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Transcript

Density functional theory Seminar Ib - 1. letnik, II. stopnja Author: Matic Pečovnik Mentor: Doc. Dr. Tomaž Rejec Faculty of Mathematics and Physics, University in Ljubljana Abstract Many-body problems in physics have been the focal point of quantum mechanics in the previous decades. Such problems are found in solid state physics, complex molecules and so on. Most of the theoretical work must be done numerically. Here we have an issue with the computational power of computers as the solving of the Schroedinger many-body equation requires computational power that scales exponential with the number of particles in the system. In this seminar I present to the reader the need to transition from wave-centered to density-centered approach when performing theoretical analysis on complex many-body systems. Then I present the basic concepts of the density functional theory (DFT) first introduced by Kohn and Sham that offers a solution to problems presented above. The second half of my seminar will be focused on simulating the behaviour of a two dimensional electron gas with the use of density functional theory. 1 Contents 1 Introduction to density functional theory 1.1 Wave-centered approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Density functional theory - A workaround the Van Vleck catastrophe . . . . . . . . 2 2 3 2 The self-consistent Kohn-Sham equations 2.1 Uniqueness of the external potential - Hohenberg-Kohn Theorem 2.2 Obtaining the ground state . . . . . . . . . . . . . . . . . . . . . . 2.3 Local density approximation . . . . . . . . . . . . . . . . . . . . . 2.4 Improvements of the local density approximation . . . . . . . . . 2.5 Numerical packages for DFT calculations . . . . . . . . . . . . . . 3 4 4 5 6 7 3 An application of DFT - 2D electron gas 4 Conclusion 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 10 Introduction to density functional theory In this first chapter I will illuminate the problem we encounter when dealing with the many-body wave function. Then I will introduce a possible solution in the form of density functional theory (DFT) which plays a major role in numerically simulating the behaviour of many-body systems in modern day physics. 1.1. Wave-centered approach Let D be the dimensionality of our system. In wave-centered computation the wave function derived from the Schroedinger’s equation is the main body of focus. Ψ = Ψ(r1 , r2 , . . . , rN ) (1) It contains N coordinates ri = ( xi , yi , zi ; σi ) so as a whole it contains ( D + 1) N parameters with which we describe the system. It has to satisfy the normalization restraint: Z V dr1 Z V dr2 . . . Z V |Ψ(r1 , r2 , . . . , rN )|2 drN = 1 (2) It has to also satisfy the antisymmetricity of a fermion function(Pauli principle): Pi,j Ψ = −Ψ (3) Here Pi,j is the parity operator that switches the i-th and j-th fermion. In systems with only a couple of particles the wave-centered approach performs really well at predicting the results of experiments and has therefore become the pillar of quantum mechanics but it faces huge problems with many-body systems such as we find in nature. Molecules, crystals, DNA molecules and other complex matter are all described with the Schroedinger equation but as we can not numerically solve it on a realistic time scale it becomes useless. Let us demonstrate this on an example. 2 In this example we will consider a molecule with N interacting electrons. We will disregard all spin degrees of freedom and ignore symmetries. In numerical calculations we normally minimize a functional of energy in accordance to the Rayleigh-Ritz minimal principle so we obtain the ground state of a system. e | H |Ψ ei E = minΨe hΨ (4) As we ignore the spin variable of the electrons the wave function consists of 3N continuous variables which we have to represent discretely by M parameters p. In this way the energy is a function of the parameters E = E( p1 , p2 , . . . , p M ). In practical calculations it turns out that we require 3 ≤ n ≤ 10 parameters per variable for the desired accuracy od δE ∼ 10−2 . We therefore need: M = n3N 3 ≤ n ≤ 10 (5) parameters of the wave function. If M = 109 is the maximum computable number of parameters and n = 3 then this means the maximum number of computable electrons is: N̄ = 1 M . ln =6 3 n (6) With the same thinking we can compute that for a system with a 100 electrons we would need M̄ = 3300 = 10150 which is more than the predicted baryons in the whole universe [1]. This is what we call the Van Vleck catastrophe. 