanni xciton ) E = ε 0 µ 136. V = 4.7 mv. (140 Å, 4. ε m 0 mv) Total xciton ngy is givn by! K E E E E E M E 1 n = g + R + = g + n (3) Using Eq. (9,16ab) t xciton wav function is givn by ikr Ψ x, u c u v ( ) 0 0 φ ( ) (4) T pobability dnsitis of s- andp-symmtic nvlop stats a dpictd in nclosd figus. It is assumd tat t ol is at = 0 and t siz of t sps indicat t pobability of finding t lcton at lattic sits aound t ol Exciton absoption 156

8 Engy consvation implis E n =!ω (5) and momntum consvation! K =! q. (6) Sinc t poton momntum is vy small in compaison wit its ngy w av fo a typical band gap (GaAs) 1.5 V!q = Js / m.tisgivs! K 005 M. activ. mv - to good appoximation only K = 0 xcitons a optically It follows tat xciton stats can absob potons only blow E g. S t dispsion cuv. Slction uls T angula momntum of t poton is tansfd to t xciton: l = 1 (7) T total angula momntum of t xciton includs t angula momntum of t loc stat and t angula momntum of t nvlop function: l = lint + l nv =+ 1 (8) W find two possibilitis to satisfy Eq. (7): Intband tansitions p s l = 1 = 0 (9) int ; l nv xciton must b catd in t s -stat. o intband s s; d s tansitions l int = 0, +, l nv =+ 1, 1 (30) 157

9 xciton must b catd in t p -stat. A mo accuat discussion would involv t us of coupld angula momnts of nvlop and loc stats. Paity consvation Paity must cang in dipol absoption and mission: T gound stat of t cystal as vn paity. Tfo t xciton stat must b odd: u c0 is vn u v0 is odd if t valnc band is p-symmtic φ is vn fo l = 04,, ; s, d, g and odd fo l = 1,,; 3 5 p, f, It is asy to not tat t s-symmtic xciton stat wit s-lik lcton and p- lik ol is asymmtic Modification of f paticl absoption by xcitons W will tun to dtaild divation lat. T xciton absoption cofficint is obtaind by making t placmnt Ncv Eg Ex n 3 3 (! ω) δd! ωi/ dπani in t f paticl absoption cofficint. H N cv (!ω) is joint dnsity of stats 3 3 and / dπa ni t squa of t ns xciton wav function at = 0. Fo continuum absoption t wav function sould b t ngy nomalizd s- symmtic ydognic continuum stat. Tis givs t Elliot fomula α 3D n= 1 ( ω) π 4π! 1 = a p cv m εv! ω 1 E 3 g ( a n) ( n) E δ! ω + + θ! ω ( Eg) Z π sin Z (31) w Z = π E /! ω E d g i and θ is t stp function. Not tat Eq. (31) is valid fo singl valnc band. 158

10 159

11 Figus a takn fom Pygambaian t al Intoduction to smicondicto optics, Pntic Hall 160

12 17. 6 Excitons in Two Dimnsions T Hamiltonian includs an additional tm V con coming fom t QW band dg confinmnt:!! H = + VCon z, z + VCoul() m m b g. (3) T QW is in xy-plan. T confinmnt potntial includs a spaat potntial box fo lctons and ols. T cnt of mass motion is spaatd now in t xy-plan only:!!! H = m z mz z Mxy X Y KJ -! m xy x +, y b V z, z Con g - 4p xy F HG I F HG H M xy and m xy a t in plan total and ducd masss M = m + m F HG I I KJ + b (33) g xy xy and = + and = mxy m mkj -. T QW av bn assumd to b so xy tin tat in t Coulomb ngy t vtical lcton-ol distanc can b placd by Æ xy. Tis is ncssay if w want t Hamiltonian (33) to av analytic solutions. Fo finit QW widt it as numical solutions only. Not also t anisotopy of ol mass. Using Hamiltonian (33) t in plan cnt of mass and and lativ motion can b spaatd: b g d xy xyi ibg jbg Ψ, = Φ, R χ z χ z (34) d i ikc xy R w Φ = uc uv n xy xy 0 0φ xy. In analogy to bulk xcitons it is assumd tat Coulomb intaction is not giving is to band mixing i.. uc0, uv0 a t band dg loc stats. T position of in plan CM is givn by 161

