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Effect of electron correlation in Kronig-Penney model
Volume 47A, number 1
PHYSICS LETTERS
EFFECT OF ELECTRON CORRELATION IN KRONIG-PENNEY
25 February 1974
MODEL
K.A. CHAO and B. MAGNUSSON Department of Physics, University of Warwick, Coventry CV4 7AL, U.K. Received 15 January 1974 A one-dimensional model is proposed to investigate the symmetry-breaking-induced transition from the band limit to the atomic limit. The model calculation shows sharper transition for stronger intra-atomic correlation effect.
In recent years, Mott transition [l] has received considerable attention. Mott [2] argues that free carriers in solids are only possible if they are present in such numbers as to give a screening constant big enough to prevent formation of electron-hole pairs. Since electron screening becomes less effective with increasing interatomic distance, a sharp metal-insulator (M-I) transition must exist. Johansson [3] has incorporated this idea of electron-hole pair with the Hubbart model [4] to illustrate the equivalence of the Hubbard’s and the Mott’s criteria for M-I transition. Mott transition is closely connected to the breaking of translational symmetry due to the electron correlation. Whilst metallic electrons sense a periodic potential, at large interatomic distance electrons no longer move in a periodic potential and therefore should be local&d. A detailed discussion on such localised states is given by March and Stoddart [ 51. In this letter we present a model to study the symmetry-breaking-induced M-I transition when the interatomic distance is increased. Since we are not interested in the magnetic ordering, the electronic spin is ignored although our model can be generalised to include the spin by including the atomic excited states. We use fig. 1 to illustrate our model. It consists of an array of one-electron atoms with the atomic potential approximated by a square welI of width a and depth V, . At the atomic limit as indicated by A, the interatomic distance b is so large that there is no overlap between neighbouring electronic wave functions. If we transfer one electron from the ith atom to the jth atom, then the jth atom is doubly occupied. Due to the strong Coulomb repulsion between the electrons, the transferred electron is less bounded in the jth atom than it is in the i th atom. Therefore in terms of the square well potential approximation, the trans-
Fig. 1. A schematic illustration of the symmetry-breakinginduced M-I transition. Details are explained in the text.
ferred electron sees a “shallower“ well. We define U, as a parameter to label the reduction of the well depth. Then U, measures the strength of electron correlation effect. Accordingly, in one-electron picture. each electron senses an impurity-type potential sketched as B. As the interatomic distance b decreases, electrons begin to tunnel through the potential barriers. We thus have a one-dimensional impurity problem which can be solved exactly [6]. The ground state wave function JI(x) is localised and its envelope function decays exponentially as exp (+.x) where P depends on V,, U, and b. This is shown in C of fig. 1. The tail of I(,(x) extends into the other atoms and screens the atomic potential. Therefore V, is reduced to Vi by screening. Since U, depends on the electronic wave functions, we should compute a new Ui in terms of the impurity states. If we neglect the interatomic correlation effect, both Vi and Q can be expressed as functions of U, 79
Volume 47A, number 1
PHYSICS LETTERS
Fig. 2. Renormalized self-consistent potential well depth Vf (dotted curves) and correlation energy Uf (solid curves) as functions of the interatomic distance b.
