Electronic Hmiltonins - prt I In this section we discuss how to del with electronic Hmiltonins, i.e. the first step in our sequence of pproimtions. Becuse we consider the ions to be frozen in their equilibrium positions, the electrons re moving in their resulting periodic potentil, nd lso eperience electron-electron interction. We will first ignore the electron-electron interctions, i.e. set them to zero. Although this my seem very strnge things to do, it is ctully firly good pproimtion in some cses. We will revisit this issue towrd the end of the section, but roughly speing, wht hppens is tht the repulsion between two electrons is nerly compensted (screened) by the ttrction with the positive chrges present in the sme region. This compenstion wors very well for electrons tht re fr prt, less so if they re sptilly close. evertheless, we will begin by ssuming tht we cn completely ignore electron-electron interctions. This llows us to study ectly some models of non-intercting electrons moving in periodic potentils, to understnd the typicl spects of electronic bnd-structures. Then, we will consider the role of (we) electron-electron repulsions using the Hrtree-Foc pproimtion, nd we will revisit screening using the Rndom Phse Approimtion. on-intercting electrons in periodic potentil For simplicity, let s ssume tht we put together mny Li toms ( s 2 2s electronic structure) in crystl, nd see how we would model this if we cn ignore electron-electron interctions. From wht we hve discussed, we epect the 2s electrons to be the vlence electrons, so we re considering vlence electron per Li + ion. For simplicity, let me consider D model of chin of such ions, plced equidistntly from ech other (of course, in relity Li would crystllize s 3D crystl, but you should be ble to del with 3D lttices if you understnd how the D cse wors). We cn lbel these with n inde n =,...,, where we will let, in the end. Let be the lttice constnt, so the position of the n th ion is R n = n. To describe the vlence electrons, we introduce cretion nd nnihiltion opertors c nσ,c nσ for n electron t site n with spin σ. To be more precise, c nσ 0 is the one-electron stte for which,σ c nσ 0 = δ σ,σ φ 2s ( R ) = δ σ,σ n, i.e. in fct the electron is creted into the 2s orbitl of the n th ion. But becuse this is the only ind of orbitl tht we include in our model, we cn sy tht the electron is t site n without creting confusion. This 2s orbitl hs some chrcteristic lenghtscle B (its Bohr rdius) over which it spreds. Our pproimtion to eep only these orbitls is resonble pproimtion if the distnce between neighbour ions is much bigger thn the size of the 2s orbitl, so tht there is no confusion to which orbitl n electron belongs. This is setched below to the left, nd is nown s tight-binding limit. One could lso consider the nerly-free electron limit, which would be vlid if the tomic orbitls spred over mny lttice sites, so tht you couldn t be sure to which orbitl n electron which is close to ion n, let s sy, belongs in relity. This is setched to the right. We will discuss this second cse lter on. tight binding limit nerly free electron limit Figure : Contrst between tightly-bound model, with orbitls smll compred to the lttice constnt (left) nd nerly-free model, with orbitls spred over mny lttice constnts.. Tight-binding models For our tight-binding model, we ssume n m = dφ 2s ( R n )φ 2s ( R m ) = δ nm. This orthogonlity is necessry in order to be ble to ssocite cretion nd nnihiltion opertors tht stisfy the proper lgebr to these sttes
(remember tht we cn do so only for bsis of orthonorml one-prticle sttes. If the orbitls were not orthogonl, we would first hve to define new bsis of orthogonl sttes). Deriving Hmiltonin for this pproimtion proceeds just lie we did in the Hubbrd cse, nd we will get the sme result. If we me the dditionl pproimtion tht we cn ignore even the on-site repulsion U 0, we obtin the D tight-binding Hmiltonin: Ĥ = ǫ n,σ c nσc nσ t n,σ ( c nσ c n+,σ +h.c. ) () The first term is just the energy to hve the electron t ny given site, which up to some smll corrections cn be thought of s the tomic energy of this orbitl. If ech tom eeps its electron 00% of the time, for instnce when nd we hve collection of isolted toms, the energy would be ǫ. The second term describes hopping: if the electron is t site n, ttrction from the neighbouring ions cn me it move either to n+ or to n. This is nown s nerest-neighbour hopping. As we discussed, the electron will feel ttrction from ions n+2 nd n 2 nd ions further wy, s well, nd could hop directly two or more lttice sites, to one of those, so in principle longer-rnge hopping should be present in the Hmiltonin. However, remember tht the hopping mtri