Electrical Conduction in Narrow Energy Bands

Transcription

1 Electrical Coductio i Narrow Eergy Bads E. Marsch ad W.-H. Steeb Istitut für Theoretische Physik der Uiversität Kiel (Z. Naturforsdi. 29 a, [1974] ; received September 9, 1974) The electrical coductivity of the sigle-bad Hubbard Hamiltoia is calculated by use of the Boltzma-Trasport-Equatio. We lea o the atiferromagetic approach to this Hamiltoia. We obtai a semicoductor with a eergy gap due to the electroic correlatios ad a semicoductor to metal trasitio at the Neel temperature Ty;. 1. Itroductio I the followig paper we give a calculatio of the electrical coductivity of the eutral Hubbard model for A-B-lattices based o the approximatio give by Dichtel, Jelitto ad Koppe 1 ' 2 (DJK). As show i a forthcomig paper by Gresig ad Koppe 3, this approximatio is oly reasoable for sufficietly small VJJ. The limitig value depeds o the type of the lattice ad is about F0//< 8. The subsequet calculatios are oly valid i this regio. We wat to calculate the coductivity with the help of the Boltzma-trasport-equatio i a costat electric field. To describe impurity-scatterig we use the relaxatio time-approximatio. I aalogy to the theory of metallic coductivity of free electros we cosider the system of free "quasiparticles", which are the elemetary excitatios of the arrow bad system. We derive a expressio for the relaxatio time ad obtai a semicoductor below ad a temperature iduced semicoductor to metal trasitio at the Neel temperature Hubbard-Hamiltoia We start with the Hubbard-Hamiltoia: H = - J 2 {U+j U + V~j V ) + V 0 2 "» V + V. T>j (1) To approximate the Coulomb-iteractio by a oe particle term oe ca itroduce magetic fields fixed at each lattice site R. The most geeral form for such a operator is Wfiew = 2 u + (H-a) u (2) where cf meas the vector build up of the Paulispi-matrices ad u the colum vector (u, v). Calculatig the free eergy of the Hubbard-model with the Bogoljubov iequality usig the test operator Wiest = - / 2 (U+j U + V+j V ) - 2 ( H ' 0 ) U j (3) the H should be variatioal parameters to miimize the free eergy fuctioal. But this program is too difficult to perform ad therefore DJK choose Afield = 2 C zlexp{ + /Q-R } A exp { i Q Rw} \ ^ j"*' (4) After Fourier-trasformatio oe obtais the grad caoical test Hamiltoia: TI +.(-ßi-Z + -A \j uk Htest= 2 \ u k Vk+Q).. f.,.,, ^ k \ -A + G + e(k + Q) j \ vk + (i (5) Q, u, s ad A are real variatioal parameters. A /^-depedet rotatio i spi space makes Tatest diagoal: (ü k\( cos{ak/2) -si(aa./2)\ / uk \ U J \si(aä/2) cos(a*/2)) \vk + J Reprit requests to Dipl.-Phys. E. Marsch, Istitut für Theoretische Physik der Uiversität Kiel, D-2300 Kiel, Olshausestraße with cos 2 (a,/2)= If e~(k)-c_ \ si 2 (aä./2) 2[ ~ + > J 1 ad e ± (k) = i[ ±e(k + Q)]. The we fid: ^test = 2 [^i (k) ük + ük + E, (k) vk + vk]. (8) The eergy spectrum is: Ei*(k) = - fi + (k) ± V[e- (k) -':] 2 + J 2. (9)