and V,. For given values of a, b, I’, and U,, we iterate this process until we obtain the self-consistent solutions Vf and Uf. As b + 0, we expect to recover the band model as shown by D. In our model calculation, we assume a = 1.2 A and increase b from the band limit to 3.4 A. The results are shown in fig. 2. The breaking of the translational symmetry is indicated by Uf/Uo, while Vf/Vo gives the strength of electron screening. At the band limit b = 0, UffUo = 0 and Vf/Vo = 1-Uo/Vo. However, at the atomic limit where there is no overlap of wave functions, we should have Uf/Uo = 1 and Vf/Vo = 1. Each curve in fig. 2 is marked by two parenthesised numbers (V, ; Uo), both in units of eV. For curves (4;0.4) and (10; l), U, is much less than Vo; corresponding to weak correlation effect. It is seen that the energy band theory is valid for this case even at large b, as is expected. On the other hand, for strong correlation where U, is comparable to V, as in curves (4;3.2) and (10;8), a sharp transition occurs between the band limit and the atomic limit. These results show a general feature that the sharpness of the transi-
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25 February 1974
tion depends on the electron correlation effect while the outset of the transition is determined by the atomic potential strength. This conclusion will be justified in a detailed paper. In Kronig-Penny model, Uf is neglected in energy band calculations. Let IVbe the so computed “correlated” band width, then it is interesting to find out the value of Uf/W for the transition. For the case of U, = 0.8 V,, we have found a transition region around Uf/ W = 1. If the correlation is weak, our calculation shows a very small Uf/W ratio. If we ignore the electron correlation, i.e. if we assume a perfect periodicity, then all the states are Bloch type. Under this condition Uf = 0 and Vf = V,- U,. Again we can use these values in Kronig-Penny model to compute the “bare” band width. The amount of band narrowing W/W, is a monotonically decreasing function of U,/W. When U, is much less than V,, W/W, in our calculation is very close to one. lherefore it supports the band model. However for U. = 0.8 V,, we obtain a rapid drop of W/W, from 0.9 to 0.65 around Uf/ W = 1. A sharp M-I transition thus occurs when the electron correlation energy is comparable to the electron hopping energy. One of the authors, (KAC), wants to express his sincere thanks to Professors Mott and March for very helpful discussion.
References [l] For a review of Mott transition, see N.F. Mott and A. Zinamon, Rep. Progr. Phys. 33 (1970) 881. [Z] N.F. Mott, Phil, Mag. 6 (1961) 287. [3] J. Hubbard, Proc. Roy. Sot. A276 (1963) 238; A281 (1964) 401. [4] B. Johansson, J. Phys. C6 (1973) L71. [5] N.H. March and J.C. Stoddart, Rep. Progr. Phys. 31 (1968) 531. [6] F. Setiz, The modern theory of solids (McGraw-Hill 1940) Sec. 70.
PHYSICS LETTERS
EFFECT OF ELECTRON CORRELATION IN KRONIG-PENNEY
25 February 1974
MODEL
K.A. CHAO and B. MAGNUSSON Department of Physics, University of Warwick, Coventry CV4 7AL, U.K. Received 15 January 1974 A one-dimensional model is proposed to investigate the symmetry-breaking-induced transition from the band limit to the atomic limit. The model calculation shows sharper transition for stronger intra-atomic correlation effect.
In recent years, Mott transition [l] has received considerable attention. Mott [2] argues that free carriers in solids are only possible if they are present in such numbers as to give a screening constant big enough to prevent formation of electron-hole pairs. Since electron screening becomes less effective with increasing interatomic distance, a sharp metal-insulator (M-I) transition must exist. Johansson [3] has incorporated this idea of electron-hole pair with the Hubbart model [4] to illustrate the equivalence of the Hubbard’s and the Mott’s criteria for M-I transition. Mott transition is closely connected to the breaking of translational symmetry due to the electron correlation. Whilst metallic electrons sense a periodic potential, at large interatomic distance electrons no longer move in a periodic potential and therefore should be local&d. A detailed discussion on such localised states is given by March and Stoddart [ 51. In this letter we present a model to study the symmetry-breaking-induced M-I transition when the interatomic distance is increased. Since we are not interested in the magnetic ordering, the electronic spin is ignored although our model can be generalised to include the spin by including the atomic excited states. We use fig. 1 to illustrate our model. It consists of an array of one-electron atoms with the atomic potential approximated by a square welI of width a and depth V, . At the atomic limit as indicated by A, the interatomic distance b is so large that there is no overlap between neighbouring electronic wave functions. If we transfer one electron from the ith atom to the jth atom, then the jth atom is doubly occupied. Due to the strong Coulomb repulsion between the electrons, the transferred electron is less bounded in the jth atom than it is in the i th atom. Therefore in terms of the square well potential approximation, the trans-
Fig. 1. A schematic illustration of the symmetry-breakinginduced M-I transition. Details are explained in the text.
