University of Ljubljana Faculty of Mathematics and Physics Seminar - Ib, 1. year II. cycle degrees Polarons Author: Jaka Vodeb Advisor: prof. dr. Viktor Kabanov Kranj, 017 Abstract After the discovery of high temperature high-t c superconductivity by Bednorz and Müller in 1986, which was at the time linked to polaron formation in the superconductor, a cascade of experimental and theoretical work was performed on the link between polarons and high- T c superconductivity over the last few decades. This lead to raising the critical temperature from 30K to 134K using the same type of materials, as were used in the original paper. The validity of superconductivity as a consequence of polaron formation is still under discussion between contemporary researchers. In this seminar, the concept of the large polaron and the non-adiabatic and adiabatic small polaron is presented, along with their main characteristics. Finally, a paper which is considered to be one of the best conformations of a link between polarons and high-t c superconductivity is also presented.
Contents 1 Introduction 1 Large polaron 1 3 Electron phonon interaction 3 4 Two site Holstein model 4 5 Non-adiabatic small polaron 6 6 Adiabatic small polaron 7 7 Evidence of polarons in high-t c superconductors 9 8 Conclusion 10 1 Introduction The concept of a polaron was rst introduced by Landau in 1933 as an electron which localises itself by interacting with the lattice deformation of the crystal it's located in. This is the so called large polaron, because even though we treat the electron with means of quantum mechanics, we still treat the lattice as a continuum. Therefore, the size of the polaron must be large compared to the lattice constant in order to satisfy the continuum approximation. This phenomenon has not been observed yet, but it's an informative and intuitive introduction to the polaron concept. The ratio between the binding energy of a polaron E p and the typical kinetic energy of the electron ma, where a is the lattice constant is called the coupling constant and is denoted by λ. It measures the strength of the coupling between the electron and the lattice deformation. As we will see in the next chapter, the polaron radius decreases as λ increases [1. This leads to the so called small polaron, with it's size comparable to a, which is believed to have been observed in high-t c superconductors [. A polaron model, which takes into accout the discrete nature of the lattice is the so called two site Holstein model. It's purpose is the description of the small polaron and it describes all of the main properties of polarons [1. Large polaron The most intuitive model of a polaron is that of an electron interacting with a crystal lattice as a dielectric polarizable continuum described by the static and high frequency dielectric constant ɛ and ɛ respectively. ψ r denotes the wave function of the electron and P the polarization eld of the crystal lattice. If the electron is the only unbound charge in the system, then there exists a non-zero electric deformation eld D in the crystal of the form D r = e 4π d r ψ r r r, 1 where e is the charge and m is the mass of the electron. The energy of the system can be written as a functional of ψ and ψ in the form [ E[ψ, ψ = d r ψ r ψ r m 1ɛ0 P r D r, 1
where ɛ 0 is the vacuum dielectric constant, is the Planck constant and the energy of the electric deformation eld was omitted, as it only contributes as a constant to E[ψ, ψ. The rst term represents the kinetic energy of the electron and the second the electron's potential energy as a consequence of an induced polarization eld. By minimizing E[ψ, ψ with respect to ψ r at xed P r, satisfying the constraint d r ψ r = 1 and substituting the expression for D r, the following equation of motion is obtained m e 4πɛ 0 d r P r 1 r ψ r = E 0 ψ r, 3 r where E 0 is the binding energy of the electron. In linear response theory, P is proportional to D with the constant being the electric susceptibility χ = ɛ 1 ɛ. In this case only the polarization related to the displacement of the lattice atoms