HW 5 SSP 601-2017. here is very relistic clcultion which uses the concepts of lttice, reciprocl spce, Brillouin zone nd tight-binding pproximtion. Go over the solution nd fill up every step nd every detil in the provided solution. No computer lgebr clcultion i s llowed. Problem 3: Bnd Structure of YB 2 Cu 3 O 7 377 Problem 3: Bnd Structure of YB 2 Cu 3 O 7 2 In 1986, fmily of oxides ws found tht proved to be superconducting with much higher criticl tempertures thn metls nd lloys. The compounds in this oxide fmily ll contin plnes of copper nd oxygen with Cu toms t the nodes of squre lttice (see Fig. P3.1). Some members of this fmily lso contin copper oxygen chins. An exmple is YB 2 Cu 3 O 7.TherelstructureisshowninFig.P3.2 (left). The simplifiction used in this problem is shown in Fig. P3.2 (right). The primitive cell of this compound contins two copper oxygen plnes (levels 2 nd 3) nd one fmily of chins, e.g., level 1. The im here is to investigte the bnd structure of YB 2 Cu 3 O 7 in simplified wy using the tight-binding pproximtion (LCAO). 3.1 exmines n isolted copper oxygen chin. 3.2 then looks t n isolted copper oxygen plne nd two Fig. P3.1 CuO 2 squre lttice Cu y O x Chins CuO Level 1 B Chins YB 2 Cu 3 O7 Level 2 Plnes CuO 2 c b Y O(2) B Cu(2) O(3) O(1) Cu(1) O(4) Plnes z y x Level 3 Level 4 Fig. P3.2 Rel structure of YBCO 7 (left) nd simplified structure used in this problem (right) 2 This problem hs been designed with C. Hermnn nd T. Jolicœur.
378 13 Problem Set coupled plnes, while 3.3 dels with the coupling between plne nd the nerest chins. Finlly, 3.4 compres the results obtined in this wy with more detiled clcultion nd experimentl dt concerning YB 2 Cu 3 O 7. Throughout this exercise, we only tke into ccount those orbitls with energy levels close to the Fermi level of the solid, i.e., the 3d orbitls of copper nd the 2p orbitls of oxygen. To simplify, we ssume tht, within the bsis, these tomic orbitls combine to form one tomic orbitl per Brvis lttice point. This orbitl will be denoted by φ 1 (r) forthechins(3.1) ndφ 2 (r) fortheplnes(3.2 nd 3.3). These orbitls re ssumed to be isotropic, rel, nd normlised. We pply the LCAO method to these orbitls. We lso ssume tht the orbitls of two neighbouring sites brely overlp, nd tke them to be orthogonl for simplicity. There is no need to write down the relevnt Hmiltonins explicitly. All the necessry mtrix elements will be given. Note: Questions 4 nd 7 in 3.2 re not essentil for tckling the neighbouring questions. 3.1: Isolted Copper Oxygen Chin Consider n isolted chin of copper nd oxygen toms. Let be the Cu Cu distnce nd y the unit vector long the chin. 1. Using Fig. P3.3, specifythebrvisltticendbsisofcopper oxygenchin. 2. Wht is the ssocited reciprocl lttice? Specify the corresponding first Brillouin zone. 3. Give the form of the Bloch functions in the tight-binding pproximtion.letk be the corresponding wve vector. 4. Considering only nerest neighbours, clculte the dispersion reltion E C (k). Express the result in terms of the mtrix elements EC 0 = d 3 rφ 1 (r)ĥ C φ 1 (r), V 2chin = V = d 3 rφ 1 (r)ĥ C φ 1 (r + y), y x Fig. P3.3 Copper oxygen chins
Problem 3: Bnd Structure of YB 2 Cu 3 O 7 379 where y is the unit vector long the chin nd Ĥ C is the Hmiltonin of n electron in the chin. 5. Plot the dispersion reltion, ssuming V > 0. If the electron occuption of this bnd is one electron per unit cell, wht is the shpe of the Fermi surfce? Wht hppens if the electron occuption is very low? Give the effective mss m of the electrons in tht cse. 3.2: Isolted Copper Oxygen Plne Consider n isolted copper oxygen plne: 1. From Fig. P3.1, specifythebrvisltticendbsisofcopper oxygenplne. Wht is the ssocited reciprocl lttice. Specify lso the corresponding first Brillouin zone. As in 3.1: consideroneorbitlperbsis,denotedφ 2 (r). Let Ĥ P be the Hmiltonin of n electron in the plne. 2. Give the form of the Bloch functions in the tight-binding pproximtion.letkbe the corresponding wve vector. Clculte the dispersion reltion E P (k), considering only nerest neighbours. Use the mtrix elements EP 0 = d 3 rφ 2 (r)ĥ P φ 2 (r) nd V P = d 3 rφ 2 (r)ĥ P φ 2 (r + x) = d 3 rφ 2 (r)ĥ P φ 2 (r + y), where x,y re unit vectors long the x nd y xes. 3. If the electron occuption of this bnd is one electron per unit cell, wht is the shpe of the Fermi surfce? If the number of electrons is 1 ± δ with δ 1, sketch the Fermi surfce in the Brillouin zone. Wht hppens if the bnd is lmost empty? Now, nd only for this question, consider second nerest neighbours in the plne using the mtrix element V P = d 3 rφ 2 (r)ĥ P φ 2 (r ± x± y), where the nottion ± indictes tht the four mtrix elements re equl by symmetry. Assume tht V P > 0.
