An introduction to the polaron and bipolaron theoretical concepts Yuri Kornyushina) Maıˆtre Jean Brunschvig Research Unit, Chalet Shalva, Randogne, CH-3975 ͑͒
A simple model for the autolocalization of a free charged particle is presented. The polarization well in the model is deep enough for only one localized level. In dielectric materials with a sufficiently large dielectric constant, two charged identical particles can be localized in one polarization potential well, forming a bipolaron. Although several localized levels can be found in more realistic self-consistent models of this type, the more realistic theories require a high level of knowledge of mathematics. Hence, the proposed model can serve as an introduction to the ideas and concepts of autolocalized states. ͓DOI: ͔
I. INTRODUCTION ized if the well is deep enough. If the well is not deep enough, the particle will not be in a localized state. Localized states of current carriers in crystals ͑polarons͒ Let us consider a particle with a wave function ͑r͒ and a have been known since the 1940s ͑see Refs. 1 and 2͒. The charge e. The charge density is (r)ϭe*(r)(r).5 Let the concept of localized states plays an important role in many corresponding electrostatic potential in the dielectric medium areas of condensed matter physics, including super- with a dielectric constant be (r)/, where (r) is the 3 0 0 conductivity. The concept of a polaron was introduced by S. electrostatic potential of the same particle in a vacuum.6 The 4 I. Pekar and subsequently developed by many authors. potential of the charged particle in a dielectric medium can A polaron is a quasiparticle that interacts with the polar- be written as ization oscillations of a crystal lattice ͑especially ionic crys- ͑ ͒ ϵ ͑ ͒Ϫ͓͑Ϫ ͒ ͔ ͑ ͒ ͑ ͒ tals͒ such that an autolocalized state of the current carriers 0 r / 0 r 1 / 0 r . 1 arises. An autolocalized state is a particular case of a local- The first term on the right-hand side of Eq. ͑1͒ represents ized state and is qualitatively different from the free state of the potential of the particle only, and the second term rep- a particle. Autolocalization occurs in a homogeneous me- resents the potential produced by the polarization of the di- dium due to the internal properties of the medium. The phe- ϭϪ electric medium. Thus, the second term, d(r) ͓( nomenon is different from localization in an external poten- Ϫ 1)/ ͔ 0(r), is the part of the potential that acts on the tial. 2,3 particle. This potential is attractive because is always The existing theory of autolocalization is a rather devel- greater than unity. oped and successful theory. But its study requires advanced mathematics, including the calculus of variations. Moreover, the theoretical calculations are very tedious, complicated, II. AUTOLOCALIZED CHARGED PARTICLE and must be done numerically. Instead, we will present a To determine the possibility of autolocalization, we as- simple model for the autolocalization of a free charged par- sume the following model for the localized wave function ticle. This model can be used to introduce the study of au- and hence for the charge density: tolocalization and help students understand the basic con- 3/2 3/4 2 cepts of the subject without performing complicated ͑r͒ϭ͑g / ͒exp͓Ϫ0.5͑gr͒ ͔, ͑2a͒ calculations. The polarization potential well in the proposed ϭe͑g3/3/2͒expϪ͑gr͒2. ͑2b͒ model is deep enough for only one localized level. In a more