needs to be taken into account. If the static dielectric constant takes into account both the eect due to the displacement of the lattice atoms and the electronic polarizability and the high frequency dielectric only takes into account the latter, then the lattice displacement polarization is related to D by P = χ χ D = D/κ, where κ 1 = ɛ 1 ɛ 1. The equation of motion is then e m 4π d r d r ψ r 1 ɛ 0 κ r r 1 r ψ r = E 0 ψ r. 4 r Using 1 r = 4πδ r, equation 4 can be reduced to m e 4πɛ 0 κ d r ψ r r ψ r = E 0 ψ r. 5 r This non-linear equation of motion can also be obtained by variational minimization with respect to ψ of the functional J[ψ, ψ = m d r ψ r ma B d rd r ψ r ψ r r, 6 r where a B = 4πɛ 0κ is the eective Bohr radius. If E me 0 = K + U is dened as the sum of the kinetic and the potential energy of the electron, which can be seen if we multiply equation 5 by ψ r and integrate over r, then J[ψ, ψ can be rewritten as J[ψ, ψ = K + 1 U. In order to variationally minimize J[ψ, ψ, the following trial function is taken ψ r = 1 e r/rp, 7 πrp 3 which is normalized and where r p represents the polaron radius. By substituting ψ r in equation 6 with the expression in equation 7, J[ψ, ψ turns out to be J[ψ, ψ = mr p 5 16ma B r p, 8 which needs to be minimized with respect to r p. This yields a value for the polaron radius of r p = 16 5 a B and a binding energy of the electron E 0 = 0.146, which turns out to be above ma B -0.5 ev in polarizable solids where 4 < κ 0 typically. The entire energy of the polaron has two interpretations in the context of the large polaron. When the electron is excited optically, the energy of the photon needs to be ω = E 0, according to the Franck-Condon principle. This
is because optical processes are considered to be fast, so the lattice deformation does not have time to relax during the excitation. In this case E p = E 0. However, during a thermal excitation, the process is considered to be much slower, so the lattice has time to relax. Here, the energy cost of the lattice deformation U d also needs to be taken into account. U d = 1 ɛ 0 d r P r D r = 3 E 0, 9 which means that the energy of the polaron is E p = 1 3 E 0 [1. In both cases E p is close to E 0, which means that if we use the denition for λ = E p / K, where K = represents the mr p typical kinetic energy of a polaron, we nd the expression for the coupling constant λ 1 1, rp 4 κ 4 1 which shows two things. Firstly, κ directly measures the coupling strength, which can also be intuitively seen from equation 5, where it determines the binding term of the equation of motion. Secondly, it shows that when the coupling between the electron and the lattice deformation increases, the radius of the polaron decreases. This may not be the most intuitive fact, but it holds true throughout the entire theory of polarons, even in the most quantum of descriptions. Of course, r p is bounded from below by the lattice constant a. In order to take a look at one of the most basic examples of a quantum model of a polaron, which is the two site Holstein model, we rst need to nd out out what's the quantum mechanical version of the interaction between an electron and the lattice. 3 Electron phonon interaction When considering an arbitrary solid in non-relativistic theory, the following Hamiltonian is considered H = i m i + e 1 8πɛ 0 r i r i i i Ze 4πɛ 0 ij 1 r i R j + Z e 1 8πɛ 0 R j j j R j j M j, where r i, R j are coordinates of electrons and nuclei, i = 1,..., N e ; j = 1,..., N; N e = ZN, N e is the number of electrons, N is the number of nuclei, i = r i, j = R, Z is the number of j protons in a nucleus and M is the nuclear mass. Because of the non-negligible strength of the Coulomb interaction and the absurd number of particles to consider in the wave function we wish to solve for, there are no methods present at the time to solve the Schrödinger equation with the Hamiltonian H. As a consequence, an adiabatic approximation is usually considered, where the motion of the nuclei is taken to be very slow compared to the electron motion, because the ratio of masses is m M < 10 3. If x l is dened as the displacement of atoms from their equilibrium positions l in the form R l = l + x l, 11 the adiabatic approximation implies an expansion of H in powers of x l and then a search for only the lowest order contributions of x l in the expansion. For the purpose of this seminar, the multi electron picture will also be omitted and the focus will only be on the one electron picture. In this picture H can be decomposed into three terms H = H e + H ph + H e ph. 10 H e = m r + V r 1 is the electron kinetic and potential energy in the presence of the nuclei in their equilibrium positions. V r = l v r l corresponds to the zero-th order in the expansion in powers of x l 3
of the third term of H, where v r l is the Coulomb attraction energy. The rst order term in displacement is H el ph = l [ x l s V s s= r l 13 and it denes the interaction between electrons in nuclei in this approximation. The interaction energy between individual nuclei is expanded up to the second or quadratic order in powers of x l, where the rst order term is set to zero, because of the assumption that all the nuclei are in their equilibrium positions. This part of the expansion, along with the kinetic energy of the nuclei is dened as H ph and it gives rise to lattice vibrations or phonons in the crystal. where H ph = l M R + 1 l, l,α,β D α,β l l = Ze 8πɛ 0 l α l β x α l x β l D α,β l l, 14 l l 1 l l. 15 [1 This general expanded Hamiltonian is simplied in the following chapter in order to be applicable to the two site Holstein model. 4 Two site Holstein model Figure 1 depicts a scheme of the two site Holstein model. The two sites contains two molecules located at sites l 1 and l, which are composed of two oppositely charged ions. The equilibrium distance between each ion is a. The only vibrational mode of the molecule considered in this model is the anti-symmetric mode, in which the signs of the displacements of the two ions are opposite and the displacements are of the same magnitude. This is taken into account by dening just one vector of displacement for each molecule x 1,. There is only one electron between the two molecules and it's wave function depends on r. After dening the two site Holstein model, each individual term of H from the previous Figure 1: A schematic representation of the two chapter can be simplied and tailored specically to this model. site Holstein model. H e = m + V r l 1 + V r l, 16 where V r l i = v r l i a/ v r l i + a/, 17 where i = 1, and v r is the Coulomb attraction energy. H e ph = i=1 x i {[ xi x i s v s s= r l i a/ + [ xi x i s v s s= r l i + a/ }, 18 4
where x i is the magnitude of the displacement of an individual ion in the i-th molecule. Finally, H ph = i=1 M x i + 1 Mω x i, 19 where ω is the eigen-frequency of the anti-symmetric mode in consideration. The goal of considering this model is to nd a simple solution corresponding to H, which will illustrate all the properties of small polarons. The ansatz will be ψx 1, x, r = a i x 1, x w r l i, 0 i=1 where w r l i are called Wannier wave functions, which are orthogonal and normalized and correspond to localized molecular states at the i-th molecule with their energy set to 0 to satisfy m + V r l i w r l i = 0 1 and a i x 1, x are parameters to be determined. By substituting ψx 1, x, r into the Schrödinger equation with the Hamiltonian H e +H e ph +H ph and multiplying it by w r l i and integrating over r, the following system of equations for the coecients a i x 1, x is obtained [ M x + 1 x + 1 mω Mω x 1 + x ω γx 1 E a 1 x 1, x + Ja x 1, x = 0 and [ M x + 1 x + 1 mω Mω x 1 + x ω γx E a x 1, x + Ja 1 x 1, x = 0, 3 where E is the eigenvalue to be obtained, γ = 1 ω mω {[ [ } d r w r x1 x1 s v s + s v s x 1 s= r a/ x 1 s= r+ a/ is the dimensionless matrix element of the electron phonon interaction and the hopping integral J is dened as J = d rv rw rw r l + l 1. 5 If the non-interacting case is considered, where γ = 0, the forms of a i can be guessed as the products of the ground states of two harmonic oscillators with coordinates x 1 and x [ a i x 1, x = A i exp 1 [ Mωx 1 exp 1 Mωx, 6 4 where A 1 and A are constants. Each oscillator contributes and the system of equations and 3 simplies to 1 ω to the energy of the system [ω EA 1 + JA = 0, [ω EA + JA 1 = 0. 7 Solutions are two energy levels of the system. E = ω + J corresponds to A 1 = A and E = ω J corresponds to A 1 = A. [1 Since the distribution of x i in both a i is centred at 0, both harmonic oscillators vibrate around displacement 0 and the electron wave function is a 5
