Instability of layered quantum liquids: 6. Strongly correlated bosons

We calculate the local-field corrections G(q, q _z) for charged bosons at zero temperature in a superlattice with interlayer distance d. An analytical expression for the local-field correction is described. The sum-rule version of the self-consistent approach for the local-field correction is used to discuss the effects of correlation. The RPAparameter r _s and the ratio d/a* determine correlation effects. a* is the effective Bohr radius. The stability region for the Bose condensate r _s < r _sc as a function of d/a* is determined: r _sc ≈ (d/a*)^3/4. The ground-state energy of the layered Bose condensate is calculated and optical and acoustical plasmons are discussed. We predict a roton structure for optical plasmons for r _s > r _sr with r _sr ≈ 0.5 (d/a*)^3/4.     

Instability of layered quantum liquids: 1. Local-field correction in superlattices

For a superlattice with periodd the Singwi et al. (Phys. Rev.176, 589 (1968)) approach for the local-field correction and the static structure factor is formulated. With two approximations we reduce the resulting three-dimensional integral-equation into a one-dimensional integral-equation. For the local-field correction we present analytical results for small wave numbers and large wave numbers. An expression of Hubbard-typ is derived for the local-field correction. Explicit results for boson superlattices and electron superlattice are given. A charge-density-wave instability in a layered Bose condensate withr _s>r_scd^3/4 is discovered.r _s is the small parameter of the random-phase-approximation. The charge-density-wave instability is due to a many-body anomaly (short-range correlations) in layered structures and is a general property of layered quantum liquids. We find the charge-density-wave instability in a layered electron gas forr _s>r_scd. Double-quantum-well structures are also considered. The effects of a finite well width is calculated. The general implications of the charge-density-wave instability for microscopically layered quantum liquids are pointed out.     

Ferromagnetism of the electron gas

The ground-state energy of the ferromagnetic electron gas is calculated for the relative polarizationζ=0−1 and the interelectron separationr _s =5−12. The method consists in describing the electron gas approximately by a quadratic boson Hamiltonian, and contains the random-phase approximation as a special case. Numerical studies show that in both the random-phase and the present approximations the paramagnetic state has the lowest energy: the energy increases withζ for all values ofr _s considered. In the present approximation instabilities are found to occur forr _s above a critical value, due to exchange processes of finite momentum transfers. Forζ=0 this critical value ofr _s is 9.4; it decreases with increasingζ. However, the fully-polarized state (ζ=1), which lies above the rest, is always stable. The conclusions are as follows: (1) Forr _s <9.4 the electron gas is paramagnetic. (2) Atr _s =9.4 it goes over to the fully-polarized ferromagnetic state. (3) This phase transition requires an energy absorption of 0.03 rydberg per electron. (4) The fully-polarized state is not obtainable as the limitζ→1.     

Effective electron-electron interactions and magnetic phase transition in a two-dimensional electron liquid

We investigate the spin-dependent effective electron-electron interactions in a uniform system of two-dimensional electrons to understand the spontaneous magnetization expected to occur at very low density. For this purpose, we adopt the Kukkonen-Overhauser form for the effective interactions which are built by accurately determined local-field factors describing the charge and spin fluctuations. The critical behavior of the effective interaction for parallel spin electrons allows us to quantitatively locate the transition to the ferromagnetic state at r_s≈27. When the finite width effects are approximately taken into account the transition occurs at r_s≈30 in agreement with recent quantum Monte Carlo calculations.     

二维电子体系在高密度下的稳定性

本文计算了高密度的二维电子体系的边缘能(将二维体系沿某一直线解离成两片时,形成单位长度新边缘所需要的能量)。结果发现,当r_ss^(c)(约0.415)时,边缘能变负,从而表明在高密度下,二维电子气的基态有可能发生不稳。我们分别讨论了二维非束缚的电子气和束缚的电子气基态的稳定性,并在一个简化的模型下给出了束缚的电子气基态稳定性的判据。 关键词：     

Ground-state properties of jellium metals with low electron densities

The ground state of a three-dimensional electron gas is theoretically investigated within the framework of the local spin density approximation with the Perdew–Zunger exchange-correlation energy. The system has been found to be in a one- or two-dimensional crystal state, when the Wigner sphere radius r_s has an intermediate value. At r_s=60, a triangular lattice with the lattice spacing 96.10 is the lowest energy state among fluids, 1D, 2D, and 3D crystals.     

A new approach to electron-electron interactions in strongly correlated systems at arbitrary spin-polarization and temperature

The uniform electron fluid is the reference model for density functional calculations. Even for this system, many-body perturbation theory, and related methods become questionable when the density parameter r_s exceeds unity. Hence, quantum Monte Carlo (QMC) simulation has been almost the only applicable method. We review a new approach, which uses a mapping of the quantum fluid to a classical Coulomb fluid, based on density-functional concepts. It is applicable at finite temperatures and arbitrary spin polarizations as well, and correctly recovers even the logarithmic terms in the exchange and correlations energies close to T=0. We show by detailed comparison with available QMC data that the method yields accurate pair-distribution functions, spin-dependent energies, static local-field factors, Landau parameter-based quantities like m∗ and g∗, for strongly coupled electron fluids.     