1.2. Density functional theory - A workaround the Van Vleck catastrophe The first density functional theory was published by Thomas [2] and Fermi [3] in the 1920s. In the centre of this approach was the density of electrons n(r) that is also easily measurable by various techniques. It was not accurate enough as it neglects the quantum corrections to the Couloumb potential which I will later introduce as exchange and correlation. It proposed an expected value of the kinetic term in the H operator that depends solely on the density. This approach is acceptable only in high electron density metals where the quantum corrections to the Couloumb potential can be neglected. Only in the 1960s a new complete formalism was proposed by Kohn and Sham [4] that sparked a new era in physics. It provides many results that are in accordance to results from experiments in various fields such as crystals, biophysics, molecular physics and many more. These systems would be otherwise inaccessible by numerical computations with wave functions as these systems contain N 1 electrons. 2. The self-consistent Kohn-Sham equations In the centre of our approach will be the density of electrons n(r) as the ground state n(r) uniquely (to a trivial constant) defines the external potential v(r). This means that if we know the external potential we in theory know n(r) and vice-versa. In the case of systems with degenerate ground states any degenerate ground state ni (r) uniquely describes the same v(r) - This is widely known as the Hohenberg-Kohn Theorem [5]. 3 2.1. Uniqueness of the external potential - Hohenberg-Kohn Theorem Let n(r) be the ground-state of a system described by the hamiltonian H1 . It corresponds to a ground-state wave function Ψ1 with energy E1 , which we assume is non-degenerate. E1 = hΨ1 | H1 |Ψ1 i = Z V v1 (r)n(r) dV + hΨ1 | T + U |Ψ1 i (7) Here U is the interaction between electrons and T is the kinetic term. Now lets assume that there exists a second potential v2 (r) 6= v1 (r) + v0 with its own ground-state wave function Ψ2 6= expiθ Ψ1 that gives the same density n(r). We can write the energy as: E2 = hΨ2 | H2 |Ψ2 i = Z V v2 (r)n(r) dV + hΨ2 | T + U |Ψ2 i (8) Since the system is non-degenerate the Rayleigh-Ritz minimal principle applies. E1 < hΨ2 | H1 |Ψ2 i = Z V v1 (r)n(r) dV + hΨ2 | T + U |Ψ2 i = E2 + Z (v1 (r) − v2 (r))n(r) dV (9) (v2 (r) − v1 (r))n(r) dV (10) V With the same reasoning we can also conclude: E2 < hΨ1 | H2 |Ψ1 i = Z V v2 (r)n(r) dV + hΨ1 | T + U |Ψ1 i = E1 + Z V If we add these equations we get a contradiction. E1 + E2 < E1 + E2 (11) Therefore it must apply that v1 (r) = v2 (r). [1] 2.2. Obtaining the ground state In this subsection we will be considering an inhomogeneous gas of electrons without spin. We can write the energy functional E[n(r)] for this system as: E[n(r)] = Z V v(r)n(r) + 1 2 Z Z V n(r)n(r‘) dr dr‘ + F [n(r)] |r − r‘| (12) The equation is written in standard atomic units, meaning the all distances are measured in Bohrs radius (a0 ) and energies in Hartrees (Ha). Ha = 2Ry = m e e4 = 27.2eV (2e0 h)2 a0 = e0 h 2 = 53pm πme e2 (13) The first part of the functional presents the interactions of electrons and the external potential and the second part is the classical Coulomb interaction between the electrons also called the Hartree interaction. F [n(r)] = Ts [n(r)] + Exc [n(r)] is a functional uniquely described by the electron density and is independent of the system that is defined by the external potential v(r). Ts [n(r)] is the functional of kinetic energy for a system of non-interacting electrons with the same density and Exc [n(r)] = Ex [n(r)] + Ec [n(r)] is by our definition the functional that describes exchange Ex [n(r)] and correlations Ec [n(r)] between electrons and contains everything about not already explicitly written out. 4 We are already familiar with the exchange part from our classes in Condensed matter on the matter of the Hartree-Fock approximation and we know the exchange comes from using only one Slater determinant to describe our wave-function. The correlation part comes from using more Slater determinants to describe the interaction between electrons. δExc [n(r)] Let f xc be the functional derivative of Exc . So f xc (r) = δn . The functional Exc [n(r)] is not (r) analytically known and neither is f xc (r). Because we have written the functional in a way where we separate the non-interacting kinetic term Ts [n(r)] from the interacting kinetic term which is hidden in Exc [n(r)] we know that the minimum of the functional will be the solution of the corresponding Schroedinger equation which is in this context knows as the Kohn-Sham equation. Z 1 n(r‘) − ∇2 + v ( r ) + dr‘ + f xc (r) − ei ϕi (r) = 0 (14) 2 V | r − r‘| ei is a Lagrange multiplication factor ensuring the conservation of particles. Because the effective potential, in which the interacting kinetic energy is hidden, is dependent on n(r) we must solve this equation self-consistently meaning that: N n (r ) = Z ∑ | ϕi (r)|2 V i =1 n(r) dr = N (15) Here the sum goes over all electrons in the system and ϕi are filled in the ascending order of their respective energies ei . As of yet there is no clear indication of the meaning of the Kohn-Sham orbitals ϕi and their energies ei . 