realistic theory several localized levels can be found in a The quantity 1/g in Eq. ͑2͒ represents the autolocalization self-consistent potential well. radius, because for r greater than 1/g, the density of the In dielectric materials with a sufficiently large dielectric particle practically vanishes. The model represented by Eq. constant, two identical charged particles can be localized in ͑2͒ was chosen because the wave function in Eq. ͑2a͒ is one polarization potential well, forming a bipolaron. A bipo- similar to those arising in a smooth potential well. Equation 2–4 laron is also a well known phenomenon and is thought to ͑2b͒ is an exact consequence of Eq. ͑2a͒.5 The total electric be a carrier of a superconducting current in many materials.3 field, calculated by using Gauss’s theorem, is The polarization potential well for a bipolaron in the simple ͑ ͒ϭ͑ 2͒⌽͑ ͒Ϫ͓ ͑1/2 ͔͒ Ϫ͑ ͒2 ͑ ͒ model that we will discuss is deep enough for only one lo- E r e/ r gr 2eg/ r exp gr , 3 calized level, in contrast to more accurate theories in which where ⌽(x) is the probability integral, that is, the integral of 2–4 several levels can exist. ͓(2/1/2)expϪy2͔ from yϭ0toyϭx. The part of the field When a charged particle is in a localized state, an electro- that acts on the particle produced by the polarization of the static field arises around it. This field polarizes the surround- dielectric medium is ing medium, so that every small volume of the medium ac- ͒ϭ Ϫ ͒ 1/2 ͒ Ϫ ͒2 quires a small dipole moment. These moments are oriented Ed͑r ͓͑ 1 / ͔͕͓2eg/͑ r ͔͓exp ͑gr ͔ so that there is an attraction between the moments and a Ϫ͑e/r2͒⌽͑gr͖͒. ͑4͒ particle in a localized state, which means that the interaction energy is negative. So a charged particle in a localized state The first term of the expansion in r of the right-hand part of creates a potential well for itself in which it could be local- Eq. ͑3͒ yields
͑͒ ͑ ͒ϭ͑ 3 1/2͒ ϩ ͑ ͒ III. THE EQUILIBRIUM VALUE OF g E r 4eg /3 r ¯ . 5 From Eq. ͑5͒, the electrostatic potential (r)is The equilibrium value of the inverse autolocalization ra- dius is determined by the minimum of the energy of the ͑ ͒ϭ͑ ͒Ϫ͑ 1/2͒͑ 3 ͒ 2ϩ ͑ ͒ system. The system consists of a charged particle and the r 0 2/3 eg / r ¯ , 6 polarized dielectric around it. Let us consider the intrinsic with the part produced by the polarization that acts on the energy of the polarization itself. The density of the electro- particle given by static energy is a scalar product of D and E divided by 8. The part of the induction, D, due to the polarization, D ,is ϭ Ϫ 1/2 3 2Ϫ ϩ ͑ ͒ p ͑r͒ ͓͑ 1͒/ ͔͓͑2/3 ͒eg r ͑0͔͒ . 7 ϭ ¯ determined by the equation div Dp 0. Because we are con- This approximate potential corresponds to a three- sidering the intrinsic energy of the polarization, we should dimensional harmonic oscillator. The model wave function not include the charge of the particle. Because Dp has a ϭ assumed in Eq. ͑2a͒ corresponds to the ground state of the radial component only, we have div Dp (dD p /dr) ϩ ϭ three-dimensional harmonic oscillator. The potential repre- (2/r)D p 0. The only solution of this equation, D p sented by Eq. ͑7͒ is formed by many electrons and ions. For ϭC/r2 ͑C is a constant͒, goes to infinity at rϭ0 when this reason we can assume that the total mass of all the par- CÞ0, which is not acceptable. Thus we conclude that D ͓ p ticles whose charges form the polarization potential repre- ϭ0. This solution