decreses eponentilly with the distnce between sites, so nerest-neighbour hopping is by fr the lrgest term. In mny cses it is good pproimtion to ignore longer-rnge hopping, nd we will do so. It cn be esily included nd you should now how to do tht nd be ble to figure out precisely wht its effects re. One issue we hve to be creful with is wht hppens t the ends of the chin. I will use the so-clled periodic boundry conditions (PBC), which stte tht if the electron is t site, it cn hop either to 2 or to, nd if the electron is t site, it cn hop either to - or to. In other words, it s s if the chin is ctully closed into circle, nd nd re nerest neighbours s well. The reson we prefer to use PBC in ll our problems is becuse it mes the clcultions simpler, s we ll see soon. The logic is tht for mcroscopic crystl, with 0 23, it should not mtter how we end the crystl, becuse the vst mjority of the contributions comes from the bul sites nd the smll contributions from the surfce must be negligible by comprison. So we cn choose whichever sitution mes clcultions esiest, nd PBC do. (As good physicist you should be septic of everything I sy nd you should chec this sttement, by repeting the clcultion we do net for chin with open ends, i.e. where n electron t site cn only hop to 2, nd one t cn only hop to -. Let me now if you need some help with tht). The nice thing bout Hmiltonins lie Eq. (), which re nown s qudrtic Hmiltonins becuse ll terms re products of two opertors, is tht they cn be digonlized ectly (in other words, we cn find ectly ll their eigensttes nd eigenenergies). Here we don t even hve to wor hrd to figure out the solution, becuse it is directly pointed t by the symmetry of the problem: if I plce one electron in the system, it is eqully liely to be found t ll sites becuse they re identicl, so we should wor in momentum spce nd therefore we define opertors ssocited with sttes with well defined momentum: c σ = n e irn c nσ c σ = n e irn c nσ which cn be inverted to (chec!): c nσ = e irn c σ c nσ = n e irn c σ The normliztion is defined such tht the new opertors lso stisfy the proper lgebr {c,σ,c,σ } = δ, δ σσ etc, in other words their corresponding wvefunctions re properly orthogonl. If we plug these into Eq. () nd do bit of clening up, we find tht Ĥ =,σ E()c,σ c,σ where E() = ǫ 2tcos(). In other words, to the one-electron eigenstte c σ 0 = n= ein c nσ 0 corresponds the eigenenergy E() = ǫ 2t cos(). Becuse the electrons re non-intercting, mny-electron solutions re trivil s well: the eigenstte c σ...c pσ p 0 hs the eigenenergy E( )+...+E( p ), i.e. the sum of individul contributions. 2
But we hve one more question to nswer: cn the prmeter be nything, or do we hve some restrictions? We sed for PBC, which cn be written more generlly s c n+,σ = c n,σ,c n+,σ = c n,σ, for ny n (we re removing/dding the electron from/to the sme site). If we put this into their dependence of c,σ, we find tht we must hve e i = = 2m,m = 0,±,±2,... So it seems tht we hve n infinite (but countble) number of eigensttes, one for ech integer. More creful thining shows tht this is not true, in fct we hve distinct vlues only for consecutive vlues of m, nd then we repet the sme eigenfunctions nd eigenvlues. Becuse we don t find something new, we cn t pretend those to be distinct sttes/opertors, so we throw them out. To see this, consider m = 0 nd m =. The first corresponds to = 0, the second to = 2/. It is esy to see tht c =0,σ = n c n,σ = c =2/,σ = c =4/,σ =..., nd in generl sttes such tht = +mg where m is n integer nd G = 2 re in fct identicl (sme cretion nd sme nnihiltion opertors). Sowectullyonlyget distinctllowed vlues. Wecouldchoosethesesbeingm = 0,,2,..., ornyother choice of consecutive integers. Becuse we lie symmetry, the trditionl choice is m = 2 +,...,,0,,... 2, i.e. vlues symmetriclly plced bout the origin. In other words, we only consider vlues such tht <. This intervl is the Brillouin zone. The llowed vlues for nd the corresponding eigenenergies in chin with = 2 sites re setched below. E() ε llowed vlue, if =2 corresponding E() Figure 2: Allowed moment nd corresponding eigenenergies E() (white nd blue circles, respectively) inside the first Brillouin zone <. For chin with sites there re precisely llowed vlues, equidistntly plced. Snity chec: is it resonble tht we hve 2 eigensttes ( possible vlues for, times 2 for spin-up nd spindown)? Mybe we didn t find ll of them we lwys hve to worry bout this when we guess the solution, lie we did here. The nswer is: ll is good. Our Hilbert spce is 2-dimensionl, becuse it ws spnned by the 2s sttes n =,..., nd σ = ±. The eigensttes form bsis, so there must be precisely 2 of them, s well. We found 2, so we hve ll of them (by the wy, I