2 1656 E. Marsch ad W.-H. Steeb Electrical Coductio i Narrow Eergy Bads It ca be show that Q = I/a {a,, 71), goig from a miimum to the maximum value of e{k), for the s. c. lattice leads to a relative miimum of the free eergy. DJK showed furthermore, that the atiferromagetic orderig always has a lower free eergy tha the ferromagetic (A = 0) oe i their approach. The chemical potetial ad magetic field are coected with ft ad through the equatios: C = H + V0sz = N p/tv 1 T. with... (10) u = //oiiem. - 2 I 0 sz = Sz/N I the eutral case = 1, which we are iterested i, the effective chemical potetial tt is idetically zero ad without a magetic field we obtai the eergies: I the followig we have the: FL2= ± V ~{k) + A 2. (11) cos 2 (a,/2) = 1 } si 2 (a,,/2) 2 I V^W +A 2 j' K 1 3. Coductivity of the Neutral Hubbard-model Now we will calculate the electrical coductivity. Electroic motio obeys the laws: V(k)= V*e(Ä), 3(^fc) =ee. (13), (14) I cubic lattices the coductivity tesor ouß reduces to a scalar ad with some simplificatios because of symmetry we obtai from the Boltzma equatio the well-kow result i relaxatio time approximatio: (15) Here the idex / umerates spi states or odegeerate bads. meas the equilibrium distributio fuctio i the absece of the electrical field. Our problem is ow to calculate the relaxatio time for the quasiparticles. For this purpose we add to the Hamiltoia oe impurity level of eergy Ej measured from the ceter of the bad [1JN T k = 0] for example at the origi of the lattice: ^impurity = Ej (U 0 + U 0 + V Q + v 0 ). (16) Trasformig this ito &-space ad writig i quasipartiele operators we have T-f _ ^' S -''impurity»7 /V kk' + Ej y ' N A COS ^ CO; a*' 2 ak. ak' cos si ak. «// + si Sm 2 2. ak a*' si - cos (U/- + life' + Vk + Vk') (Ük + Vk' - Vk + Uk'). (17) Our approximative Hamiltoia was Htest=2[ i(a;)ä& + ü* +Es(k)vk + vk]. (18) k Uder the ifluece of the impurity the system of quasiparticles as a whole ca make trasitios from a eergy level E Em with the probability (first order perturbatio theory) : W(~+m) =i2jz/h)\(\mimptity\m)\ 2 d(e-em). The state-fuctio y has the form T» = I W I W 0 > with A,B^ kea keß " J 0)" is the quasipartiele vacuum. (19) But oly those matrix elemets of Ffjmpurjty where y' ad y'm differ i oe ^-vector are ozero; therefore Fermi's "Golde-rule"' holds for the idividual scatterig process of oe quasipartiele ad w r e obtai W(k, k') = (2 7l/h) I (<pk' I ^impurity ^fc) P d[e,'(k')-ex(k)]. (20) The first term of 3~fimpurity describes itra-bad scatterig, the secod iter-bad-scatterig; oly terms of the first kid cotribute because the eergy gap of 2A makes El(k')=E2{k) impossible. So we get for both bads: v-k a k' COS - COS - + si - si d[e(k')-e(k)] (21) /here E(k) =+y 2 {k) A 2.

3 1657 E. Marsch ad W.-H. Steeb Electrical Coductio i Narrow Eergy Bads Usig: the relatio <3 [E(k') E(k)] E (k) {<5 [ (&') + <5[e(*') +]} (22) ad assumig Nj impurity sites, we obtai fially for the trasitio probability per uit time (Ej meas the mea value of all impurity eergy levels) : W(k,k')= 2 * ^E/ E(k) d[e(k')-] +,E(k) ow d[e(k')+ (k)]. (23) W{k,k') has the properties [because k) ] : W(k, k') = W( k, k') W (k, k') ad is symmetric W (k, k') W{k', k) ; therefore it is possible to itroduce a time of relaxatio, which is well-kow to be: r(k) =r(-k) = 1/2 W(k,k'). (24) We calculate r(k) (Vc = Volume of the elemetary cell of the lattice) 1 = 2a Nj, V, E(kJ s k' d[e{k') e(k') t (k) h N J j d (2 a) 3 -] ( With the desity of states has: r(e) K 9(e) = 79~V {2jl) 0 f i.b.-z. Vkö[e-] 2a (Nj N A d[e(k') + (/c)] (25) (26) (27) It should be metioed, that the relaxatio time approximatio does't cotradict to particle coservatio i first order of the perturbatio; we used the Asatz: fk^f -rx(k)v,(k)-ee 3 h kl 3E,(k) ' Summig over the whole Brilloui-zoe oe gets: = \ 2 fki = \j 2 f/j = JV kx i> A-;. because it is - I r M v M ) 3 ^ = 0. (28) The equilibrium distributio fuctios are give by: ßi,2 = l/[exp{± VeHk) + Ä 2 } + 1]. (29) The velocities are: v(k)\ = \\ 1 ye 2 (k) +A 2 h V*e(*) (30) for both bads; therefore w T e have total hole-particle symmetry ad the coductivities for both bads are equal. At this poit we are able to calculate the coductivity: f rd Sk i\\7k\ \j(m\ {2)*NjEj*2h J [Ve*(k)+A 2 ] 2 {k)\ve 2 {k) +A 2 cosh 2 ( Ve 2 {k) + A 2 ) g[e{k)] [> 2 (k) + 2 A 2 ] (31) Fially we have / O = O / dt s h(e)ß o Ve 2 + A 2 ( 2 + 2A 2 ) g (E) cosh 2 ( VE 2 + A 2 (32)