13 b g b g Rxy = m + m / m + m w, a t in plan position vctos. T vtical nvlop stats a dnotd by indics i and j. Fo infinit QW and j = i (lcton and ol in t sam QW subband) T ignvalus of Hamiltonian (33) a givn by E nij D! π j! π i E = Eg + + m L m L ( n 1/ ) z z z z n = 1,, 34, (35) In (35) E is absolut valu of t bulk xciton binding ngy E 4 m! = = ( ) 4πε! ma (36) It is sn tat t binding ngy of gound stat xciton is givn by ( n = 1) E D 4 m 1 = E = 4 4 ( πε )! ( 1 1/ ). (37) T nancmnt of xciton binding ngy in QW in compaison to bulk valu is wll confimd by xpimnt. A D o adius is dfind by E D D w! = m a d i a D 4πε! a = = m 3 D. T position dpndnc of dilctic constant may cang t sult givn by (37). Rlativ dilctic constants of III/V matials a aound If t bai matial as a muc low dilctic constant t fild lins lik to concntat in bai and fo a tin QW blz < ag tis will lad to nancmnt of Coulomb intaction by facto 16

14 E D ε Wll E ε ( n / ) = F H G I K J 1 ai (38) T atio of dilctic constants in bai and wll lay can b of t od of 10 wic sults in two ods of magnitud nancmnt of xciton binding ngy Optical absoption in D T Elliot fomula fo D absoption is givn by bg bg d i d i D α ( ω) p χ z χ z Φ = 0 δ! ω E D nij (39) cv i j n xy xy ijn,, bg bg = and t absoption cofficint is Fo infinit QW χi z χj z δij govnd by t lativ motion wav function. Fo D t ydognic gound stat is givn by φ d i= xy 1s xy 8 a πa xy / (40) w a is t 3D o adius. In analogy to t 3D bulk xciton cas t absoption cofficint is obtaind by making t placmnt a f j cv D g x n 3 N! w Æd E - E -! w pa n-1 / ( / ),w N 1 m = W p! cv D is t D joint dnsity of stats. A modification of Elliot fomula fo D xciton absoption in infinit QW is givn fo a paticula tansition btwn valnc and conduction subband j by a D af 4p w = a p m w c L 3 n 1 pa N MÂ = if F HG I a f j 1 d w gj D E q w gj D /! - E + E ( n- 1/ ) n - 1 KJ +! / - cos DP D O P Q (41) w 163

15 E g D j = Eg +! π ml z (4) and j (43) D = p /! w - Eg D / E T sum is takn ov all stats of t lativ motion. Fo t total absoption cofficint tis as to b summd ov t subband indx j. T baviou of D xciton absoption is dpictd in t nclosd figus. 164

16 165

17 XVIII Exciton wav functions in k-spac W will now tun to mo dtaild discussion xcitons in tms of f lcton and f ol stats. Sinc t analytic considations in pvious capt involvs sious limitations gading spcially t tatmnt of valnc band w will now cay out an analysis of xciton wav functions in k- spac. Paticula advantag of ts discussions is tat t k-spac xpansion povids a lvant stating point fo numical calculations. W fist consid a consqunc of loc s tom tat is lvant fo xcitons. Accoding to loc s tom a wav function tat is solution of lattic piodic Hamiltonian fo two paticls tansfoms in tanslation of a avais lattic as Y x ik R 1 x 1 (,) = x Y ( + R, + R) (O1) w R is any vcto in avais lattic. Tfo if wn w wit t xciton wav function as a sum of t poducts of f lcton and ol stats ik ik u u ck vk (O) ac tm in t xpansion must fulfill k + k = Kx. Fo optical xcitons K x = 0 and accodingly k =- k. W can account fo tis condition by witing t xpansion ov f paticl stats as b g af af ik i i Y x A - k k, = Â k uckuv- k = ÂA k uckuv- k. (O3) k k w t lativ distanc is = -. Not tat in t many-lcton pictu Eq. (O3) psnts a final stat of absoption in wic t lcton obital ik uvk is not occupid Calculation of tansition pobabilitis In t absoption t initial stat is considd to b a many-paticl stat wit ik all lcton stats u in t valnc band filld and all t conduction band stats ik u ck vk mpty. T final stat includs a wigtd supposition of many paticl stats wit singl xcitations fom valnc to conduction band. 166