Correlation energy of 2-D electrons

The correlation energy and the Fermi momentum of an electron gas in 2-D are evaluated explicitly as functions of density. The ring diagram and first- and second-order exchange contributions are treated. In comparison with the 3-D case, the kinetic energy for the same r_s is approximately one-half and the exchange and correlation energies are somewhat larger. The ground state energy plotted against r_s shows a minimum at around r_s = 1.65 with a minimum value of ?0.9858 Ryd. If the third-order ring contribution is added, the curve is shifted upward. The correlation energy is ?0.6258 to order e⁴. The third-order ringw contribution increases this value almost linearly with r_s. The Fermi momentum decreases with r_s due to the contribution. Different from the 3-D case, no ln r_s term appears in the correlation energy within the approximation.     

Rigorous evaluation of the ring diagram contribution to the ground state energy of an electron gas

The ground state energy and the correlation energy of an electron gas are evaluated rigorously without using the smallr _s expansion and the small momentum-transfer approximation in the ring diagram contribution and taking into consideration the first order and second order exchange graphs. The Fermi momentum is determined by solving the number density equation without using iteration and is compared with that obtained by iteration. The ground state energy is found to stay positive in contrast to the iterative solution which becomes negative beyond a certain value ofr _s .     

Green function theory of Curie and order—order transitions in ferromagnetic systems

Dipolar critical temperatures in ferromagnetic systems with isotropic bilinear and biquadratic exchange are investigated by means of the Green function technique. Expressions are found for both the familiar Curie temperature, T_c, and the less well known order-order transition temperature, T_o, at which, under appropriate conditions, the magnetic ordering undergoes a change between fully aligned and canted ferromagnetism. At T = 0, a fully aligned state has <s_i^z = s for spin s and all lattice sites i, while a canted state has 〈s_i^z〉<s. It is shown independently of the Green function analysis that the T = 0 ground state is fully aligned if α, the ratio of biquadratic to bilinear exchange integrals, obeys ?[2s(s?1)]^?1<α< [2s²?2s+1]^?1. The region below the lower limit is identified as the range in which canted ferromagnetism can occur and is a range that does not appear to have been considered previously via the Green function formalism.The temperature dependence of the magnetic ordering is investigated by means of the double-time temperature-dependent Green function formalism. A new decoupling scheme is derived and used to reduce higher order Green functions to lowest order. It is found that a canted state, occuring at low temperatures, undergoes a transition to a fully aligned state at a temperature T₀ and subsequently becomes disordered at temperature T_c. Transitions to paramagnetism are found to be second order for α<α_c and first order for α?α_c where α_c is a critical value that depends on the atomic spin and weakly on the lattice structure. A phase diagram is given to illustrate the results, and a comparison is made with the corresponding results found in mean field theory.     

On the ground state energy of a two-dimensional electron fluid

A new theory of the ground state energy of a two-dimensional electron fluid is presented. It is shown that the ring diagram contribution changes its analytical behavior atr _s =2^1/2, wherer _s is the usual density parameter defined by r_S = 1/a ₀(π n)^1/2,a ₀ being the Bohr radius andn is the electron density. For smallr _s , a high density series is obtained in agreement with the previous calculation. For larger _s , a hitherto unknown low density series is obtained. In the low density region, the first order exchange energy is completely cancelled out by a term from the ring contribution so that the ground state energy decreases in proportion tor _s ^?2/3 , followed byr _s ^/?4/3 and higher order terms. The energy is found to be minimum atr _s=1.4757, the minimum value being ?0.481915 Rydbergs.     

A New Entropy Function to Analyze Isentropic Processes of Ideal Gases with Variable Specific Heats

A new entropy function s⁺ is defined in terms of the existing entropy function s° and temperature as s⁺ = s° − R lnT to facilitate the analysis of isentropic processes of ideal gases with variable specific heats. The function s⁺ also makes it possible to calculate the entropy changes of ideal gases during processes when volume information is available instead of pressure information and the variation of specific heats with temperature is to be accounted for. The introduction of the function s⁺ eliminates the need to use the dimensionless isentropic functions relative pressure P_r and relative specific volume v_r of ideal gases and to tabulate their values. The P_r and v_r data are often confused with pressure and specific volume, with an adverse effect on the study of the second law of thermodynamics. The new s⁺ function nicely complements the existing s° function in entropy change calculations: the former is conveniently used when volume information is given while the latter is used when pressure information is available. Therefore, the introduction of the new entropy function s⁺ is expected to make a significant contribution to the thermodynamics education and research by streamlining entropy analysis of ideal gases.     

