The Hartree Equations So far we have ignored the effects of electron-electron (e-e) interactions by working in the independent electron approximation. In this lecture, we shall discuss how this effect may be treated without moving too far afield from the approach we have been using. That is, we shall still construct the problem in the one-electron Schrodinger equation with a proper choice of the potential U(r). Clearly, we should include the potential of the ions, U ion (r): ion 1 U ( r ) Ze. () 4 r R To incorporate the fact that the electron feels the electric fields of all the other electrons, we treat the remaining electrons as a smooth distribution of negative charge with charge density. The potential energy of the given electron in their field would be U el R 3 ( r ') ( r ) e d r' 4 r R Furthermore, if we persisted in an independent electron picture, the contribution of an electron in the quantum level i to the charge would be. (1) (3) The total electronic charge density would then be (4) where the sum extends over all occupied one-electron levels in the metal. (Here, we neglect over-counting the contribution by the given electron itself as it is small compare to that by the rest of the 1 or so electrons.) Placing (5) in (3) and letting U = U ion + U el we arrive at the one-electron equation: ion 3 1 i ( r ) U ( r ) i ( r ) e d r' j ( r ) i ( r ) i i ( r ) m (6) j 4 r R Note that (6) represents a set of equations, one for each occupied one-electron level i. It is known as the Hartree equations. These equations need to be solved self-consistently (for the one-electron wave functions and energies). Typically, a form is guessed for U el on the basis of which the equations are solved. A new U el is then computed from the resulting wave functions, i (r), and a new Schrodinger equation is solved. This procedure is continued until further iterations do not alter the potential within some tolerance. (5) Mostly reproduced from A&M 1
In essence, equation (6) simplifies the e-e interactions by describing the given electron as interacting with the remaining electrons through the mean field obtained by averaging over the other electrons positions with a weight determined by their wave functions. Although this is a crude approximation, it still leads to a task of considerable numerical complexity. To improve upon the Hartree equations is quite difficult. A perhaps more important issue is that some important physical features of e-e interactions simply cannot be treated by using the simple approach discussed above. We shall survey some of them below, which include: 1. The extension of the self-consistent field equations (6) to include what is known as exchange.. The phenomenon of screening, which is important in developing a still more accurate theory of e-e interactions, and in accounting for the response of metallic electrons to charged particles such as ions, impurities and other electrons. 3. The Fermi liquid theory of Landau, which provides a phenomenological way of predicting the qualitative effects of e-e interactions on the electronic properties of metals, as well as offering an explanation for the quite extraoridinary success the independent electron approximation has had. Exchange: The Hartree-Fock Approximation The Hartree equations (6) have a fundamental inadequacy that is not at all evident from the above derivation. To appreciate the inadequacy, we return to the exact N-electron Schrodinger equation and cast it into the equivalent variational form, which asserts that a solution to H = E is given by any state that makes the following quantity stationary. (7) where (8) and s i labels the spin of electron i. The ground-state wave functionis that that minimizes (7). In practice, approximate ground states are constructed by minimizing (7) not over all, a limited class of wave functions chosen to have a more tractable form. It can be shown that the Hartree equations (6) follow from minimizing (7) over all of the form: where the i are a set of N orthonormal one-electron wave functions. So, (6) give the best approximation to the full N-electron wave function that can be represented as a simple product of one-electron levels. The wave function (9) is, however, incompatible with the Pauli principle, which requires the sign of to change when any two of its arguments are interchanged: (9) Mostly reproduced from A&M
The simplest generalization of the Hartree approximation that incorporates the antisymmetry requirement (1) is to replace the trial wave function (1) by a Slater determinant of one-electron wave functions: It is a linear combination of the product (9) and all other products obtainable from it by permutation of the r j s j among themselves, added together with weights +1 or -1 so as to warrant condition (1). This sum can be written compactly as the determinant of an N x N matrix: (1) (11) (1) It can be shown that if the energy (7) is evaluated in a state of the form (1), and the result is minimized with respect to the i *, one obtains a generalization of the Hartree equations known as the Hartree-Fock equations: ion el i ( r ) U ( r ) i ( r ) U ( r ) i ( r ) m 3 e d r' j * ( r ') i ( r ') j ( r ) sisj i i ( r ), (13) 4 r R j These equations differ from the Hartree equations (6) by an additional term on the LHS, known as the exchange term. The complexity introduced by the exchange term is considerable. Like the self-consistent field U el (often referred to as the direct term), it is nonlinear in, but unlike U el, it is not of the form V(r)(r). Instead, it has the structure, d 3 r V(r, r )(r ) i.e., it is an integral operator. As a result, the Hartree-Fock equations are in general quite intractable. The one exception is the free electron gas and the Hartree-Fock equations can be solved exactly. As we shall see, the free electron solution suggests a further approximation that makes the Hartree-Fock equations in a periodic potential more manageable. Hartree-Fock Theory of Free Electrons We adopt i to be the set of free electron plane waves, (14) Mostly reproduced from A&M 3