2.3. Local density approximation To use this formalism effectively we must come by some approximation for the functional Exc [n(r)]. The first approximation is the so-called local density approximation or LDA made by Kohn and Sham in their original paper [4]. It assumes that the density n(r) is slowly varying on the scale of the Fermi wavelength that we derive from k F which corresponds to the radius of the Fermi sphere. It also works very well if the density is very high because the Fermi vector is small in this regime and the variations of density on this scale are therefore small. We can express the LDA in the following way: Z LDA Exc [n(r)] = V n(r)(ex [n(r)] + ec [n(r)]) dr (16) exc [n(r)] = ex [n(r)] + ec [n(r)] is the exchange-correlation energy per particle of a uniform electron gas of density n. This functional is well known. The exchange (ex ) can be analytically calculated [6] and we can estimate the correlation (ec ) with an analytical fit obtained with interpolation. A rough estimate is presented here: ex = − 0.458 +... rs ec = − 0.43 +... rs + 7.8 (17) 1 3 ) 3 and is a Here rs is defined as the radius of a sphere that contains one electrons rs = ( 4πn dimensionless quantity. In recent decades a better approximation for the correlation per electron was obtained by using an analytical fit for the data gathered by Monte-Carlo simulations that has the required behaviour when rs 1 and rs 1 which are analytically known. 5 2.4. Improvements of the local density approximation As we have seen in the previous section we obtain adequate results with the LDA approximation if the system has a very high density or if the local fluctuations of density are very small. But in realistic example this is rarely the case. So we need a new better approximation to describe these systems. At the centre of our approach in LDA was that we can write the energy functional simply as a functional of density as we can see in equation 16 but we must realize this is only the first order of expansion of the general functional Exc that describes effects of correlation and exchange between electrons. We can get a better approximation with the next orders in the expansion series that also include the change of density with ∇n(r). We can write the expansion to the second order as: GGA Exc [n(r)] = Z V f (n(r), ∇n(r)) dr (18) This second order is called the Generalized Gradient Approximation (GGA for short). For it to be of any use we must find a analytical fit to the f (n(r), ∇n(r)) function. The best choice is still a matter of debate. We can see the improvement that we obtain with the GGA approximation in Fig. 1. Figure 1: Errors of bond lengths computation for a GGA functional and LDA functional. We can see that GGA is better or comparable than LDA in these calculations. Taken from [7]. Also we can generalize the DFT by including a finite temperature. You can see the whole approach in [4], while here I will only cite the results. The Kohn-Sham equation is the same only how we self-consistently produce the density changes: N n (r ) = ∑ | ϕi (r)|2 i =1 1 exp ei − µ kT (19) +1 k is the Boltzmann constant, T is the temperature and µ is the chemical potential derived from the number of particles by integrating the equation 19. We can see that the only change is the Fermi function. These approaches can also be further generalized to study excitation energies, photo-absorption spectra and frequency responses using time-dependent external potentials like oscillating electric fields and magnetic fields in a theory we call time-dependent density functional theory - TD-DFT for short. From experience it is known that the accuracy of modelling the energy of bonds in molecules is about 10-20% and the sizes of the bonds is about 1%[1] as we can see in Fig. 1. We get the 6 bond lengths by introducing these lengths as parameters into the energy functional E[n(r)] and minimizing the corresponding energy function of parameters by using the Rayleigh-Ritz principle. We can not make a general statement for the quality of the DFT approach for all systems. For instance in semiconductor systems the DFT gives quantitatively wrong results for the energy gap while accurately predicting the energy band structure. In some systems like two dimensional electron gas in specific cases the LDA is good enough to get quantitative results. Also systems with very strong electron-electron interactions are better described by dynamical mean-field theory [8]. 