forgot to mention tht these eigensttes must be orthonorml. You cn chec by direct clcultion tht this is indeed the cse). So wht is the solution for our chin of Li, i.e. when we hve electrons in the system so tht precisely hlf of the sttes re filled? We hve to plce the electrons in these one-prticle sttes, obeying Puli s principle. In the ground-stte they occupy the lowest vilble levels, to minimize the totl energy. Tht mens tht electrons occupy the hlf of the sttes with energies E() < ǫ (these re the sttes with moment 2 < 2 ), nd we leve empty the other hlf of the sttes, with energy E() > ǫ. So clerly the totl energy is lower thn the ǫ we would hve if ech Li ept its electron. So the ground-stte eigenstte is: GS = c,σ 0 2 2,σ 3
nd the ground-stte energy is: E GS = 2 2,σ E() = ǫ 4t 2 <ǫ< 2 cos() In the limit, the sum cn be turned into n integrl. This should not be surprising, becuse the llowed vlues of come closer nd closer together. As you should be ble to show, in this limit: m f() = 2 = min m min df() where min nd m re the limiting vlues for the llowed, in our cse ±/2. If you do the integrl, you ll find tht: E GS = ǫ 4 t < ǫ So there is gin of 4t/ per tom in cohesion energy, becuse the vlence electrons become deloclized over the whole system (ech one cn be found nywhere in the system with equl probbility). This gin increses if we bring the toms closer nd t becomes lrger, but then we strt to py the price due to repulsion between the clouds of core electrons (this is included in ǫ) so there is some equilibrium distnce between ions where the totl energy is minimized. For other types of lttices (nd in higher dimension), the epression of E() will be different, nd s result the prefctor multiplying t chnges, so different lttices led to different cohesion energies. The one with lowest energy will be the preferred structure tht will be observed in relity..2 Free electrons (jellium model without interctions) Remember tht the jellium model ssumes tht the ions re spred out in uniform density of positive chrge. If we ignore the electron-electron interctions, the resulting Hmiltonin (see previous chpter, but now for D system) becomes: Ĥ =,σ ǫ,σ,σ where ǫ = h2 2 2m is the free-electron inetic energy, nd the opertors re ssocited with plne-wves with momentum :,σ,σ 0 = δ σ,σ ei / L, where L is the length of the D crystl. ote tht these opertors re very different from the c,σ we found in the previous section: those creted/nnihilted electrons in very different sttes, which were liner combintions of 2s orbitls, not these plne-wves! This is why I ll cll them insted of c. This difference is lso cler if we loo t the llowed vlues in this cse. If we gin impose PBC, this mens tht nothing should chnge if we trnslte + L. From the definition bove, this obviously hppens only if e il = = 2 L m,m = 0,±,±2,... This loos similr to wht we hd before (there L = ) ecept now there is no restriction on the vlues of m! For the tight-binding chin, the restriction ppered becuse the vlues of the plne-wves only mttered t R n = n. However, in this jellium model they mtter t ech [0,L], so ech llowed vlue gives different, cceptble solution. Thus ny integer m nd so ny = 2 L m is llowed, in other words there is no restriction to Brillouin zone lie before. If you thin bout it, this jellium model ctully mimics well the nerly-free electrons limit shown on the right-hnd side of Fig.. The limit B is obtined when we let 0 indeed, in this cse insted of distinct ions, we see continuous distribution of positive chrge. Hving 0 lso mens tht the Brillouin zone etends to cover the whole is since ± ±, which is wht we just found; nd finlly, in this limit we find E() = ǫ 2tcos() ǫ 2t + (t 2 ) 2. If we define n effective mss m such tht h2 2m = t2, then up to n unimportnt constnt, E() ǫ s well (it is resonble tht t 2 might sty finite s 0, becuse remember tht hopping integrls increse with decresing distnce). So formlly the results mp onto one nother, lthough the wvefunctions re somewht more problemtic. If we hve electrons in this system, in the GS they will gin occupy ll sttes up to F, so: GS = < F,σ,σ 0 4
