Quantum diffusion monte carlo method for low-dimentional systems

Abstract. We will show that Schr¨odinger equations for low-dimensional systems can be solved by the method of quantum Diffusion Monte Carlo (DMC). The wave function and energy in a ground state are found for the two-dimensional harmonic oscillator, quantum wells, quantum wires and quantum dots. This is great approach for problems of low-dimensional systems. In this paper, the sign of the wave function is not treated.

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2015-0036 Mathematical and Physical Sci., 2015, Vol. 60, No. 7, pp. 81-87 This paper is available online at QUANTUM DIFFUSIONMONTE CARLOMETHOD FOR LOW-DIMENTIONAL SYSTEMS Nguyen The Lam Faculty of Physics, Hanoi Pedagogical University No. 2 Abstract. We will show that Schro¨dinger equations for low-dimensional systems can be solved by the method of quantum Diffusion Monte Carlo (DMC). The wave function and energy in a ground state are found for the two-dimensional harmonic oscillator, quantum wells, quantum wires and quantum dots. This is great approach for problems of low-dimensional systems. In this paper, the sign of the wave function is not treated. Keywords: Schro¨dinger equations, method of quantum Diffusion Monte Carlo, low-dimensional systems. 1. Introduction Nowadays, low-dimensional materials are produced with high technologies. Examples of these are thin film, InAs/AlSb super lattice [1], InGaAs/GaAs quantum well [2], CdTe, GaAs and GaP quantum wire [3,4], PbS quantum dot [5] and nano materials. In general, these materials are called low-dimensional systems and in theory, these systems are governed by Schro¨dinger equations. The solving of the Schro¨dinger equations for these systems must be done for theoretical problems. In fact, these equations can be solved analytically in some simple cases, for complex energy potentials, they can not be solved. In this paper, the Quantum diffusion Monte Carlo (DMC) method will be applied for low-dimensional systems. With this method, the wave function and energy in the ground state may be found. The solution of a time-dependent Schro¨dinger equation may be written as a linear superposition of stationary states in which the time-dependence is given by phase factor exp(−iEnt/~) where En is the energy in the n-th level of the quantum system. An energy scale may be chosen such that all energies are positive. In the DMC method, the time-dependent Schro¨dinger equation is considered as assuming imaginary time τ after replacing t by -iτ . The solution is then given as the sum of transients of the form exp(Enτ/~) and energy in the ground state E0 < En with n = 1, 2, 3,. . . [6]. The DMC method may be formulated in two different ways. In the first, the time-dependent Schro¨dinger equation may be considered as a generalized diffusion equation in which, the kinetic (potential) energy term of the time-dependent Schro¨dinger equation corresponds to the diffusion (source/sink) term in the generalized diffusion equation [7]. In this way, the time-dependent Received December 3, 2014. Accepted October 12, 2015. Contact Nguyen The Lam, e-mail address: nguyenthelam2000@yahoo.com 81 Nguyen The Lam Schro¨dinger equation may be solved by simulating random walks of particles which are subject to birth/death processes imposed by source/sink term. The probability distribution of the random walks is an identical wave function. This is possible only for wave functions which are positive everywhere. This limits the range of applicability of the DMCmethod. In the second, the Feynman path integral solution of the time-dependent Schro¨dinger equation is an important formulation of the DMC method. By properties of path integrals, the wave function can be expressed as a multidirectional integral which may