superposition of the two molecular levels, which is usually interpreted as hopping between them and J is associated with the kinetic energy of the electron. This is a two site analogy to the formation of a band in the tight binding model. More generally, the coecients a i x 1, x can be taken as a product of two arbitrary excited harmonic oscillators a i x 1, x = x 1 n 1 x n, 8 where x i n i represents the coordinate representation of the harmonic oscillator wave function determined by the number operator n i x i n i = 1 n n! π 1/4 exp Mωx i Mωxi H n, 9 where H n x is the Hermite polynomial. The procedure for γ = 0 can be repeated with any choice of n i 0, which in this case only means that both harmonic oscillators can vibrate at higher frequencies while still in the same mode and for every choice of n 1 and n the energy level splits into two in the same way as shown above. This will be more important in the discussion in chapter 7. The system of equations and 3 cannot be solved analytically, which is why the following chapters will present two dierent approximations of the two site Holstein model. 5 Non-adiabatic small polaron In the non-adiabatic approximation the lattice moves very fast compared to the electron, which is why J is set to 0. The system of equations then decouples into two independent equations, which describe two decoupled harmonic oscillators of which one is displaced and the other isn't. If a i x 1, x are dened in the form a 1 x 1, x ; n 1, n = x 1 n 1 d x n 30 and a x 1, x ; n 1, n = x 1 n 1 x n d, 31 where the index d represents a displaced harmonic oscillator x i x i + γ Mω, the system of equations reduces to [ ω n d 1 + 1 + ω n + 1 1 ωγ E a 1 x 1, x ; n 1, n = 0 3 and [ ω n 1 + 1 + ω n d + 1 1 ωγ E a x 1, x ; n 1, n = 0. 33 The two solutions of this system of equations are either a 1 x 1, x ; n 1, n is non-zero and dened as in equation 30 and a x 1, x ; n 1, n = 0 or vice versa, where a x 1, x ; n 1, n is dened as in equation 31. They correspond to the polaron either sitting on molecule 1 or molecule. In both solutions the eect of the electron phonon interaction is the shift of one harmonic oscillator by γ the polaron binding energy Mω and lowering the energy of the system by E p = 1 ωγ, which now corresponds to E = ω n 1 + 1 + ω n + 1 E p, 34 6
where the index d is omitted, because it isn't needed. If a 1 x 1, x ; n 1, n is denoted by a l x 1, x ; n 1, n and a x 1, x ; n 1, n by a r x 1, x ; n 1, n, then a new a 1 x 1, x ; n 1, n and a x 1, x ; n 1, n can be constructed in the context of rst order perturbation theory [ [ [ a1 x 1, x ; n 1, n al x = α 1, x ; n 1, n 0 + β, 35 a x 1, x ; n 1, n 0 a r x 1, x ; n 1, n where α and β are constants. By inserting the expression for a i x 1, x ; n 1, n into the original system of equations and 3, multiplying by a l x 1, x ; n 1, n and 3 by a r x 1, x ; n 1, n and integrating them both over the coordinates x 1 and x just like in the derivation of the system of equations and 3 the secular equation reduces to [ ω n d 1 + 1 + ω n + 1 E p E α + β J = 0 36 and [ ω n 1 + 1 + ω n d + 1 E p E β + α J = 0, 37 where J = J dx 1 dx a l x 1, x ; n 1, n a r x 1, x ; n 1, n. 38 Each state of the system is again split into a symmetric and anti-symmetric level with energies E ± = ω n 1 + 1 + ω n + 1 E p ± J. 39 The dierence between the previous non-interacting example is the fact that now the hopping is attributed to the polaron, and the kinetic energy associated with it is J. The integral in equation 38 turns out to be e g, where g = γ. Small polarons can only be formed in the strong coupling regime λ = Ep J γ g 1, which will be argumented in the next chapter and means that the kinetic energy of the polaron is exponentially reduced J = Je g. 40 The eective mass of the non-adiabatic small polaron is therefore very large and hence the polaron is most likely localized. 