corresponds to the absence of a free ͑ ͔͒ sented by Eq. 7 is much larger than the mass of the par- charge and the presence of a bound charge.6 The value of the ticle under investigation. Hence, it is possible to approximate ϭ ϩ ϭ electric field is not zero, but D p E p 4 P 0(E p is the the reduced mass by the original value of the mass of a electric field due to the polarization, and P is the density of particle. This point is a crucial one for the phenomenon of the electric dipole moment of the dielectric, which is formed autolocalization. Only the polarization of a medium created ͒ ϭ by the bound charge . Because D p 0, the intrinsic energy of by heavy enough particles can lead to the autolocalization of ϭϪ Þ a charged particle, because the structure of the potential, the polarization is also zero, even though E p 4 P 0. which keeps the particle localized, should be relatively stable The intrinsic self-interaction energy of the particle should with respect to the motion of the localized particle. be excluded. What is left is the energy of the interaction of An autolocalized current carrier in a crystal together with the particle with the polarized medium induced by the charged particle. The energy at the bottom of the potential the surrounding polarization of the medium created by heavy ͓ ͑ ͔͒ inert particles is usually called a polaron.1–4 When the mo- well is W(0) see Eq. 8 , but the particle occupies its ͑ ͒ប ͑ ͒ ͑ ͒ tion of a polaron is considered, we have to take into account ground state with energy 3/2 . Thus, from Eqs. 8 – 10 , that the motion of the surrounding cloud of polarization the interaction energy is charges accompanies the motion of the localized particle. ϭ 1/2 1/4͒ Ϫ ͒ 1/2 ប 1/2͒ 3/2 Wi ͑3 / ͓͑ 1 / ͔ ͑e /m g This phenomenon is expressed by the larger effective mass of a polaron compared to the mass of a bare particle. Hence, Ϫ͑2/1/2͓͒͑Ϫ1͒/͔e2g. ͑12͒ the polaron current carriers have low mobility and the The right-hand side of Eq. ͑12͒ has a minimum at sample has high electrical resistance. Now let us calculate ϭ ϭ͑ 1/2͓͒͑Ϫ ͒ ͔͑ 2 ប2͒ ͑ ͒ the depth of the potential well ͑0͒. For r→ϱ, the potential g ge 16/27 1 / me / . 13 (r) goes to zero, which means that the value of ͑0͒ is ϭ ϭ 2 ប2 For the typical value 5, ge 0.267(me / ). equal to the integral of E(r) from zero to infinity. Hence, we ϭ ͑ ͒ 4 Ͻ ϭ 1/2 At g ge the inequality 11 yields ( 9) 1. The ratio of have (0) 2eg/ . The electrostatic potential energy of ͑ 3͒ប Ϫ ϭ 2 2 to W(0) at g ge is equal to 3, which means that at a charged particle is equal to its charge multiplied by the ͑ ͒ potential due to the polarized medium, and thus from Eq. ͑1͒ least one level the ground state level could be formed in the it is given by potential well. But the energy of the first excited level at g ϭ ͑ 3͒បϩបϭ ប ge is 2 2.5 , which is larger than the depth of W͑0͒ϭϪ͑2/1/2͓͒͑Ϫ1͒/͔e2g. ͑8͒ the potential well ϪW(0) ͑the ratio of 2.5ប to ϪW(0) is 10 ͒. Hence, the potential well in our model is not deep ͑ ͒ ͑ ͒ 9 The energy of the particle is, according to Eqs. 7 and 8 , enough to have a first excited level. Moreover, the approxi- ͑ ͑ ͒ϭ͑ 2 1/2͓͒͑Ϫ ͒ ͔͓͑ 3 ͒ 2Ϫ ͔ϩ mate model potential which is derived from the model wave W r 2e / 1 / g /3 r g ¯ function͒ is not correct for the wave function of the first ϭϪ͑2/1/2͓͒͑Ϫ1͒/͔e2gϩ0.5m2r2ϩ , ͑9͒ excited state of a three-dimensional oscillator. In a more re- ¯ alistic theory several localized levels often can be found