18 T wigt factos a t xpansion cofficints Aaf k of t xciton wav function. Tfo t dipol tansition amplitud fo tansition involving xcitation of an lcton fom valnc to conduction band can b calculatd spaatly fo ac stat in t xpansion. Aft tat t patial amplituds a summd wit t cofficints A( k) as wigt factos. T tansition amplitud is calculatd btwn t atomic loc stats at G point. Fo optically activ stats K x = 0. It can b sown tat fo ac tm in t xpansion (O3) t singl paticl potoabsoption amplitud tat is obtaind wn w convt t lcton-ol stats in (O3) fist into many-lcton pictu and tn tak t matix lmnt of t lcton poton intaction btwn ts stats and t smiconducto gound stat givs t tansition amplitud 3 ik ik * if ( k) = ( k) ck p vk T A d u a u. (O4) W us uck ª uc0; uvk ª uv0 again nglcting band mixing fo bvity. Sinc t dynamic potoabsoption amplitud is takn btwn t atomic loc stats a f = 3 Tif k A k d uc 0 a pu * v 0. (O5) ( )z Finally summing ov t wav vcto valus w obtain 3 * Tif = A k d uc 0 a puv 0. (O6) k Â ( )z Compaing tis wit t wav function (O3) w can also wit t tansition amplitud in a fom Â af, Tif = a pcv A( k) = a pcvy x 0 k (O7) w t nvlop pat of t xciton wav function y x is dfind by ikx R Y x b, g= u c u v xaf 0 0 y (O8) Not tat sinc w nglctd t band mixing t xciton wav function (O8) is xactly t analytic solution obtaind ali fo bulk xciton. As a sult t 167

19 tansition amplitud is popotional to t vlop wav function fo t lativ lcton-ol distanc qual to 0. W now tun in diving t Elliot fomula. W must stat by considing t nomalization of t xciton wav function. W assum tat t f cai stats a nomalizd in t sam way as fo f cai absoption. Tn t xcitonic ffct will lad to mixing of loc stats wit diffnt k. T sum ov k is owv takn contly insid t amplitud and yilds t facto y x af 0 in t absoption cofficint. T tansition lads to final stat tat as disct ngy and tfo w av to includ ngy consvation via d E - E n -! w g x j. Futmo w do not av an incont sum ov k valus fo t absolut squa of t amplitud and accodingly w do not av t joint dnsity of stats in t absoption cofficint. W consid t ydognic cas (singl isotopic paabolic conduction and valnc band). T gound stat nvlop function at zo is givn by a f =. (O9) 1 fnlm 0 dl d m0 pan Eq. (O9) mans tat only s-lik ydognic obitals av nonzo adial pat at oigin. Insting (O9) in (O7) givs fo t non-zo amplituds T = a p i ns cv 1 πan 3 3 (O10) t absoption cofficint is obtaind by placing in t f cai absoption t squa of t absolut tansition amplitud tim t dnsity of stats by z uc * 0 a p uv 0 d 3 g(!ω) (O11) 1 Ti ns = cv Eg Ex n 3 3 a p δ! ω πan d i (O1) In Eq. (O1) it is sn tat t joint dnsity of stats cv g x n 3 3 a f j j N! w Æd E - E -! w / pa n wn w go fom f cai 168