4 1658 E. Marsch ad W.-H. Steeb Electrical Coductio i Narrow Eergy Bads with A = 2 J 8/ B = 3 1 for s.c for b.c.c. latti ad (a = lattice costat) : A e 2 ti h a K h(e) = (2 7l) 3 d 3 kö Nj A EJ V**(*)l f 1 (a'a) 2 A, ß = 1 /ä:b T are measured i uits of A. The fuctio h(e) is the mea value of the ordiary bad velocity over the eergy surface!a which was calculated umerically. The factor e 2 /(2 Tiha) =e, has the dimesio of a coductivity ad about the value «s 10 6 i2 _1 cm _1. We eed the gap equatio too (DJK) : Fig. 1. Eergy gap A as a fuctio of temperature for s.c. lattice. 1: V 0/J=6; 2: VJJ=4.5; 3: F0// = 3. 1 = F, f dc ]/e 2 + A 2 * J ß (34) log(ct/cq We ca ow calculate o as a fuctio of temperature ad for fixed temperature as a fuctio of the ratio VJJ. This will be doe i the last chapter. 4. Discussio of Results For very high temperatures ad A = 0 the formula (32) would lead to o~l/f. This is of course a cosequece of the fact, that a oe-bad model becomes uphysical at high temperatures. As soo as k T is much larger tha the width of the bad, all states are occupied with equal probability, ad the bad behaves as if fully occupied. For low temperatures, (32) leads to Fig. 2. Coductivity as a fuctio of temperature for s.c. lattice. 1: VJJ=6; 2: V QfJ=4.5; 3: V 0/J = 3; 4: A= 0. o-d/vexvi-ajkzt} (35) w here A0 is the gap at T = 0. Results of a detailed calculatio are show i Figs. 1 ad 2 for the s.c. lattice (the b.c.c. lattice shows qualitatively the same behaviour). As ca be see, the disappearace of the gap causes a very sudde icrease of the coductivity. I Fig. 3, the coductivity has bee plotted as a fuctio of A for ß fixed. The depedece is almost expoetial. As it should be, for F/ 0 the coductivity teds to ifiity ad just so for vaishig impurity cocetratio Nj/N-^-0. Fig. 3. Coductivity as a fuctio of the eergy gap for some temperature values (/? = //ä:b T) ad for the s.c. lattice. 1: ß =0.5; 2:^=2.5; 3: ß = 5; 4: =10.

5 1659 E. Marsch ad W.-H. Steeb Electrical Coductio i Narrow Eergy Bads A remark is ecessary about the rage of temperature. Oly for low temperatures the resistace is domiated by impurities; the high (room) temperature depedece of the coductivity is due to electro-phoo iteractio. This fact is ot cosidered i our model. Ackowledgemet We would like to thak Professor Koppe for stimulatig ideas ad kid iterest i this work. Thaks are due to the Rechezetrum der Uiversität Kiel, where the umerical calculatios have bee performed. 1 K. Dichtel, R. J. Jelitto, ad H. Koppe, Z. Physik 246, 248 [1971]. 2 K. Dichtel, R. J. Jelitto, ad H. Koppe, Z. Physik [1972]. D. Gresig ad H. Koppe, Thermodyamics of the Hubbard Model i the Case of Strog Couplig (to be published). E. Marsch, Diplomarbeit, Kiel R. J. Jelitto, J. Phys. Chem. Solids 30, 609 [1969].