in a where m is the original effective mass of the particle and is self-consistent potential well.2–4 So our model is not suitable the angular frequency of a harmonic oscillator, which is for the description of such a case. given by ϭ͑2/31/21/4͓͒͑Ϫ1͒/͔1/2͑e/m1/2͒g3/2. ͑10͒ IV. THE BINDING ENERGY ͑ ͒ ͑ ͒ In the ground state the energy of the three-dimensional The binding energy is determined from Eqs. 12 and 13 : ͑ 3͒ប ϭ ͒ϭϪ ͒ Ϫ ͒ 2 4 ប2͒ ͑ ͒ oscillator is 2 , and it is smaller than the depth of the Wb Wi͑ge ͑32/81 ͓͑ 1 / ͔ ͑me / . 14 Ϫ potential well W(0). From this inequality it follows that ϭ ϭϪ 4 ប2 ϭϪ For 5, we have Wb 0.0805(me / ) 2.19 eV, gϽ͑4/31/2͓͒͑Ϫ1͒/͔͑me2/ប2͒. ͑11͒ which corresponds to a temperature of 25415 K, which is much larger than the ambient temperature. Because the depth Equation ͑11͒ expresses the condition for the autolocalization of the potential well is much larger than kT, all the electrons of the charged particle in a dielectric medium. are in an autolocalized state. So in an autolocalized state, the
2 electrons in a conductivity band have considerably lower en- follows from Eq. ͑9͒ that the potential for the two particles, ergy than in a free ͑plane wave͒ state. This lower energy each one with charge e ͑the total charge is 2e), is influences the concentration of the electrons in a conductiv- ͒ϭϪ 1/2͒ Ϫ ͒ 2 ity band, because their concentration depends exponentially W͑r ͑4/ ͓͑ 1 / ͔e g 7 ͑ on the gap energy. The gap energy is the difference be- ϩ͑4/31/2͓͒͑Ϫ1͒/͔e2g3r2ϩ . ͑19͒ tween the electron energy at the bottom of the conductivity ¯ band and at the top of the valence band.͒ The concentration The last term on the right-hand side of Eq. ͑19͒ is equal to 2 2 of the current carriers significantly influences the electrical 0.5m br . It follows that conductivity and optical properties. Optical properties are in 7 ϭ͑ 3/2 1/21/4͓͒͑Ϫ ͒ ͔1/2͑ 1/2͒ 3/2 ͑ ͒ a great extent dependent on the plasma frequency, which is b 2 /3 1 / e/m g . 20 1/2 ͑ proportional to (n/ma) n is the number density of the The direct interaction energy of the identical particles current carriers in a conductivity band and ma is the effective should be taken into account. It is determined by the integral ͒ mass of an electron in an autolocalized state . So autolocal- of 2E2(r) over all space divided by 8. This integral yields ization significantly affects various physical properties. 0 ϭ ͒1/2 2 ͑ ͒ Wi ͑2/ e g. 21 V. CURRENT CARRIERS ELECTRONS AND The energy of the two particles in the ground state of a „ ប potential well is 3 b . In an autolocalized state the sum of HOLES… IN A SEMICONDUCTOR this energy and Wi is smaller than the depth of the potential The present approach is valid when it is possible to con- well (4/1/2)͓(Ϫ1)/͔e2g ͓see Eq. ͑19͔͒. It follows that sider a separate localized current carrier and to neglect the 1/2 3/2 1/2 1/4 1/2 1/2 presence of the other current carriers. This assumption is 0Ͻg Ͻ͑1/2 3 ͓͒͑Ϫ1͒/͔ ͕͓͑4Ϫ2 ͒Ϫ4͔/ valid when the Debye screening radius 1/gD is much larger ͑Ϫ1͖͒͑m1/2e/ប͒. ͑22͒ than the localization radius of the current carrier 1/ge . When the current carriers present in the sample are of one type only Equation ͑22͒ is the condition for the existence of at least and are not degenerate,7 which means that the density of the one localized level in a self-consistent potential well. Note current carriers is not too high and the temperature is not too that g1/2 is positive when Ͼ4/(4Ϫ21/2)ϭ1.55 only. When low, then g1/2 is negative, there is no localization because the wave function is not localized. The direct interaction of the two g2 ϭ4e2n/kT. ͑15͒ D particles results in an increase of the energy of the system. We have from Eqs. ͑13͒ and ͑15͒ the inequality The attractive interaction of the particles with the polariza- Ӷ ϭ ͓͑Ϫ ͒2 ͔ ͑ 2 2 ប4͒ ͑ ͒ tion of the medium leads to a decrease of the energy. But if n nu 0.00890 1 / kT m e / . 