6 Adiabatic small polaron In the adiabatic approximation where J ω, the electron can hop many times between molecules without inducing a local deformation like in the non-adiabatic case and when it does, the eective mass of the polaron is much smaller. Therefore, the system of equations and 3 can be solved by rst treating x 1 and x as being frozen in the hopping part of the wave function and then looking for the rest of the solution. To do this, a i x 1, x need to be of the form [ a1 x 1, x a x 1, x [ ψx1, x = ξx 1, x φx 1, x, 41 where ψ and φ satisfy H without the phonon contribution while still assuming a non-zero deformation of the molecules. The system of equations they satisfy is [ mω ω γx 1 Ex 1, x ψx 1, x + Jφx 1, x = 0 4 7
and [ mω ω γx Ex 1, x φx 1, x + Jψx 1, x = 0, 43 where the the eigenvectors represent two linear super positions of the two molecular states, which basically just means that the electron is allowed to hop between molecules for now. The eigenvalues are of more interest and they turn out to be E ± x 1, x = 1 ω mω γx 1 + x ± 1 4 Mω3 γ x 1 x + J. 44 Using the two solutions of equations 4 and 43 ψ ± x 1, x and φ ± x 1, x in the system of equations and 3 the following equation is obtained for ξx 1, x [ M x + 1 x + 1 Mω x 1 + x E ± E ξx 1, x = 0. 45 By dening two new variables and separating them in ξx 1, x X = 1 x 1 + x, x = x x 1, ξx 1, x = F Xχx, 46 and dening FX as X n d +, where d' now represents a displacement by γ the operators which operate with respect to the X coordinate as [ M + X + 1 M +ω X + γ M + ω M + ω, FX reduces [ F X = ω n d + + 1 F X, 47 where M + = M is the sum of the two masses of the harmonic oscillators and therefore χx satises [ µ x + U ±x + ω n d + + 1 E p E χx = 0, 48 where µ = M is the reduced mass of the two harmonic oscillators and the third term comes from the energy of the oscillator with coordinate X. U ± x is dened as U ± x = 1 µω x E p µω x + J = E p x x + 1 4λ, 49 where x = µω x γ. Figure shows a plot of U ± x. The third and fourth term in equation 48 are taken only as constants added to the energy of states of χx, which means that if U x is approximated by a harmonic oscillator with ω = ω 1 1, which comes from expanding 8λ 3 U x around x = 0, χ x states are non-displaced harmonic oscillator states with the energy ω n + 1. States of χ +x however represent the solutions of the quantum double potential Ep well problem. The minima of the double well are located at x min = ± 1 1 and µω 4λ the potentials around the two minima can be approximated by harmonic potentials U + x = U min + 1 µω x x min, where ω + = ω 1 1. If the barrier between the two minima is 4λ introduced, the solutions of the entire potential then become the symmetric and anti-symmetric combinations of the harmonic oscillator states belonging to the two minima, which have the same energy χ + x x n d + + ± x n d + +, where d ± represent displaced harmonic oscillators by ± x min and the numbers n d + and n d + + must be the same. The energy dierence between the symmetric and anti-symmetric states is E E pωe g. 8
To sum up, it follows from FX, that a constant vibration of the mean of the two molecular displacements around γ M + ω is always present, which basically means that both molecules deform a little because of the electron hopping between them. In addition, there exist two states of the dierence between both molecular displacements for every n d ± +, which correspond to hopping of the lattice deformation between the two molecular sites together with the electron. The d state corresponds to molecule 1 being more deformed than molecule and the d + state corresponds to the exact opposite state. The previous J in the non-adiabatic model now corresponds to E, which means that the lattice deformation can hop from site to site, where the hopping parameter is E. Another