20 absoption to xcitonic absoption. Tis givs t xciton absoption cofficint (fo xcitation to t ns xciton stat) α(! ω) I a p d i. (O13) 4π! 1 F 1 = cv ε! ω H G m v πa n 3 3 δ Eg Ex! ω KJ n T total absoption cofficint is obtaind by summing ov t quantum numb n : 4π! 1 LF 1 α(! ω) = M δ! ω ε! ω H G I O a pcv 3 3 π KJ deg Ex n i P (O14) m v n= 1 an Tis is t disct pat of t Elliot fomula. N Q Fom (O14) on can div t limit of Elliot fomula fo!ω E g as follows. W can div tis by stating fom!ω < E g W calculat t dnsity of xciton stats as n. T ngis a disct but t a vy many of tm fo unit ngy ang sing t intlvl distanc appoacs zo. Tfo w can tink of tm as a quasi continuun of stats. T ngy diffnc btwn stats two stats is givn by de d E n En 3 dn dn de = F H I K = = 3 n E (O15) Insting tis in (O14) givs t asymptotic limit 4p! 1 1 a(! w) = a p cv m v! w pa n F HG 3 3 I n = KJ a 3 1/ pe a F 1 /! w - Eg d i (O16) wa f is t f paticl absoption cofficint. It is sn tat at band dg xcitonic absoption obtains a constant valu (not tat a F includs also t joint dnsity of stat, tis will cancl t facto d!w - E g i 1 / ). Tis is in contast to f cai absoption wic is govnd at band dg by t / DOS lim d! ω Egi 1 = 0.! ω E g 169

21 T continuum pat of xciton absoption is obtaind in t sam way as t disct pat. W obtain t s-symmtic ngy nomalizd continuum stats of x t ydogn wav function y s a = 0 f. Tis absolut squa givs x y Z s a = 0f = (O17) cos Z w Z = p E / d! w - Egi. Insting tis in Eq. (O13) instad of g x j j d E E n ! w / pa n lads to 3D Elliot fomula. Simila pocdu lads to t D Elliot fomula. 18. Excitons in QW: Lutting Kon modl W now consid numical solution of xciton stats in a QW aving finit widt and stong HH-LH coupling. T conduction band f paticl stats a givn by n, k = f ( z) n ik ^ (O18) and t f cai stats fo t coupld valnc band by m, k = Â g ( k, z). (O19) v m v i k ^ In Eqs (O18-19) t functions fn, g includ bot t nvlop function and t cosponding atomic loc stat at zon cnt and n and m a t subband indics. T Coulomb intaction givn by n v (, ;, ) V z z = ε + ( z z) 1/ (O0) is wittn in t k psntationas 170

22 b g a f b g z 1 i k k V k k ; z, z = d V, z ; 0, z ( ) π = πε k k k k z z (O1) T full Hamiltonian is givn by 0 0 H = H + H + V (O) 0 w H and 0 Hv giv t E( k) atio fo t f cai stats. T xciton wav function wittn as an xpansion ov t f cai stats ψ x z = dkn, k m, k G ( k ). (O3) n m 3 nm Tisxpansionisusdtofindsolutionof Hy x x x = E y. Fo naow QW t matix lmnt (O1) fulfill appoximatly t condition n, k m, - kv m,- k n, k ª0 unlss m= m ; n= n. Tis implis tat t Coulomb intaction is not abl to mix t QW subbands. Tis implis lag ngy spacing of QW lvls and accodingly a naow QW. Wn xpansion (O3) is instd in t wav quation on obtains matix ignvalu poblm E - E ( k) + E (- k) G k = x n m nm z z 3 * d k dz dzfnbgbg z fn z m v * m v nm v z Â af b g b g b g af g -k, z g - k, z V k - k ; z, z G k. (O4) Eq. (O4) is solvd by numical diagonalization tcniqu. It poducs two diffnt st of xcitonic stats on cosponding to HH lik xcitons and anot cosponding to xcitons wit t LH nvlop function dominating. T absoption cofficint fo D xciton absoption lading to a disct D xciton stat aving E j x is givn by a gnalization of Eq. II (10) : 171

23 α D 4π ( ω)= Gnm( k) a pif δ! ω Eg + E m ωη c knm d x j i, (O5) w t tansition amplitud is a gnalization a p = a if c v v u 0 pu 0 fn() z g () z. (O6) m v Not tat Eqs. (O5-6) av bn wittn in a gnal fom allowing fo band mixing by Coulomb intaction. T sum in Eq. (6) is takn ov t fou componnts of t Lutting-Kon nvlop wav function. 17

Free carriers in materials

Free carriers in materials Lctu / F cais in matials Mtals n ~ cm -3 Smiconductos n ~ 8... 9 cm -3 Insulatos n < 8 cm -3 φ isolatd atoms a >> a B a B.59-8 cm 3 ϕ ( Zq) q atom spacing a Lctu / "Two atoms two lvls" φ a T splitting