16 is not big enough, this decrease is not sufficiently large to Here nu is the upper limit of the concentration of the current compensate for the energy increase due to the repulsion of carriers, defined by Eq. ͑16͒. On the other hand, to consider two particles. In this case the resulting potential well is not the current carriers as noninteracting, we should find much deep enough and the ground state cannot be formed. In such less ͑on the average͒ than one current carrier inside the materials a bipolaron does not exist. sphere of a Debye screening radius. In this case every current The total energy of the system of two particles follows carrier is screened from the influence of the others. Thus from Eqs. ͑19͒–͑21͒ and is given by 3 Ӷ 4 n/3qD 1, and it follows that W͑g͒ϭ͑23/231/2/1/4͓͒͑Ϫ1͒/͔1/2͑eប/m1/2͒g3/2 ӷ ϭ͑ ͒͑ 2͒3 ͑ ͒ n nl 1/36 kT/e . 17 ϩ͑2/͒1/2e2gϪ͑4/1/2͓͒͑Ϫ1͒/͔e2g. ͑23͒ Here nl is the lower limit of the concentration of the current The minimum of this energy corresponds to carriers, defined by Eq. ͑17͒. The inequalities ͑16͒ and ͑17͒ 1/2ϭ͑ 1/2 3/21/4͓͒͑Ϫ ͒ ͔1/2͕͓͑ Ϫ 1/2͒Ϫ ͔ ͖ can be combined to give ge 1/2 3 1 / 4 2 4 / Ӷ Ӷ ͑ ͒ nl n nu . 18 ϫ͑m1/2e/ប͒. ͑24͒ ϭ ϭ For m equal to the electron mass, 5, and T 300 K, we We can see that g1/2 is positive for Ͼ4/(4Ϫ21/2)ϭ1.55, in ϭ ϫ 15 Ϫ3 ϭ ϫ 20 Ϫ3 e have nl 6.41 10 cm and nu 1.82 10 cm . Be- accord with the previous result. Equation ͑24͒ can be rewrit- cause the concentration of the current carriers in semicon- ten as ductors is about 1017–1018 cmϪ3,6 the inequalities in Eq. ϭ 1/2͒ Ϫ ͒ Ϫ 1/2͒Ϫ 2 2 ប2͒ ͑18͒ are satisfied by the properties of many materials. ge ͑1/54 ͓͑ 1 / ͔͕͓͑4 2 4͔/ ͖ ͑me / . ͑25͒ ϭ ͑ ͒ ϭ 2 ប2 VI. BIPOLARON For 5, Eq. 25 yields ge 0.0416(me / ). This value of the inverse of the autolocalization radius is much smaller Let us consider two identical particles in a potential well. than that calculated from Eq. ͑13͒, which means that the Two current carriers in a self-consistent potential well are localization radius of a bipolaron is much larger than that of called a bipolaron.2–4 Bipolarons play an important role in an autolocalized single charged particle. solid state physics, and they can be superconducting current The equilibrium energy of the system according to Eq. carriers in some materials.3 Both of the charged particles of a ͑23͒ is bipolaron occupy the ground state ͑they could be bosons or W͑g ͒ϭϪ͑1/162͕͓͒͑4Ϫ212͒Ϫ4͔3/ fermions with opposite spins in general͒. Let , , E(r), e 2 Ϫ ͒ 4 ប2͒ ͑ ͒ E0(r), and Ed(r) be the same as before for each particle. It ͑ 1 ͖͑me / . 26