We argue that the Slater determinant constructed from these plane waves, in which each wave vector less than k F occurs twice (once for each spin orientation), gives a solution to the Hartree-Fock (H-F) equation for free electrons. It is because for i that are plane waves, U el as constructed from (5) will be uniform. At the same time in the free electron gas the ions are represented by a uniform distribution of positive charges with the same density as the electronic charge. Hence the ion potential U ion is precisely canceled by U el. So, only the exchange term survives, which can be evaluated by writing the Coulomb interaction in terms of its Fourier transform: (15) If this expression is substituted into the exchange term in the H-F equation (13) and the i are taken to be plane waves (14), the LHS of (13) assumes the form where (16) (17) and (18) This shows that plane waves do indeed solve the H-F equation, and that the energy of the one-electron level with wave vector k is given by (17). The function F(x) is plotted in Fig. 17a, and the energy (k) in Fig. 17b in A&M and reproduced here. Mostly reproduced from A&M 4
Screening The phenomenon of screening is one of the simplest and important manifestations of e-e interactions. Here we only consider screening in a free electron gas. Suppose a positively charged particle is placed at a given position in the electron gas and rigidly held there. It will attract electrons, creating a surplus of negative charge in its neighborhood, which reduces or screens its field. In treating this screening, it is convenient to introduce two electrostatic potentials. The first, ext, arises solely from the positively charged particle itself, and thus satisfies Poisson s equation: - ext (r) = ext (r), (19) where ext (r) is the particle s charge density. The second potential, (r) produced by both the positively charged particle and the cloud of screening electrons it induces. It thus satisfies where (r) is the full charge density, - (r) = (r), () (r) = ext (r) + ind (r) (1) and ind (r) is the charge density induced in the electron gas by presence of the external particle. By analogy with the theory of dielectric media, one assumes that and ext are linearly related by an equation of the form ext (r) = d 3 r (r, r )(r ). () In a spatially uniform electron gas, can depend only on the separation between r and r, but not their absolution position. Thus, () becomes With this, the corresponding Fourier transforms satisfy where the Fourier transforms are defined by ext (r) = d 3 r (r - r )(r ). (3) ext (q) = (q)(q), (4) (q)= d 3 r e -iqr (r), (5) (r)= 1/() 3 d 3 q e iqr (q), (6) Mostly reproduced from A&M 5
with similar equations for and ext. The quantity (q) is called the (wave vector dependent) dielectric constant of the metal. When written in the form, (r) ext (q) / (q) (7) the kind of relation expressed in (7) is familiar in elementary discussions of dielectrics, where however, the fields are generally uniform so that the dependence on wave vector does not come into play. The quantity that turns out to be the most natural to calculate directly is the induced charge density ind (r). In sufficiently weak, ind and are linearly related. Then their Fourier transforms will satisfy a relation of the form ind (r) (q)(q) (7) We can relate to as follows. The Fourier transforms of the Poission equations are Together with (1) and (7) these give q ext (q) = ext (q), q (q) = (q), (8) or q (q) - ext (q)] = (q)(q), (9) (q) = ext (q)/[1 - (q)/q ]. (3) Comparing this with (7) leads to ind ( q) 1 ( q) (q) = 1 1. (31) q q (q) Up to now, we have only assumed that ind and are linearly related. To calculate, one needs to make serious approximations. Two common theories of are employed. Both of them are essentially simplified Hartree calculation of the charge induced by impurities. The first the Thomas-Fermi method is basically the classical limit of the Hartree theory; the second the Lindhard method is basically an exact Hartree calculation of the charge density in the presence of the self-consistent field of the external charge plus electron gas. The former has the advantage that it holds even when ind and are not linearly related. But it has the disadvantage that it is reliable only for very slowly varying external potentials. When the problem is linearized, the two methods give the same result at small values of q; but when q is not small, the Lindhard method is more accurate. Mostly reproduced from A&M 6