nd where the Fermi momentum is such tht: E GS = ǫ = 2 ǫ < F,σ < F = = 2 < F,σ < F Agin, in the limit L, the llowed vlues of become dense nd we cn replce the sums with integrls: 2 f() = L 2 2 df(). We then find: = 2 F L/ F = n/2, where n = /L is the electron density, nd we cn then go nd clculte the GS energy EGS = h2 2 n 2 24m nd ny other quntities tht we re interested in. ote: I epect you to be ble to do similr clcultions in higher dimensions. I epect you sw them before in undergrdute-level solid stte nd lso sttisticl mechnics courses. If they give you trouble, come nd tl to me..3 Free electrons in periodic potentil: bnd structures At this point you re probbly feeling bit confused becuse these two cses loo very different from one nother, so it s not ll tht cler wht they hve in common nd how one my go from one to the other by vrying the rtio B /. I thin the best wy to understnd this is to ctully solve problem we ll do this net for the simplest non-trivil emple tht I now (still in D). Let s gin ssume tht we hve simple crystl with single tom in the bsis, lthough gin, things generlize strightforwrdly to more complicted cses. Suppose tht we hve ions plced t their lttice loctions X n = n, where is the lttice constnt nd n =,..,, nd vlence electrons whose behvior we re trying to understnd. We will gin ssume PBC nd t the end we will let, so tht the boundry condition becomes irrelevnt. Then, the totl potentil felt by ny of the vlence electrons is: V tot () = V ( n) n= where V () is the potentil creted by n ion locted t the origin, nd the sum is simply the contribution of ll the ions. Becuse there re no electron-electron interctions we cn solve the problem for single electrons, nd then me Slter determinnt from the one-electron eigensttes. So we need to solve Schrödinger s eqution: [ h2 d 2 ] 2md 2 +V tot() φ() = Eφ() with the periodic boundry condition φ(+) = φ(), for ny. At this point I epect you remember tht such periodic potentils lwys led to so-clled Bloch eigensttes. I could jump directly to using them, but becuse this is ey point, let me briefly review one wy to derive them (if this is not enough to jog your memory, plese do some reding!). Let s try to guess wht the solution of this Schrödinger eqution must loo lie: (i) the solution must hve plne-wve lie chrcter, becuse the symmetry of the problem tells us tht the probbility to be in ny unit cell must be the sme, irrespective of which unit cell we re tling bout. However, (ii) we epect tht the wvefunction vries from plce to plce inside the unit cell; for emple, the probbility to find the electron should be higher ner n ion thn in between two ions. Putting these together, let s try the guess: φ () = e i u () where u () = u ( + ) is periodic function. This hs the right chrcter, becuse it leds to probbility φ () 2 = u () 2 which is periodic function, just s we wnt it to be. Another wy to see this is tht, with our guess: φ (+) = e i(+) u (+) = e i e i u () = e i φ (), 5
i.e. moving one unit cell over leves the mgnitude of the wvefunction unchnged, but it dds phse, just s it should (remember, we wnt mny eigenfunctions, not just the one with φ(+) = φ(), nd we need the plne-wve for tht). By the wy, in 3D, the guess would be φ ( r) = e i r u ( r), where now u ( r) is periodic in the 3D unit cell. It turns out tht this guess wors perfectly. In fct, one cn prove tht ny single-prticle Schrödinger s eqution with periodic potentil hs eigenfunctions of this type, which re nown s Bloch wvefunctions. Wht is left to do is to figure out wht is u () nd its corresponding eigenenergy E, so tht the eqution is stisfied. This is mtter of mth. Before discussing how to do tht, let s loo more crefully t the momentum. For our D crystl, the wvefunction must lso stisfy the periodic boundry condition φ(+) = φ(). For Bloch wvefunction, though, φ(+) = e i φ(), so we must hve = 2 m, where m is n integer. One cn gin show tht only of these solutions re distinct (just lie the discussion we hd for the tight-binding model), so we must restrict (, ], i.e. to the first Brillouin zone. This is lwys the cse when we hve model with lttice. Before looing t how this wors in simple but non-trivil emple, let s first me sure tht we regin the correct solution if U () 0. This, of course, leds us bc to free electrons, but zero is lso periodic potentil so we should be ble to put the free electron solution in the Bloch form, nd with the restriction on the vlues of if we dopt the periodic boundry condition. The wy this wors is illustrted in the figure below. On the left, I show the dispersion in free spce: the usul prbol with ny momentum llowed. If I choose to view the free spce s periodic lttice (without ny potentil), then the correct description is shown on the right. I plced prbol t ech reciprocl lttice point, so now the eigensttes re periodic but I only count the sttes inside the first Brillouin zone, since the other unit cells re just repeting these nd don t give nything new. So now we ve restricted the momentum, but for ech vlue of we hve mny solutions E n (0) (), n = 0,,2,... Overll we hve the sme spectrum, it s just different wy to inde the sttes. By the wy, this re-rrngement of the sttes to fll ll inside the Brillouin zone is now s folding. E() E() 4 2 Brillouin zone 2 4 Figure 3: Free spce eigensttes. Left: the usul prbol for free