ferred electron sees a “shallower“ well. We define U, as a parameter to label the reduction of the well depth. Then U, measures the strength of electron correlation effect. Accordingly, in one-electron picture. each electron senses an impurity-type potential sketched as B. As the interatomic distance b decreases, electrons begin to tunnel through the potential barriers. We thus have a one-dimensional impurity problem which can be solved exactly [6]. The ground state wave function JI(x) is localised and its envelope function decays exponentially as exp (+.x) where P depends on V,, U, and b. This is shown in C of fig. 1. The tail of I(,(x) extends into the other atoms and screens the atomic potential. Therefore V, is reduced to Vi by screening. Since U, depends on the electronic wave functions, we should compute a new Ui in terms of the impurity states. If we neglect the interatomic correlation effect, both Vi and Q can be expressed as functions of U, 79
Volume 47A, number 1
PHYSICS LETTERS
Fig. 2. Renormalized self-consistent potential well depth Vf (dotted curves) and correlation energy Uf (solid curves) as functions of the interatomic distance b.
and V,. For given values of a, b, I’, and U,, we iterate this process until we obtain the self-consistent solutions Vf and Uf. As b + 0, we expect to recover the band model as shown by D. In our model calculation, we assume a = 1.2 A and increase b from the band limit to 3.4 A. The results are shown in fig. 2. The breaking of the translational symmetry is indicated by Uf/Uo, while Vf/Vo gives the strength of electron screening. At the band limit b = 0, UffUo = 0 and Vf/Vo = 1-Uo/Vo. However, at the atomic limit where there is no overlap of wave functions, we should have Uf/Uo = 1 and Vf/Vo = 1. Each curve in fig. 2 is marked by two parenthesised numbers (V, ; Uo), both in units of eV. For curves (4;0.4) and (10; l), U, is much less than Vo; corresponding to weak correlation effect. It is seen that the energy band theory is valid for this case even at large b, as is expected. On the other hand, for strong correlation where U, is comparable to V, as in curves (4;3.2) and (10;8), a sharp transition occurs between the band limit and the atomic limit. These results show a general feature that the sharpness of the transi-
80
25 February 1974
tion depends on the electron correlation effect while the outset of the transition is determined by the atomic potential strength. This conclusion will be justified in a detailed paper. In Kronig-Penny model, Uf is neglected in energy band calculations. Let IVbe the so computed “correlated” band width, then it is interesting to find out the value of Uf/W for the transition. For the case of U, = 0.8 V,, we have found a transition region around Uf/ W = 1. If the correlation is weak, our calculation shows a very small Uf/W ratio. If we ignore the electron correlation, i.e. if we assume a perfect periodicity, then all the states are Bloch type. Under this condition Uf = 0 and Vf = V,- U,. Again we can use these values in Kronig-Penny model to compute the “bare” band width. The amount of band narrowing W/W, is a monotonically decreasing function of U,/W. When U, is much less than V,, W/W, in our calculation is very close to one. lherefore it supports the band model. However for U. = 0.8 V,, we obtain a rapid drop of W/W, from 0.9 to 0.65 around Uf/ W = 1. A sharp M-I transition thus occurs when the electron correlation energy is comparable to the electron hopping energy. One of the authors, (KAC), wants to express his sincere thanks to Professors Mott and March for very helpful discussion.
References [l] For a review of Mott transition, see N.F. Mott and A. Zinamon, Rep. Progr. Phys. 33 (1970) 881. [Z] N.F. Mott, Phil, Mag. 6 (1961) 287. [3] J. Hubbard, Proc. Roy. Sot. A276 (1963) 238; A281 (1964) 401. [4] B. Johansson, J. Phys. C6 (1973) L71. [5] N.H. March and J.C. Stoddart, Rep. Progr. Phys. 31 (1968) 531. [6] F. Setiz, The modern theory of solids (McGraw-Hill 1940) Sec. 70.