380 13 Problem Set 4. For one electron per unit cell, wht is the new vlue of the energy for those vlues of k tht corresponded to the Fermi surfce in the lst question? Deduce the pproximte position of the new Fermi surfce in the region 0, /. In the compound YB 2 Cu 3 O 7,theplnesreinfctcoupledintopirs[levels2nd 3recoupledinFig.P3.2 (right)]. We tret the cse of n isolted double plne. This complictes the squre lttice bsis considered bove. We now consider tht there re two orbitls per site of this bsis, one for ech plne. These re φ 2 (r)ndφ 2 (r+ cz), where z is the vector joining the two plnes. Use the following LCAO function: ψ k (r) = [ ] exp(ik R j ) A k φ 2 (r R j ) + B k φ 2 (r + cz R j ), j where the sites R j run over the points of the Brvis lttice of the plne z = 0 with j s index. The coefficients A k nd B k re djustble prmeters. 5. Show tht the LCAO function stisfies Bloch s theorem. The Hmiltonin of n electron in the double plne is denoted by Ĥ DP. Project Schrödinger s eqution Ĥ DP Ψ k = E(k) Ψ k onto the functions φ 2 (r) nd φ 2 (r + cz). Simplify the problem by neglecting overlps between distinct sites, i.e., ssume tht the orbitls φ 2 (r)ndφ 2 (r + cz)reorthogonl. 6. Derive homogeneous liner system of equtions in which the unknowns re the coefficients A k nd B k. Specify the coefficients of the system in the form of mtrix elements of Ĥ DP.Assumethtthemtrixelementsinvolvingonlytheorbitlsof givenplnerethesmesthoseof Ĥ P introduced in question 2. For mtrix elements involving orbitls from both plnes, keep only the one involving the sme site, viz., d 3 rφ 2 (r R n )Ĥ DP φ 2 (r + cz R n ) = T. 7. How mny bnds re there? Give their dispersion E(k) s function of E P (k) nd T. Plot the result for 0 = /. Assuming T smll, nd when there re two electrons per unit cell, one from ech plne, locte the Fermi energy on the bnd digrm E(k) for the given direction. Deduce the shpe of the Fermi surfce in the region 0, /, with the help of question 3. 3.3: Chin nd Plne In YB 2 Cu 3 O 7,thererelsocopper oxygenchins,sdiscussedin3.1 nd shown in Fig. P3.2 (right). Consider now plne coupled with lttice of chins. To describe the combined CuO 2 plne in level 2 nd Cu O chins in level 1, use the sme Brvis lttice s in 3.2, butconsidertwoorbitlsperprimitivecell,viz.,φ 1 (r) of the chin nd φ 2 (r)ofthessocitedplne.