Thomas-Fermi Theory of Screening In principle, to find the charge density in the presence of (r) = ext (r) + ind (r), we must solve the one-electron Schrodinger equation, and then construct the electronic density from the one-electron wave function using (5). The Thomas-Fermi approach simplifies this procedure by assuming that (r) is a very slowly varying function of r so that it is meaningful to write the electron energy (in the presence of (r)) at position r to be: To calculate the charge density produced by these electrons we place their energies into the expression for the electron number density (with = 1/k B T): (3) (33) The induced charge density is en(r) + en, where the second term is the density of the uniform positive background, which is the density of the system when ext (r), and thus (r) is zero: Combining (34) and (35), we find This is the basic equation of the nonlinear Thomas-Fermi theory. As mentioned above, we treat the problem where is small enough that we can expand (36) in leading order (34) (35) (36) (37) Comparing (37) with ind (r) (q)(q) in (7), we find: (38) Substituting this in (31) gives the Thomas-Fermi dielectric constant e n ( q ) 1. (39) q Mostly reproduced from A&M 7
Defining a Thomas-Fermi wave vector k, n k e,. (39) (39) can be expressed as k ( q) 1. (4) q To illustrate the significance of k, consider the case where the external potential is that of a point charge. ext Q ext Q ( r ), ( q) (41) 4 r q The total potential in the metal will then be 1 ( q) ( q) ext ( q) Q ( q k. ) (4) The Fourier transform can be inverted to give ( 3 d q iqr Q Q ik r r ) e e 3 ( ) ( q k ) 4 r. (43) This is known as a screen Coulomb or Yukawa potential, with a screening length of ~ 1/k. To estimate k, note that for a free electron gas, when T << T F, n / is simply the density of levels at the Fermi energy, g( F ) = mk F /ħ. Therefore, Since r s /a to 6 for metals, k is of the order of k F, i.e., disturbances are screened in a distance similar to the interparticle spacing. Thus the electrons are highly effective in shielding external charges. Lindhard Theory of Screening In the Lindhard approach, one returns to the Schrodinger equation (3), and exploits from the outset the fact that the induced density is required only to linear order in. It is then a Mostly reproduced from A&M 8
routine matter to solve (3) only to linear order by perturbation theory. Once the wave functions are known, one can compute the change in electronic charge density by (6). Below we quote the result, which states that where f k = 1/{exp[(ħ k /m - )] + 1}denotes the equilibrium Fermi function for a free electron. When q << k F, the numerator of the integrand can be expanded about its value at q = : The term linear in q in this expansion gives the Thomas-Fermi result as expected. But when q becomes comparable to k F, there is considerably more structure in the Lindhard dielectric constant. At T =, the integrals in (44) can be performed explicitly to give (44) (45) (46) The quantity in square brackets, which is one at x =, is the Lindhard correction to the Thomas-Fermi result. Note that at q = k F, the dielectric constant = 1 /q is not analytic. Frequency-Dependent Lindhard Screening If the external charge density has time dependence e -it, the induced potential and charge density will also have such a time dependence, and the dielectric constant will depend on both frequency and wave vector. In the limit where collisions can be ignored, one can show that the static result (44) need be modified only as follows: (47) This result is of considerable importance in the theory of lattice vibrations in metals, as well as in the theory of superconductivity. Note that when q approaches zero at fixed, the Lindhard dielectric constant reduces to the Drude result for spatially uniform disturbance. Mostly reproduced from A&M 9
Screening the Hartree-Fock Approximation We have discussed screening by metallic electrons of an externally imposed charge distribution. But screening will also affect the interaction of two electrons with each other. By viewing the two electrons as external charges, one can improve the H-F equations for screening. Recall that the H-F equations (13) are: ion el i ( r ) U ( r ) i ( r ) U ( r ) i ( r ) m 3 e d r' j * ( r ') i ( r ') j ( r ) sisj i i ( r ), (13) 4 r R j One cannot tamper with the Hartree self-consistent field term, since this is the term that gives rise to the screening in the first place. However, one may be tempted to replace the e-e interaction occurring in the exchange term in (17) by its screened form, i.e., 1. ( k k ') This eliminates the singularity responsible for the anomalous divergence in the one electron velocity at k = k F, for in the neighborhood of q = the screened interaction approaches not e /q but e /k. Fermi Liquid Theory Here we examine some subtle arguments, primarily due to Landau, that (a) explain the success of the independent electron approximation and (b) indicate how, in many cases, the consequences of e-e interactions can qualitatively be taken into account. Landau s approach is known as the Fermi liquid theory, which was designed to deal with the liquid state of He-3, but has been applied to the theory of e-e interactions in metals. We first observe that up to this point our analysis of e-e interactions has led to a substantially modified (k) relation for the one-electron levels (17), but has not in any substantial way challenged the basic structure of the independent electron model. Specifically, in the H-F approximation we continue to describe the stationary electron states by specifying which one-electron levels i are present in the Slater determinant. The N-electron wave function therefore has the same structure as that for noninteracting electrons. It is not clear that this is a sensible way to describe the stationary states of the N-electron system, even with suitably modified energies. There is, however, reason to expect that this may be the case for electrons with energies near the Fermi energy. Consider a set of noninteracting electrons. If we imagine gradually turning on the interactions between electrons, they will lead to two kinds of effects: Mostly reproduced from A&M 1