2.5. Numerical packages for DFT calculations There are many numerical packages that one can acquire for DFT calculations. To know which package suits one best we should note the important differences between these packages. • Electrons We can divide the programs by how they treat electrons far from the Fermi surface. Some consider all electrons as equally important for understanding the behaviour of the system. Others use pseudo-potentials. When using pseudo-potentials we consider only the electrons near the Fermi surface accurately while the electrons far bellow the Fermi level are considered only as a background that effects the electrons near the Fermi surface. • Basis set The numerical calculations are done in some basis set of functions. There are two widely used variants. The Gaussian basis set (SIESTA) is local and the functions differ if we are in solids or molecules. The plane-wave basis set (Quantum-Espresso, VASP, ABINIT) is non-local and the functions are universal in all systems. 3. An application of DFT - 2D electron gas We can fabricate a 2D electron gas at the junction of two semiconductors. One such example is at the heterojunction of AlGaAs and GaAs. The semiconductors have a different energy gap between the valence and conducting bands and also their chemical potentials are not equal. When we put the semiconductors in contact a current starts to flow, because of different chemical potentials. It flows until both chemical potentials are equal. The result is that the energy bands bend to accommodate this requirement. In this process of bending a potential well about 10 nm wide forms at the junction and some electrons get trapped as their movement in the direction perpendicular to the junction gets quantized. The chemical potential is usually lower then the energy of the first excited state so electrons get trapped in the ground state of the well. Therefore their movement is reduced to two dimensions in which they are almost completely free. We can see the junction in Fig. 2. This system of electrons is of particular interest to scientist because of its high mobility in the two dimensions in which the electrons are free as it is comparable or even higher than in most metals. We can study this system using a quantum point contact (QPC) which is a narrow constriction made by the potential of two metallic gate electrodes that separate two electron reservoirs of 2D electron gas as see in Fig. 3a. The gate electrodes lie about 100 nm above the electron gas and are separated from it by an insulator so the entire effect on the electron gas is the potential the electrodes form. With such a device we can measure the conductance G in the QPC, by measuring 7 Figure 2: A junction of two semiconductors. On the top we can see the energy band structure before we create the junction. The red lines represent their respective chemical potentials. In the lower part of the figure we can see the energy band structure after we create the junction with the chemical potential in equilibrium. We can see the trapped electrons depicted in blue. the current through such a device, that is proportional to the conductance, while changing the voltage: I = GV (a) (b) Figure 3: (a) An image of a QPC through a microscope. (b) An ELS (highlighted with the white circle) that formed in a QPC calculated by DFT using LSDA approximation. Taken from [9]. In the constriction a saddle in the potential forms. When the Fermi energy of the electrons reaches the value of the potential saddle the density of electrons at the saddle is low and high in the reservoirs. Now lets look at a 2D jellium system, where the density of electrons is constant, in the 2D Hartree-Fock approximation. Also let the electrons have spin. We can define the polarization of n −n such a system as ζ = n↑ +n↓ where n↑,↓ are the densities of electrons with spin up or down. We ↑ ↓ can write the energy of such a system with the parameters rs and ζ [10] as: E (r s , ζ ) = √ 3 3 1 + ζ2 4 2 (1 − ζ ) 2 + (1 + ζ ) 2 − 3π rs rs2 8 (20) Lets look at both limits for polarization ζ = 0 and ζ = ±1. Their respective