Problem 3: Bnd Structure of YB 2 Cu 3 O 7 381 1. Strt by exmining the lttice of Cu O chins in level 1 using the tight-binding pproximtion. These chins re brely coupled together in YB 2 Cu 3 O 7 nd we my completely neglect mtrix elements involving different chins. Using 3.1, give the dispersion E CL (k). Plotthisfunctionfork = (0, ) nd / +/. Plot the constnt energy curves of E CL (k) in the region /, +/. With one electron per unit cell in this bnd, sketch the Fermi surfce. 2. Now investigte the coupled problem of the plne nd the chins using the wve function Ψ k (r) = [ ] exp(ik R j ) C k φ 1 (r R j ) + D k φ 2 (r + cz R j ). j The one-electron Hmiltonin is now Ĥ PC.Assumethtthemtrixelements within given plne re the sme s those of Ĥ P (see question 2 of 3.2: Isolted Copper Oxygen Plne) ndtht,inthechinlttice,theyrethesmes those of Ĥ C (see question 4 of 3.1 nd question 1 of 3.3). Using the rguments in 3.2, show tht there re two bnds E ± (k) nd express them in terms of E P (k), E CL (k), ndthemtrixelement T = d 3 rφ 1 (r R n )Ĥ PC φ 2 (r + cz R n ). 3. Assume tht T = 0 nd tht there re two electrons per unit cell, one per unit cell of the plne nd one per unit cell of the chin lttice. Assume lso tht E 0 P = E0 C. Plot the Fermi surfce of the ensemble in the squre 0, +/. 4. Now consider the cse T = 0, still with E 0 P = E0 C.Thiscouplingisonlyimportnt where the Fermi surfces of the plne nd the chins used to intersect. By exmining the neighbourhood of the point k = (/2,/2), mkequlittive sketch of the Fermi surfce for the electron occuption of question 3 in 3.3: Chin nd Plne, then for n occuption number close to this. Wht simple remrk cn be mde bout the wve functions t the edge of the region 0, +/? This my be importnt for explining the Josephson effect in YB 2 Cu 3 O 7 [see Combescot, R., Leyrons, X.: Phys. Rev. Lett. 75, 3732 (1995)]. 3.4: Relistic Models of YB 2 Cu 3 O 7 1. In fct the structure of YB 2 Cu 3 O 7 comprises two wekly coupled plnes (levels 2nd3)ndchinlttice(level1coupledto2)intheprimitivecell.Ech isolted CuO 2 plne is described by the dispersion obtined in question 7 of 3.2. This system is more wekly coupled to the chins by T T in question 4 of 3.3. Figure P3.4 (left) shows the results of more detiled clcultion. Give qulittive interprettion of the different prts of the Fermi surfce.
382 13 Problem Set 0,, 0,, photoemission positrons (0,0 ),0 (0,0),0 Fig. P3.4 Left: Fermi surfce obtined from detiled clcultion, dpted from results of Yu, J., et l.: Phys. Lett. A 122,203(1987)Right:Dttkenbyphotoemissionndpositronnnihiltion dpted from Pickett, W.E., et l.: Science 255, 46(1992) 2. In Fig. P3.4 (right), blck points represent photoemission mesurements of the bnd structure nd white points correspond to results obtined by nother technique using positron nnihiltion. Wht prts of the bnd structure re these techniques ble to revel?
Problem 3: Bnd Structure of YB 2 Cu 3 O 7 383 Solution Isolted Copper Oxygen Chin 1. The Brvis lttice of chin is the set of points R n = ny,wheren Z nd y is unitvectorlongthechinxis.thebsisthencomprisesonecoppertomt R 0 = 0 nd one oxygen tom t y/2. 2. The reciprocl lttice is then K p = (2/)py, where p Z. The first Brillouin zone is the intervl [ /,+/]. 3. Using the nottion dopted in this book, we hve where N n is the number of unit cells. 4. We obtin ψ k (r) = 1 e ikl φ 1 (r ly), Nn l E C (k) = EC 0 2V cosk. 5. Restricting to the first Brillouin zone, this reltion is shown in Fig. P3.5. The function E C (k)issymmetricunder k k, E C(k) 2E 0 C E C(k). So hlf the sttes re in the sub-intervl [ /2, +/2]. One electron per unit cell corresponds to hlf-filled bnd, nd the Fermi energy is then E 0 C.TheFermi surfce reduces to the two points k =+/2 nd k = /2.Iftheoccuption is very low, only the bottom of the bnd close to k 0isoccupied.Then E C (k) E 0 C 2V ( 1 1 2 k2 2 ), whence m = h 2 2V 2. V E C (k) E 0 C + k Fig. P3.5 Dispersion reltion of the chin obtined in question 4, identicl to the one given in Chp. 1 V