1. The energies of each one-electron level will be modified, as illustrated by the H-F approximation.. Electrons will be scattered into and out of the single-electron levels, which are no longer stationary. This does not happen in the H-F approximation, where oneelectron levels continue to give valid stationary states of the interacting system. Whether this scattering is able to invalidate the independent electron picture depends on its rate. If it is sufficiently slow, we can introduce a relaxation time and treat the scattering the same way we did before. If it should happen (and we shall see that it usually does) that the e-e relaxation time is much larger than other relaxation times, then we can safely ignore it and use the independent electron model with more confidence, subject only to modifications required by the altered (k) relation. One may expect the e-e scattering rate to be quite high, since the Coulomb interaction, even when screened, is quite strong. However, the exclusion principle drastically reduces the scattering rate in many cases. This reduction occurs when the electronic configuration differs only slightly from its thermal equilibrium form (as is the case in all of the transport processes we have discussed). To illustrate this effect, suppose that the N- electron state consists of a filled Fermi sphere (equilibrium at T = ) plus a single excited electron in a level with 1 > F. In order for this electron to be scattered, it must interact with an electron of energy, which must be less than F, since only those levels are occupied. The exclusion principle requires that these two electrons can only scatter into unoccupied levels, 3 and 4 whose energies must therefore be > F. So we require that In addition, energy conservation requires that < F, 3 > F, 4 > F. (48) 1 + = 3 + 4 (49) When 1 = F, (48) and (49) can only be satisfied if = 3 = 4 = F. Thus the allowed wave vectors for electrons, 3, and 4 occupy a region of k-space of zero volume (on the Fermi surface), and thus give a vanishingly small contribution to the process. In the language of scattering theory, one says that there is no phase space for the process. When 1 is slightly above F, the other three energies can now vary within a shell of thickness of order 1 - F about the Fermi surface, and remain consistent with (48) and (49). This leads to a scattering rate of order ( 1 - F ). The order is note cubic because once and 3 have been chosen from the allowed phase space, energy conservation allows no further choice for 4. In the above, we have considered the situation at T =. Suppose T is nonzero. Then there will be partially occupied levels in a shell of width k B T about F. This provides an additional range of choice of order k B T in the energies satisfying (48) and (49), and thus leads to a scattering rate going as (k B T), even when 1 = F. Altogether, we conclude that at temperature T, an electron of energy 1 near the Ferm surface has a scattering rate 1/ that depends on its energy and the temperature in the form Mostly reproduced from A&M 11
where the coefficients a and b are independent of 1 and T. Thus the e-e interaction relaxation time can be made as large as one wishes by going to sufficiently low temperatures and considering electrons sufficiently close to the Fermi surface. Since it is only the electrons within k B T of the Fermi energy that affect most transport properties, the physically relevant relaxation time goes as 1/T. As estimated by A&M, the considered above is of the order of 1-1 s at room temperature. As one may recall from Chapter 1, the relaxation times at room temperature is of the order of 1-14 s. We therefore conclude that at room temperature e-e scattering proceeds at a rate 1, times (within orders of magnitude accuracy) slower than the dominant scattering mechanism, and thus is negligible. Since increases as 1/T with decreasing temperature, it is possible that it can be of little consequence at all temperatures. (5) Fermi Liquid Theory: Quasiparticles The above argument indicates that if the independent electron picture is a good first approximation, at least for levels near the Fermi energy, e-e scattering will not invalidate that picture even if the interactions are strong. However, if the e-e interactions are strong it is not at all likely that the independent electron approximation will be a good first approximation, and it is therefore not clear that our argument has any relevance. Landau acknowledged that the independent electron picture was not a valid starting point. He emphasized, however, that the argument described above remains applicable, provided that an independent something picture is still a good first approximation. He christened the somethings quasiparticles. If the quasiparticles obey the exclusion principle, the argument we have given works as well for them as it does for independent electrons, acquiring thereby a much wider validity. For more details, please refer to A&M s book. Mostly reproduced from A&M 1