spce, where ny is llowed nd we hve single eigenstte for given. Right: the corresponding bnd-structure if we te the vcuum to be periodic crystl with lttice constnt (nd no potentil). ow the moment re restricted to the BZ, but for ech we hve n infinite number of bnds E n (0) (). The thic coloured lines show the n = 0,,2 bnds. For ny given in the BZ, we cn see to wht K of the originl prbol tht prticulr energy corresponds. I showed 2 such correspondences (horizontl lines with rrows). To be bit more precise, let s consider the first 3 bnds, n = 0,,2, for given (see right figure). I illustrted the corresponding sttes in both representtions in the usul free spce lnguge, they correspond to different moment, K 0 =,K = 2,K 2 = + 2. The energies, then, re E(0) 0 () = h2 2 2m = E 0(K 0 ); E (0) () = h2 ( 2 )2 2m = E 0 (K ); E (0) 2 () = h2 (+ 2 )2 2m = E 0 (K 2 ), nd so on nd so forth, I hope you gree tht we cn mp ll the points in one spectrum to points in the other. How bout the wvefunctions, cn we put them in the Bloch form? For n = 0 this is trivil, since here φ (0) K 0 () = e ik0 = e i = φ 0, (), so it hs the epected Bloch form φ 0, () = e i u 0, () with u 0, () =, which is certinly periodic. How bout for K? In this cse, the eigenstte is: φ (0) K () = e ik = e i( 2 ) = e i u, () = φ, () 6
where for the n = bnd, it follows tht we must hve u, () = e i2. Is this periodic function? The nswer is yes: u, ( + ) = e i2 (+) = e i2 e i2 = u, (). And so on nd so forth, you cn chec tht for ech bnd n, the phse difference between the corresponding K n nd is multiple of 2/ nd gives different periodic prt u n, () for the Bloch wvefunction φ n, () = e i u n, () of tht bnd. By the wy, it is good tht different bnds hve different u n, (), becuse wvefunctions with different n nd/or different hve to be orthogonl. So this scheme wors for free spce, if we choose to view it s periodic potentil. As the rule sys, we find tht is indeed restricted to the Brillouin zone. The price is tht we hve mny bnds (different eigensttes for the sme ), ech of which hs Bloch wvefunction with different periodic prt. Turns out tht this is wht hppens in the generl cse, too we will hve (infinitely) mny bnds in the Brillouin zone, however their energies E n () will, of course, depend on the prticulr potentil we choose, s will the periodic prts of their Bloch wvefunctions, φ n, () = e i u n, (). To the find the specific E n () nd u n, (), we need to solve Schrödinger s eqution. Let me show you how it wors in simple cse when we hve the simplest, δ-function ttrction between ions nd electrons, i.e. V () = U 0 δ() (By the wy, why does the constnt in front hve units of energy distnce if V () is n energy? Figure tht out. Once you do tht, you ll see tht we need to use the lttice constnt, which is our nturl lengthscle in this problem, s the length. Tht leves U 0 s n energy which we cn djust t will: the lrger it is, the stronger the ttrction). First, let me quicly review the solution for single ion. I epect you hve solved this problem in n introductory QM course, nd even if not tht you cn figure it out but s for pointers if you need help. Of course, there re eigensttes for ll positive energies E > 0 (when the electron is not bound to the ion), but it turns out tht there is one bound stte with wvefunction (up to normliztion constnt): nd energy φ B () e κ = e B E B = h2 κ 2 2m = U2 0. 2 h 2 m 2 where the Bohr rdius is B = κ = h2 mu 0 defines the spred of the wvefunction. This tells us tht the lrger U 0 is (the stronger the ttrctive potentil), the more negtive the energy of the bound stte, nd the smller B is. This mens tht by decresing U 0, we cn go from the limit where B, which is the tight-binding limit where our tight-binding model should wor well, to the limit with B, which is the nerly free electron limit. These re setched below. B B n n n+ n n n+ tight binding limit >> B nerly free electron limit << B Figure 4: Left: tight-binding limit, where B. Right: nerly free electron limit, where B. For single ion, then, the spectrum for negtive energies hs only this one level t energy E B. Let s see wht hppens for lttice of such ions, t negtive energies E = h2 λ 2 2m ; for wht vlues of λ do we now find eigensttes? We need to solve the SE for the potentil U tot () = U 0 n=0 δ( n). The Bloch theorem gurntees tht the wvefunctions must be such tht φ ( + ) = e i φ () this is very nice, becuse it mens tht we only need solve SE inside one unit cell, nd mtch the boundries ccordingly. So consider the unit cell 0 < <. The potentil t ll these vlues is zero, so the wvefunction in this intervl must be φ () = αe λ +βe λ. According to Bloch s theorem, the wvefunction in the net unit cell, < + < 2, 7