energies are: √ √ 1 8 21 2 16 2 1 E (r s , ζ = 0) = 2 − E (r s , ζ = ±1) = 2 − 3π rs 3π rs rs rs (21) We can see that for low densities (rs 1) the system will prefer to be polarized because the energy of the polarized state is lower. Now lets return to the QPC. Even though this system is not consistent with the jellium model as the density is not constant we expect that we should get qualitatively similar behaviour. We said that when the Fermi energy approaches the potential of the saddle the density there will be low so we expect that we will get a spin-polarized local state at the saddle. We can study systems where spin is relevant with DFT using a generalization of LDA named local spin density approximation or LSDA where we can write the exchange and correlation energies as: Z LDA Exc [n↑ (r), n↓ (r)] = V n(r)exc (n↑ (r), n↓ (r)) dr (22) This DFT approximation also adds correlations previously mentioned in Sec. 2.3 to the HartreeFock approximation we studied earlier. An article [9] has reported that by using LSDA the authors have found a single electron that gets caught in the constriction and so forms a localized spin-polarized state often named emergent localized state - ELS. We can see such an ELS in Fig. 3b. Now we can generalize our findings from this relatively simple system to a more complex system. To get high values of conductivity in this system one of the mentioned semiconductors is doped with a n-type material like Si. These dopants contribute even more electrons into the 2D electron gas but it comes at a cost. The dopants form a random potential seen in Fig. 4. Such a disordered potential also contains saddles on which ELS could form. We can see these ELS in Fig. 4. Figure 4: Upper left: A random disordered potential. Other pictures: Polarization calculated with the Octopus code using the LSDA approximation as a function of the space coordinates for various electron densities represented by the number of electrons N on top of every picture. For different densities different ELS are prevalent. This is because an ELS forms when the Fermi energy is the same as the energy of the saddle in the potential. Taken from [11]. 9 Such systems are still under study as we want to understand how the disordered potential influences the conductivity of the 2D electron gas. 4. Conclusion The transition from the wave-centered approach to the density approach is paramount to being able to quantify complex systems such as crystals and molecules as it allows us to predict behaviours of these complex systems. With this new-found understanding we can put these systems to use from nano physics such as the conductance of a 2D electron gas on the border of two semiconductors to various other examples not mentioned in this seminar. It is also very useful in chemistry and biology. Many improvements can still be made to the DFT formalism as the exact functional that describes the kinetic energy of interacting particles and the exchange and correlation of these particles is not known yet. It may never be known exactly but more elaborate methods of approximating this functional are to be expected in the future as more precision is demanded of the DFT formalism. References [1] W. Kohn, “Nobel lecture: Electronic structure of matter—wave functions and density functionals,” Reviews of Modern Physics, vol. 71, pp. 1253–1266, October 1999. [2] L. H. Thomas, “The calculation of atomic fields,” Proceedings of the Cambridge Philosophical Society, vol. 23, no. 5, p. 542–548, 1927. [3] E. Fermi, “Un metodo statistico per la determinazione di alcune prioprietà dell’atomo,” Rend. Accad. Naz. Lincei., vol. 6, pp. 602–607, 1927. [4] W. Kohn and L. J. Sham, “Self-consistent equations including exchange and correlation effects,” Physical Review Letters, vol. 140, pp. 1133–1138, November 1965. [5] P. Hohenberg and W. Kohn, “Inhomogeneous electron gas,” Physical Review Letters, vol. 136, November 1964. [6] G. D. Mahan, Many particle physics. Springer, 1981. [7] M. Fuchs, “Comparison of exchange-correlation functionals: from lda to gga and beyond,” in A Hands-On Computer Course, 2005. [8] D. Vollhardt, K. Byczuk, and M. Kollar, Dynamical Mean-Field Theory. Springer, 2011. [9] T. Rejec and Y. Meir, “Magnetic impurity formation in quantum point contacts,” Nature, vol. 442, pp. 900–903, August 2006. [10] B. Tanatar and D. M. Ceperley, “Ground state of the two-dimensional electron gas,” Physical Review B, 1989. [11] R. Levy, Spontaneous Formation of Magnetic Moments and Dephasing in Two-Dimensional Disordered Systems. PhD thesis, Ben-Gurion University of the Negev, 2013. 10