384 13 Problem Set Isolted Copper Oxygen Plne 1. The Brvis lttice is squre: R n,m = nx+my, n,m Z.Thebsiscnbetken s CuO 2 with Cu t (0,0) nd two oxygens t x/2 ndy/2. The reciprocl lttice is squre: K p,q = (2/)px+ (2/)qy, p,q Z. ThefirstBrillouinzone is then the squre ([ ] ) ([,+ long ] ),+ long. 2. We hve ψ plne k (r) = 1 e ik R j φ 2 (r R j ), Nn where N n is the number of unit cells nd j indexes the sites of the Brvis lttice. Further, E P (k) ψ plne k Ĥ P ψ plne k = 1 e ik (R j R i ) d 3 rφ 2 (r R i )Ĥ P φ 2 (r R j ). N n i,j The digonl terms re ll equl to E 0 P. A given site R i hs four nerest neighbours: R i + x, R i x, R i + y, ndr i y. Forjustthesecses,wehve nonzero mtrix element equl to V P.Hence, nd E P (k) = E 0 P V P( e ik x + e ik x + e ik y + e ik y), j E P (k) = E 0 P 2V P(cos + cos ). 3. The constnt energy terms hve the form We hve the symmetry cos + cos = C.,, C C. The hlf-filled stte thus corresponds to C = 0, which reduces to + =. The Fermi surfce comprises four stright-line segments [see Fig. P3.6 (center)]. The generl shpe of the constnt energy curves hs been given in Chp. 3, Fig. 3.7. Ifthenumberofelectronsis1+ δ, theimmeditelydjcentsttes
Problem 3: Bnd Structure of YB 2 Cu 3 O 7 385 + + + + + + Fig. P3.6 Fermi surfce clcultions for 2D squre lttice with nerest neighbour hopping, for one electron per unit cell (center), (1 + δ)electronperunitcell(right)ndfor(1 δ)electronper unit cell (left) re filled [see Fig. P3.6 (right)] nd if the number is 1 δ the surfce becomes connected [see Fig. P3.6 (left)]. If the bnd is lmost empty, k 0ndE P (k)hs lmost circulr level curves s seen in Fig. 3.7. 4. In the clcultion for question 2, for ech site R i,wetkeintoccountsecond neighbours, viz., R i ±x±y.thisproducesnextrtermin ψ plne k Ĥ P ψ plne k equl to [ e ik (x+y) + e ik (x y) + e ik (x y) + e ik (x+y)], nd then +V P E P (k) = E0 P + 2V P(cos + cos ) + 4V P coscos. In the region 0, /, thefermisurfceofquestion3is + = /. With second nerest neighbours, the energy on this stright line is equl to E P = E0 P 4V P cos2.
386 13 Problem Set Fig. P3.7 Fermi surfce tking into ccount second nerest neighbour contributions E = E 0 P E F (0,0 ) When V P > 0, this energy remins unchnged t the center of the squre, but is lwys reduced for the other vlues of.theconstntenergycurveep 0 thus moves towrd the corner of the Brillouin zone. For the hlf-filled bnd, the new Fermi surfce tkes the form shown in Fig. P3.7.(NotethtitdividestheBrillouinzone into two equl res.) 5. If r is trnslted to r + R 0 for R 0 in the Brvis lttice, the dummy sum chnges by R j R j + R 0, 6. First project onto φ 2 (r)toobtin ψ k (r + R 0 ) = e ik R 0 ψ k (r). φ 2 E(k) ψ k = E(k)A k, since we neglect non-locl overlps. Then we hve φ 2 Ĥ DP ψ k = A k e ik R j d 3 rφ 2 (r)ĥ DP φ 2 (r R j ) j +B k j e ik R j d 3 rφ 2 (r)ĥ DP φ 2 (r + cz R j ). The first sum only involves mtrix elements within plne. These re equl to the mtrix elements of H P.ThisfirstsumisthusequltoE P (k) sobtinedin question 2. In the second sum, only the term R j = 0 is nonzero nd equl to T. Therefore, φ 2 Ĥ DP ψ k = E P (k)a k + TB k = E(k)A k. As the two plnes enter the expression for ψ k (r) insymmetricwy,theprojection onto φ 2 (r + cz) ledsinsimilrmnnerto φ2 (r + cz) E(k) ψ k = TAk + E P (k)b k = E(k)B k.