must be φ (+) = e i φ () = e i( αe λ +βe λ). All we need to do now, is to sew together the wvefunctions t =, where we hve delt function in the potentil. Continuity t = implies: αe λ +βe λ = e i (α+β) while the jump in the derivtive becuse of the δ-function results in: h2 [ e i (αλ βλ) ( αλe λ βλe λ)] = U 0 e i (α+β) 2m So we hve two homogeneous equtions for α,β, nd we now tht non-trivil solution is possible only if the determinnt is zero. After some boring lgebr, this condition becomes: cosh(λ) sinh(λ) λ B = cos() where cosh() = 2 (e +e ), sinh() = 2 (e e ). So wht hppens here is tht for ech vlue of (inside the Brillouin zone), this eqution will give us one or more solutions λ n (),n =,2,.. (however mny solutions we get) nd therefore we find the llowed energies E n () = h2 λ n() 2 2m. This eqution is too complicted to dmit simple nlyticl solutions, but we cn still figure out lot bout its solutions in vrious symptotic cses. Let me define the function: f() = cosh() sinh B Then λ = / is the solution when f() = cos(). Since cos(), we need to figure out wht s the shpe of f(), nd in prticulr when it hs vlues between,. At the origin, f(0) = B. This is definitely less tht. If > 2 B, then f(0) <. As increses, you should be ble to convince yourself tht this function increses monotoniclly. In prticulr, t lrge we cn discrd the e eponentils s being smll, nd we find f() e 2 ( ), which diverges s. B f() =0 f() =0 B = 2 λ( 2 ) λ(0) λ( ) λ( ) 2 λ(0) = 2 B = Figure 5: f() setched when (left) < 2 B, nd (right) > 2 B. The solutions for f() = cos() with (, ] lie in between between -,. This function is setched bove, for both < 2 B nd > 2 B cses. Let me strt with the second cse which contins the tight-binding limit B. Here, there is single solution for f() = cos() [,], so we epect single solution for λ = /, nd therefore single bnd t negtive energies E = h2 λ 2 2m. Let s find it in the limit B, i.e. in the tight-binding limit. ow f(0) strts t very negtive vlue, so by the time it crosses the [,] intervl the vlues of re very lrge nd we cn use the simpler symptotic epression, i.e. here: f() e 2 ( ) B = cos() = B [ 2e cos()] This is still trnscendentl eqution so we don t hve n ect solution, but we cn get n ccurte one by itertions. If the eponentil wsn t there, the solution would be = B in this limit (which is lrge vlue, consistent with 8
our symptotic pproimtion). As result, e is smll nd leds to tiny correction, so the ctul solution won t be too fr from B. So I cn replce the in the eponentil with / B, to find: = λ() [ ] [ ] +2e B 2e B cos() B cos() B (this is just Tylor epnsion, since the eponentil continues to be smll). Remembering tht / B = κ for the bound stte, we find: ] λ() = κ [+2e B cos() leding to the energies: E B () = h2 λ() 2 2m E B ) (+4e B cos = E B 2tcos() if we define the hopping t = 2 E B e B. Before continuing, note tht I dropped the term tht goes lie e 2 B from the squre this is becuse I now tht it is much smller thn the ones I ept, nd moreover I lredy neglected such terms when I did the Tylor series bove, so it s simply not meningful to eep it here. So for B we regin the tight-binding eigenenergy we epected to be vlid here. Moreover, we lso hve the correct epression for the hopping t (if you remember how this is defined s the overlp between neighboring orbitls, you cn chec this sttement). The left pnel below shows the llowed energy bnd, i.e. the energy intervl where we find solutions in the limit when, so tht ll inside the BZ re llowed. The right picture shows the dispersion for one prticulr vlue of / B, in the tight-binding limit. The lrger / B is, the nrrower the bnd becomes, becuse the hopping between nerest sites becomes eponentilly smll. In the limit / B we bsiclly hve isolted toms, ech with their llowed level t E B. E 2 llowed energies B E 0() E 2tcos() B E B Figure 6: Left: llowed bnd of energies with E < 0, s function of / B. Right: For prticulr vlue of / B, indicted by the dshed line, I showed the dispersion E B () vs, in the Brillouin zone. How bout the wvefunctions, do they come out correctly? In the tight-binding limit we obtined the solutions c,σ 0 = n ein c n,σ 0, whose wvefunctions re (ignoring the spin) φ () = c 0 = n=0 ein φ B ( n). ote tht these re indeed Bloch sttes! (s they should be, the tight-binding model corresponds to periodic potentil.) Remember tht the bound level eigenfunctions re simple eponentils. For 0 < <, the ones tht contribute most re the e /B prt from the ion t n = 0, nd the e B prt from the ion t n = ll other ions contributions re very much smller thn these terms, since they re further wy from this intervl. Keeping only these two lrgest contribution, the tight-binding pproimtion sys tht for 0 < < we should find: φ () e B +e i e B 9