3 Ͼ ͑ ͒ Again the minimum energy W(ge) is positive for 4/(4 and c is the speed of light in vacuum. Equation 27 is valid Ϫ21/2)ϭ1.55, in accord with our previous result. For ϭ5, for r smaller than ប/mc.8 The first term in Eq. ͑27͒ describes ϭϪ 4 ប2 the minimum energy W(ge) 0.0140(me / ), which is the part of the potential produced by the bare charged par- much smaller than the binding energy of two separate par- ticle, while the second term describes the part of the poten- ticles. The autolocalized state of two particles ͑bipolaron͒ is tial, produced by the polarization of the vacuum. a metastable state because its minimum energy is higher than Because there is no self-interaction in nature, the part of that of the two separate particles. the electrostatic potential that acts on the particle is ͓see Eq. ͑27͔͒ VII. DISCUSSION ͒ϭϪ ␣ ͒ Ϫ ប ͒ ͑ ͒ a͑r ͑2 e/3 r ͓1.41 log ͑ /mcr ͔. 28 We have proposed a model that describes the concept of autolocalization. The advantage of this model is that it allows To analyze this potential we introduce the notation x ϭប us to study the basic concepts of autolocalization without /mcr. Then using complicated and mainly numerical calculations. ͑x͒ϭ͑2␣emc/3ប͓͒͑x log x͒Ϫ1.41x͔. ͑29͒ We have shown that localized states can form in a regular a dielectric or semiconductor with a sufficient ͑but not too ϭ ϭ ϭ This function has a minimum at x xe exp (0.41) 1.51, high͒ density of current carriers. Only one ͑ground͒ level can ϭ ប which corresponds to re 0.664( /mc). The minimum en- be formed in a polarization potential well. If the dielectric ϭϪ ␣ 2 ប ␣ϭ ergy is e e 3.01 e mc/3 .At 1 for an electron, constant is greater than 1.55, bipolaron states could exist as ϭϪ metastable states, because the binding energy of two separate e e 1191 eV. It looks like that at such a deep energy mi- current carriers is larger than the binding energy of the two nimum the wave function of the particle can be localized wi- carriers in one bipolaron well. Thus, it is obvious that there thin the sphere of radius re . But the potential of the well, exists only one state in our model of a bipolaron polarization a(r), in the vicinity of the energy minimum is almost zero Ͼ well. The localization radius of a bipolaron is considerably for r re . So the well is very narrow, kinetic energy is very Ϸ larger than that of a polaron. high and the localization at r re is impossible as expected. Of course, reality is much more complicated than the ͒ model discussed here. Several excited levels of a polaron and a Electronic mail: [email protected] bipolaron can exist, which also influence the properties of 1H. Frohlich, H. Pelzer, and S. Zienau, ‘‘Properties of slow electrons in ͑ ͒ solids.3,4 polar materials,’’ Philos. Mag. 41, 221–242 1950 . 2I. G. Austin and N. F. Mott, ‘‘Polarons in crystalline and non-crystalline materials,’’ Adv. Phys. 50, 757–812 ͑2001͒. VIII. SUGGESTED PROBLEM 3A. S. Alexandrov and N. F. Mott, Polarons and Bipolarons ͑World Scien- tific, Singapore, 1995͒. Consider a charged particle in a vacuum. Calculate and 4S. I. Pekar, Investigation on the Electronic Theory of Crystals ͑Nauka, analyze the part of the electrostatic potential that acts on the Moscow, 1951͒. particle. 5L. D. Landau and E. M. Lifshitz, Quantum Mechanics ͑Pergamon, Oxford, Solution: It is well known in quantum electrodynamics 1984͒. 6L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media that near the point charge in a vacuum, the electrostatic po- ͑ ͒ tential has the form8 Pergamon, Oxford, 1984 . 7C. Kittel, Introduction to Solid State Physics 7th ed. ͑Wiley, New York, ͑r͒ϭ͑e/r͒Ϫ͑2␣e/3r͓͒1.41Ϫlog ͑ប/mcr͔͒, ͑27͒ 1996͒. 8E. M. Lifshitz and L. P. Pitaevskii, Relativistic Quantum Theory ͑Perga- where ␣ is a constant, m is the ͑bare͒ mass of the particle,8 mon, Oxford, 1986͒, Part 2.
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