be evaluated using the Monte Carlo method. In this paper, the DMCmethod will be applied for the two-dimensional harmonic oscillator, GaAs/InAs quantum well, CdTe quantum wire and CdSe/CdS quantum dot. 2. Content 2.1. Basic theory The time-dependent Schro¨dinger equation for one particle with energy potential V(x) is in the form i~ ∂ψ ∂τ = Hψ. (2.1) The Hamiltonian has the form H = − ~ 2 2m ∂2 ∂x2 + V (x). (2.2) where m is mass of the electron. According to [6], the shift energy ER is defined such that all energy En > 0 by replacing V(x)→ V(x) - ER and En→ En - ER. From the time-dependent Schro¨dinger equation, replacing t → -iτ , the imaginary time Schro¨dinger equation is given in the form: ~ ∂ψ ∂τ = ~ 2 2m ∂2ψ ∂x2 − [V (x)− ER]ψ, (2.3) where Ψ(x, τ) = ∞∑ n=0 cn φn(x)e En−ER ~ τ . (2.4) When τ →∞ we have - If ER > E0 then limτ→∞ψ(x, τ) =∞ and the wave function diverges exponentially fast. - If ER < E0 then limτ→∞ψ(x, τ) = 0 and the wave function vanishes exponentially fast. - If ER = E0 then limτ→∞ψ(x, τ) = C0φ0(x) and the wave function converges up to a C0 constant and to the ground state. In other hand [8, 9], the solution of the Schro¨dinger equation may be written as a path-integral formalism Ψ(x, τ) = ∞∫ −∞ dx0K(x, τ |x0, 0)Ψ(x0, 0). (2.5) 82 Quantum diffusion monte carlo method for low-dimentional systems where the propagator K(x, τ |x0, 0) is expressed in terms of path-integral, modified by a replacement t→ −iτ . K(x, τ |x0, 0) = lim N→∞ ∞∫ −∞ dx0... ∞∫ −∞ dxN−1 ( m 2pi~∆τ )N 2 ×exp { − ∆τ ~ N∑ j=1 [ m 2∆τ 2 (xj − xj−1) 2 + V (xj)− ER ]} . (2.6) Here,∆τ = τ /N is a small time step. Setting xn ≡ x, the wave function may be rewritten in the form Ψ(x, τ) = lim N→∞ ∞∫ −∞ N−1∏ j=0 dxj  N∏ n=1 w(xn)× P(xn, xn−1)Ψ(x0, 0), (2.7) where P(xn, xn−1) ≡ ( m 2π~∆τ ) 1 2 exp [ −m(xn − xn−1) 2 2~∆τ ] (2.8) and w(xn) ≡ exp [ − [V(xn)−ER]∆τ ~ ] . (2.9) The P (xn, xn−1) is related to kinetic energy and is thought of as a Gaussian probability density for random variable xn with the mean equals xn−1 and the variance σ = √ λ¯∆τ/m. The w(xn) is called the weight function and it depends on both potential energy and reference energy ER. The vector x = (x0, x1, x2, xn−1, xn), the xn and xn−1 are related by equation xn = xn−1 + σρn (2.10) where σ = √ λ¯∆τ/m and ρn is Gaussian random number with the mean equal to zero and the variance equal to 1. The initial state is interpreted as a delta function ψ(x, 0) = δ(x− x0) where x0 is position and where the ground state of the quantum system is expected to be large. Then the diffusive displacement is started with a vector generated by Eq. (2.10). The calculation of wave function ψ(x, τ) is regarded as a simulated diffusive-reaction process of some imaginary particles (replica). In the replication process, each particle is replaced by a number mn = min{int[w(xn) + u], 3} particles where int(x) is an integer part of x and u is a uniformly distributed random number in the interval (0,1). If mn = 0, the particle is deleted and the diffusive process stops, and this is considered as a ‘death’ of a particle. If mn = 1, the particle is unaffected and continues with next step. If mn = 2, 3, it continues with next step but it also begins a new series of diffusive displacements starting at xn. The latter case is referred to as the ‘birth’ of a particle with mn = 2 (of two particles withmn = 3). From Eq. (2.9) we averaged all replicas = 1− −ER ~ ∆τ, (2.11) where the average potential energy is given as = 1 N ∑ i V (xi), (2.12) 83 Nguyen The Lam xi are generated by Eq. (2.10) in the case three dimensional system, the Hamiltonian is given as following H = − ~ 2 2m 3∑ α=1 ∂2 ∂x2α + V (x1, x2, x3) (2.13) and expression (2.8) is also redefined in the form P(xn,1, xn−1,1...xn−1,3, xn,3) ≡ 3∏ α=1 ( m 2π~∆τ ) 1 2 exp [ −m(xn,α − xn−1,α) 2 2~∆τ ] . (2.14) 2.2. Results and discussion For convenience, in the Schro¨dinger equation (2.13) for a 2D harmonic oscillator, we choose m = 1 and ~ = 1. The potential for a 2D harmonic oscillator is given as V (r) = 1 2 m.