Let sseewhttheectsolutionpredictsinthislimit. Thisrequiresustofindα,β sothtwehvethewvefunction. We cn use either eqution, let s use the simpler one: αe λ +βe λ = e i (α+β) α β = ei e λ e λ e i But in this limit λ B e λ = e B, thus we cn simplify the frction: α β ei e λ α = βei e B. Putting this into our solution on the 0 < < intervl, φ () = αe λ + βe λ αe B + βe B (since λ B ) gives us, up to normliztion constnt, the epected solution. So our simple tight-binding model gives the correct eigenenergies nd eigenfunctions in the correct limit. Wht hppens when we decrese / B, i.e. we bring the ions closer nd closer together? For one, the hopping increses so the bndwidth 4t becomes lrger. From the discussion of the ect f(), you cn see tht something hppens t / B = 2 only bove it re we gurnteed to find solutions for ll in the BZ. Wht hppens for smller / B, s the figure bove hopefully suggests, is tht we need to worry bout positive energies (so fr we only discussed E < 0 eigensttes). So let s loo t the spectrum there, too. The E = h2 K 2 2m > 0 cse is studied just lie before. Everything mirrors wht we did for E = h2 λ 2 2m < 0 if we replce λ ik. In prticulr, we now find tht for ech, the llowed vlues of K (which determine the energy) re given by the eqution: cos(k) sin(k) = cos() K B Let s define gin function g() = cos() sin B, nd solve g() = cos() K = /. Wht does g() loo lie? Its two components re setched below; the cos function oscilltes, while the sin oscilltes but lso decys slowly with incresing. cos 2 3 2 2 sin 2 Figure 7: Left: cos() vs. Right: sin()/ vs. Let s see first wht hppens for B. In this cse, the sin term is lrge ner the origin nd domintes there, so g() loos lie in the net plot. I lso drew the lines for g() = (this gives solutions for = 0) nd g() =, which corresponds to solutions for = ±. As we increse we sweep between the two. Unlie for E < 0, where we found single solution for ech, here we hve n infinity of them, so they predict n infinity of bnds t positive energies. The full spectrum loos lie setched on the right, with the tight-binding bnd t negtive energy, nd ll these other llowed bnds seprted by gps t positive energies. ote how the curvture of the bnds lterntes. O, so now finlly let s see wht hppens s / B becomes smll. In this limit, the function g() loos lie below. The contribution of the sin term is very smll, so g() brely crosses bove or below -. We still get llowed bnds seprted by forbidden energy rnges, but they re wider. The bottom of the lowest bnd nicely mtches up with wht we found there for the negtive spectrum, so the whole spectrum here loos lie shown in the right pnel: This is strting to loo lie the free electron spectrum, which it should evolve into when / B 0 (for B when U 0 0, i.e. when there is no potentil). Indeed, s we me / B smller, the bottom of the lowest bnd comes closer to zero while the gps bove strt to close, nd we regin the folded bnd structure from Fig.. If you wish to chec this sttement, you cn clculte, for emple, the energies of the first two bnds t = /. I did this, nd found E 0 ( = ) = ( [ ] h2 2 2m ) 4 2 B, while for the second bnd, I find E ( = ) = ( [ h2 2 2m ) + 4 2 B ]. So the 0
E() E g() 0000 0000 0000 0000 0000 0000 0000 0000 0000 2 3 4 =0 = /2 = / bnd of llowed energies forbidden gp B E B Figure 8: Left: Solution of g() = cos() in the limit / B. Right: Full spectrum in the limit B. ( first bnds ends just below the h2 ) 2 2m epected in vcuum for =, while the second bnd strts just bit bove it. The gp between them is proportionl to / B, nd indeed closes s U 0 0. This clcultion tells us tht in the presence of periodic potentil, we genericlly epect to see spectrum consisting of llowed energy bnds which re seprted by gps. Of course, rel toms hve mny bound sttes, not just one lie our pretend toms. However, this is not going to chnge things significntly. Ech level hs chrcteristic distnce over which it decys eponentilly, nd deeper levels decy fster becuse they re locted closer to the nucleus. For given lttice constnt, the low-lying levels decy much fster thn so they re well-described by tight-binding models. For higher-energy orbitls, which re more etended, the tight-binding pproimtion is not going to be ccurte nymore, nd we ll need to do better. In ny event, if we consider wht hppens s we decresed the lttice constnt from infinity (corresponding to isolted toms, ech with its own tomic spectrum), we should get something lie setched below: The precise loctions of the llowed energy bnds will, of course, depend on wht type of crystl we hve, nd wht is the equilibrium vlue for, etc etc. But there will certinly be energy bnds chrcterized by momentum restricted to the first Brillouin zone, seprted by forbidden