Analytic theory of correlation energy and spin polarization in the 2D electron gas

We present an analytic theory of the pair distribution function and the ground-state energy in a two-dimensional (2D) electron gas with an arbitrary degree of spin polarization. Our approach involves the solution of a zero-energy scattering Schrödinger equation with an effective potential which includes a Fermi term from exchange and kinetic energy and a Bose-like term from Jastrow-Feenberg correlations. The form of the latter is assessed from an analysis of data on a 2D gas of charged bosons. We obtain excellent agreement with data from quantum Monte Carlo studies of the 2D electron gas. In particular, our results for the correlation energy show a quantum phase transition occurring at coupling strength r_s≈24 from the paramagnetic to the fully spin-polarized fluid.     

Spatial correlations in the electron gas: Path integral Monte Carlo simulation

Thermally excited states of the three-dimensional electron gas in a neutralizing background are computed by path integral Monte Carlo simulation for values of the Wigner-Seitz radius within the interval 5 < r _s < 15. Coulomb and exchange interactions, permutation symmetry, and spin state are treated explicitly. Variation of electron correlation functions with density and temperature is analyzed. Quantum effects suppress and enhance spatial correlation at low and high densities, respectively. Transition between the electron-gas states characterized by these opposite trends corresponds to a density of approximately 2.5 × 10²¹ cm^?3. A transition line between liquid-like and gaslike phases is determined in the temperature-density diagram. Weak anisotropy of many-body correlations in the liquid-like state stimulates excitation of spherically symmetric collective rotational modes. The effective short-range pseudopotential exhibits strong temperature dependence due to exchange effects. For strongly correlated systems, the characteristic screening length deviates from that predicted by the Thomas-Fermi screening model ( $ \sim \sqrt {r_s } $ ), approaching a linear function of r _s. The effective short-range interaction substantially differs from the Yukawa potential in mean field theory. Coulomb interaction shifts the Fermi level up by an order of magnitude or higher, and this effect becomes stronger with decreasing density.     

Some perturbation calculations for 1s2p, 1s 22p,and 1s 22s

The Dalgarno interchange theorem is used, together with the U function approach, to evaluate the first order perturbation corrections to <r ₁ ^-1+r ₂ ^-1> and <r ₁ + r ₂> for the two-electron states 1s2p ¹ P and 1s2p ³ P. The results for <r ₁ + r ₂> are extended by using a screening approximation, and are compared with the results of accurate variational calculations. The first order perturbation correction to spin-weighted expectation values of type <ΣV(r_j )s_zj > is given for the three-electron states 1s ²2p ² P and 1s ²2s ² S. The case V(r_j ) = r _j ^-1 is treated in detail.     

Memory function for cyclotron resonance of two-dimensional electron systems

Coulomb effects on cyclotron resonance in two-dimensional electron systems are investigated based on a self-consistent approach which improves the random-phase approximation. The memory function in the dynamic magnetoconductivity depends on the electron density not only through the filling factor v but also on its combination with the dimensionless density parameter r_s in the form $\text{r}{}_{\mathrm{s}}\text{v}{}^{\mathrm{<mtext>3</mtext><mtext>2</mtext>}}$. The memory function reproduces the data of Wilson, Allen and Tsui for intermediate densities.     

Local-field correction of the charged Bose condensate for two and three dimensions

The local-field correction for the dielectric function of the two-dimensional and of the three-dimensional Bose condensate is calculated within a sum-rule version of the Singwi et al. (Phys. Rev.176, 589 (1968)) approach. We derive analytical expressions for small and large wave numbers and give analytical expressions for the density dependence. We compare the results of the groundstate energy for the three-dimensional system with Monte-Carlo computations. In two dimensions a roton structure in the plasmon dispersion is found at low boson density. The plasmon density of states is calculated. A correlation induced charge-density-wave instability in layered structures of two-dimensional Bose gases is discussed.     

The concept of screening length in lifetime and relaxation semiconductors

It is shown that, contrary to normal practice, the most appropriate criterion for distinguishing between lifetime and relaxation semiconductors in the presence of traps is the ratio of the screening length L_s, to the ambipolar diffusion length L_Da,. L_s, is calculated. Its significance is not limited to zero current, even though it reduces to the conventional Debye length L_D when the trap concentration is zero. (With traps, we always have L_s < L_D.) The dielectric relaxation time itself is unaffected by traps, but in steady state situations, a material behaves as if it had an effective lifetime τ_0s = τ₀η, where η depends on the concentration and energetic position of the traps. τ_o, may be orders of magnitude greater than τ₀, the conventional diffusion length lifetime. Typical values of L_s, are presented as a function of trapping parameters. L_s>L_Da leads to relaxation behavior; L_s < L_Da, to lifetime behavior.