ω.r2, (2.15) where r = √ x2 + y2 , m is the mass and ω is the angular frequency. For convenience, we choose m = 1 and ω = 1. Fig. 1. The energy of the 2D oscillator converges to its ground state energy when imaginary time increases upwards when m = 1 and ~ =1 Fig. 2. The ground state wave function of a 2D harmonic oscillator when m = 1 and ~ =1 In Figure 1, with ∆τ = 0.01 and after N = 2000 steps of imaginary time, the energy of the 2D oscillator converges to -6.5. In Figure 2, originally, all replicas are located in (50, 50) and after 2000 steps of imaginary time, the diffusive displacement of replicas show the wave function in ground state. For a GaAs/InAs quantum well with a width wd = 7 nm, m = 0.067m0 where m0 is the mass of the free electron [10], the potential is given as V (x, y, z) = ∣∣∣∣ 0eV with 0 ≤ x ≤ wd ; ∀y, z1.024eV with other. (2.16) 84 Quantum diffusion monte carlo method for low-dimentional systems Fig. 3. The energy of a particle in the GaAs/InAs quantum well, with width wd = 7 nm, converges to its ground state energy when imaginary time increases upwards and the effective mass m = 0.067m0 [11] Fig. 4. The ground state wave function of a particle in the GaAs/InAs quantum well is shown by replica density. The well width wd = 7 nm and the effective mass m = 0.067m0 [11] In Figure 3 we see that, after 2000 steps with∆τ = 0.01, the energy converges to 0.495 eV, this being the ground energy of a particle in the well (experimental data is about 0.4 eV [10]). In Figure 4, the wave function is flat and its value is shown by the replica density in the well. Fig. 5. The energy of a particle in quantum wire, with wx × wy = 5 nm × 5 nm, converges to its ground state energy when imaginary time increases upwards and the effective mass m = 0.14 m0 [11] Fig. 6. The ground state wave function of a particle in quantum wire is shown by the replica density with size wx × wy = 5 nm × 5 nm and effective mass m = 0.14 m0 [11] For the CdTe quantum wire, we introduce a box with size in the x, y dimensions being very small, wx × wy = 5 nm × 5 nm [11]. The potential is given in the form: V (x, y, z) = ∣∣∣∣ 1.5eV with 0 ≤ x ≤ wx; 0 ≤ y ≤ wy;∀z∞ with other. (2.17) In Figure 5 the energy converges to 1.85 eV, the energy in ground state when imaginary time increases upwards. This energy is in good agreement with other calculations [11]. In Figure 6, the wave function is shown by replica density in the wire. 85 Nguyen The Lam For the CdSe/CdS quantum dot, we introduce a box with the size is wx × wy × wz = 5.5 nm × 5.5 nm × 5.5 nm [12]. The potential is given in the form V (x, y, z) = ∣∣∣∣ 0 with 0 ≤ x ≤ wx; 0 ≤ y ≤ wy; 0 ≤ z ≤ wz0.9eV with other. (2.18) The energy and wave function are shown in Figures 7 and 8. The energy is in quite good agreement with experimental data (0.48 eV [12]) Fig. 7. The energy of a particle in quantum dot of size wx × wy ×wz =5.5 nm × 5.5 nm × 5.5 nm converges to its ground state energy when imaginary time increases upwards and effective mass m = 0.13 m0 [11] Fig. 8. The ground state wave function of a particle in a quantum dot is shown by the replica density with size wx × wy × wz= 5.5 nm × 5.5 nm × 5.5 nm and effective mass m = 0.13 m0 [11] 3. Conclusion In this paper, we have written the program in MATLAB and obtained results. We have found the energy and wave function in the ground state for a two-dimensional harmonic oscillator, a GaAs/InAs quantum well, CdTe quantum wire and a CdSe/ZnS quantum dot. In using this method, the sign of the wave function has not been treated. 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