gps. Higher bnds hve lrger hopping constnts becuse their orbitls re more spred-out, so their width increses fster; they re lso closer together, so we epect tht they strt overlpping t some point or other (when this hppens, the simple tight-binding pproimtion definitely fils nd we need to wor hrder to find n ccurte solution). So this is the generic picture for the single-electron spectrum of crystl. ow we hve to fill the lowest levels with ll the electrons contributed by ll the toms. There re two possible outcomes: (i) the lst bnd tht contins electrons is only prtilly full. In this cse, the mteril is metl if we pply smll electric field, there re free levels just bove the occupied ones, so we cn esily ecite electrons nd me them move preferentilly in the direction of the field, to get n electric current (we ll discuss this more crefully very soon). (ii) the lst bnd tht contins electrons is completely full, in the ground-stte. In this cse, the mteril will not E() E bnd of llowed energies g() 0 00 0 00 00 0 00 0 00 00 0 0 00 00 0 0 00 =0 B 00 3 00 00 0 00 0 00 00 0 00 0 00 00 0 00 2 0 4 00 00 0 00 0 00 00 0 00 0 = /2 = / forbidden gp Figure 9: Left: Solution of g() = cos() in the limit / B. Right: Full spectrum in the limit B.
E gps E E 2p 2s bnds of llowed energies E s tomic lie core sttes Figure 0: Generl evolution of the energy bnds in solid, s the lttice constnt is decresed. be metl; to ecite electrons to empty levels, we need to give them n energy t lest equl to the gp so tht we cn move them into the net bnd. This is why pplying smll electric field will not strt current flowing, so this is not conductor. If the gp is lrger thn 2eV, we cll this n insultor; if it is less thn 2eV, we cll such mteril semiconductor. Bsed on the solution for the tight-binding model, you might epect tht ny time we hve one (or n odd number) of vlence electrons per ion we should get metl. This is not true! To obtin the ground-stte with the lower-hlf of the tight-binding bnd full, we mde the crucil pproimtion tht we cn ignore electron-electron interctions. If those interctions cnnot be ignored, then the mny-electron wvefunction is not Slter determinnt of one-electron sttes nd these rguments fil, nd we hve to clculte the mny-body wvefunction somehow else. It turns out tht for Hubbrd model in 3D, if U is sufficiently lrge, gp opens up nd splits the occupied sttes from the empty sttes, so the system is n insultor (in D, this hppens for ny U > 0). We distinguish such insultors from the ones discussed bove, which occur even in the bsence of interctions. Those re clled bnd insultors (nd re bit boring lthough etremely importnt for current technologies), wheres the ones tht rise becuse of electron-electron interctions re clled Mott insultors. We ll discuss bit more bout these lter on. Before finishing this section, let me mention tht we cn mesure directly the energy bnds using photo-emission spectroscopy. This is when bem of high-energy photons is shined onto smple, nd some of them re bsorbed by electrons which therefore receive enough energy to escpe the crystl. Detectors mesure the energy nd momentum of the escping electrons, nd from conservtion of energy nd momentum, one cn etrct informtion bout wht ws the energy nd momentum, i.e. E n (), of the electrons while in the crystl. This technique is nown s Angle-Resolved Photoemission Spectroscopy (ARPES), nd the Dmscelli lb here t UBC does such mesurements. You cn loo t their webpge for more informtion nd some nice pictures of eperimentlly mesured electronic bnds. In terms of clculting bnd structures going beyond simple models lie tight-binding, one uses density functionl theory, lso nown s b-initio methods. These re bsed on the Hohenberg-Kohn theorem, which is n mzing result plese te bit of time to red bout this in stndrd tetboo (for emple, Tylor nd Heinonen pp. 82-92). These dys there re free (nd commercil) softwre pcges to del with this pproch, nd some of you my end up using some of these. Even if you don t, you should hve bsic understnding of the underlying ides, which re firly simple to grsp. Let me just sy tht lot of cre is needed with the interprettion of these results, though, becuse this theory only gurntees to predict the (totl) ground-stte energy nd density of electrons in the ground-stte, nd this only if certin functionl is nown. We do not now tht functionl but we cn compute n pproimtion for the jellium model, nd tht is used for ny model (the so-clled LDA pproimtion). This wors pretty well for systems with we electron-electron interctions, but not for systems with strong interctions. Vrious improvements nd wys to fi the problem re proposed for the ltter problems, but this is still very much wor in progress. 2