Problem 3: Bnd Structure of YB 2 Cu 3 O 7 387 E(k) E + E + E 0 P = E E (0,0 ) Fig. P3.8 Two bnds obtined in the tight-binding pproximtion for bilyer of CuO 2 plnes. Dispersion reltion of the two bnds obtined in the direction = (left). Fermi surfce obtined for one electron per CuO 2 unit cell per plne (right) The coefficients A k nd B k thus stisfy the system { Ak E P (k) + B k T = A k E(k) A k T + B k E P (k) = B k E(k). 7. There re eigensttes if the determinnt of this system is zero: E P(k) E(k) T T E P (k) E(k) = 0. There re two solutions, hence two bnds of dispersion E ± (k) = E P (k) ± T. Along =, E P (k) = E 0 P 4V P cos. The two bnds E ± re relted to one nother by trnsltion [see Fig. P3.8 (left)]. By symmetry, the Fermi energy for one electron per plne remins equl to E 0 P.ThisFermienergycutsthesurfce E + t level curve of question 3 with occuption 1 δ,butcutsthesurfcee t levelcurvewithoccuption1+ δ. TheFermisurfcethuscomprisestworcs [Fig. P3.8 (right)]. Chin nd Plne 1. Use the LCAO function ψk CL (r) = 1 e ik R j φ 1 (r R j ). NCL j
388 13 Problem Set E CL (0, ) / / + Fig. P3.9 Bnd structure for lttice of chins: Dispersion reltion E CL (k) (left) ndconstnt energy level curves in the region /, +/ (right) Only the mtrix elements of Ĥ C come in when clculting E CL (k) (see question 3of3.1), becuse there is no coupling between chins. As consequence, E CL (k) = E C ( ), s shown in Fig. P3.9 (left). The constnt energy curves of E CL (, )rethus stright lines t fixed [see Fig. P3.9 (right)]. According to question 5 of 3.1, with one electron per unit cell, the Fermi surfce thus comprises two stright lines =+/2 nd = /2, sshowninfig.p3.10. 2. Project Ĥ PC Ψ k = E(k) Ψ k onto φ 1 (r)ndφ 2 (r+cz). The clcultion is similr to the one in question 6 of 3.2. Forφ 1 (r), we obtin nd with φ 2 (r + cz), we obtin C k E CL (k) + D k T = C k E(k), C k T + D k E P (k) = D k E(k). +/2 / Fig. P3.10 Fermi surfce of the chins for one electron per unit cell /2
Problem 3: Bnd Structure of YB 2 Cu 3 O 7 389 Setting the determinnt equl to zero, this yields two bnds: E ± (k) = 1 { [EP E P (k) + E CL (k) ± (k) E CL (k) ] } 2 + 4T 2. 2 3. Since E 0 P = E0 C,theplnendchinbndsrefilledeqully.TheFermisurfce then comprises the surfces of these two ensembles [see Fig. P3.11 (left)]. 4. The surfces intersect in (/2,/2)whenT = 0. If T = 0, the result of question 2 bove shows tht the equlity E P = E CL no longer holds for E + nd E :the levels repel one nother. Level crossing disppers nd we obtin the qulittive result shown in Fig. P3.11 (right). At the edge of the Brillouin zone, the Bloch sttes correspond to sttes completely within the plne or completely within the chins. / +/2 (0, 0 ) (0,0) / Fig. P3.11 Chin nd plne Fermi surfce for different hopping T between chin nd plne (left) T = 0, (right) T = 0 Relistic Models of YB 2 Cu 3 O 7 1. A system of two plnes with V P = 0willhveFermisurfcemdeupoftwo segments in [0,/] 2,sshowninFig.P3.12 (left). If there is chin lttice s +/ Double plne Double plne (0,0) +/ Chin Fig. P3.12 Fermi surfces for the bilyer of CuO 2 plnes (left) ndforthebilyercoupledto CuO chin (right)
390 13 Problem Set well, it will lso give brnch tht will void crossing the brnches of the double plne by the hybridistion phenomenon of questions 3 nd 4 of 3.3. Thisgives Fig. P3.12 (right), which grees with Fig. P3.4 (left). 2. The dispersion points due to photoemission [see Fig. P3.4 (right)] coincide with contributions coming from double plnes. In contrst, positron nnihiltion sees the chin contribution. A single technique ws not enough initilly to investigte the whole Fermi surfce. It hs since been viewed by higher resolution ARPES experiments, nd mtches the results of full clcultion illustrted in Fig. P3.13. Fig. P3.13 Full clcultion of the 3D bnd structure. Surfces due to double plnes re very close together, while those due to chins re widely spced. The Fermi surfce is lmost cylindricl, becuse the hopping integrls between cells re very smll in the c direction. Imge courtesy of O. Andersen nd I. Mzin from results published in Andersen, O.K., Liechtenstein, A.I., Rodriguez, O., Mzin, I.I., Jepsen, O., Antropov, V.P., Gunnrsson, O., Gopln, S.: Physic C 185 189, 147 155(1991)