fact of the adiabatic model is that it is only possible when λ > λ c, which is determined by the existence of the two minima λ c = 0.5. Corrections to E have been found to be 4E pω -0.40-0.45-0.50-0.55-0.60 Ux Ep -1.5-1.0-0.5 0.0 0.5 1.0 1.5 x.5.0 1.5 1.0 0.5-1.5-1.0-0.5 0.5 1.0 1.5 x -0.5 Ux Ep Figure : a Plot of U x for λ = 1. b Plot of U x and U + x for λ = 1. E π e g, where g = g [1 1 ln4λ 1. This means that the non-adiabatic 4λ 8λ model produces the exact same result for λ 1, where g = g. The adiabatic approximation is therefore also valid in the so called intermediate coupling regime, where λ λ c. In this regime the eective mass of the polaron is also much smaller than in the λ 1 regime. U + U + U - 7 Evidence of polarons in high-t c superconductors When polaron transport theory is applied to the optical conductivity of a material σω, which is dened with the frequency dependent Ohm's law Jω = σω Eω, where Jω is the electric current density and E is the electric eld, the following formula is obtained σω = ne a πt a[1 e ω/k BT ω E a T [ exp ω 4E a 16E a T, 50 where n is the electron number density, T a is the hopping kinetic energy in the a direction, k B is the Boltzmann constant, T is temperature and E a is the activation energy required to excite an electron into the conducting band, which turns out to be about Ep. This represents an asymmetric Gaussian peak centred at ω = 4E a with the half-width 4 E a k B T. This formula can be understood in the context of the adiabatic small polaron model. Optical conductivity's microscopic picture process is understood as the excitation of a charge carrier electron in this case by a photon into the conducting band. In the context of the adiabatic model, this means exciting an electron from states χ +, where the electron hops between molecules with a larger than free eective mass due to the lattice moving with it, to states χ, which are de-localized in the sense that the electron, on average, has an equal probability of being on either molecule. The Franck-Condon principle states that the lattice does not have time to relax during an optical excitation and that the most likely transitions will be the ones with minimal changes to the molecular coordinates. 9
If T E p is assumed, then only the states near the ground state of χ x will contribute signicantly and their uncertainty in molecular coordinate will contribute to the broadening of the possible absorption energies. The transition requires the photon energy ω = E p, which agrees with the maximum of the Gaussian and the broadening also matches. Figure 3 shows the comparison between experimentally measured and theoretically calculated optical conductivity of three high-t c superconductors in their normal state. It's obvious from the comparison, that for the top and middle plot, theory ts experimental measurements perfectly. The bottom plot shows discrepancy at ω < 3000 cm 1, which is the authors of the article explain with them having bad samples [. Figure 3: Comparison of the optical conductivity σω between experimental measurements thick line and theoretical calculationsdashed line for T l Ba CaCu O 8 top, Y Ba Cu 3 O 7 middle and La x Sr x CuO 3 bottom [. 8 Conclusion The theory of polarons is still under discussion in regard to explaining high-t c superconductivity. Nevertheless, there exists strong experimental evidence, that polarons are present in high-t c superconductors and they explain many properties of such materials, aside from superconductivity. Although they are a consequence of just the rst term in the expansion of the general crystal Hamiltonian with respect to the displacements of nuclei, they are only present in some materials. Nevertheless, they are an important contribution to solid state theory and a key to understanding a little bit more about the world around us. References [1 Polarons and bipolarons. World Scientic, 1995. [ F. C. V. K. Mihailovic, D. and A. Heeger. Application of the polaron-transport theory to sigmaomega in tlbaca1minusxgdxcuo8,ybacu3o7minusdelta, and laminusxsrxcuo4. Physical Review B, 1990. 10