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Physical review. B, Condensed matter

Two-dimensional electrons in a magnetic field. III. Many-body effects

Y. Shiwa, A. Isihara

Published Apr 15, 1983• 24 citations
Publisher

Abstract

A many-body theory is given to show that the magnetic properties of a two-dimensional electron system are dependent on the filling factor ${\ensuremath{\gamma}}_{0}^{\ensuremath{-}1}$ defined as the ratio of the ideal Fermi energy against the field energy. Explicit results are obtained for absolute zero and large values of this factor. The internal energy is evaluated explicitly in consideration of the ideal, first-order exchange and ring contributions. The result converges for all ${r}_{s}$ because for the ring contribution no small-momentum-transfer assumption is made and for low densities a rigorous analytical continuation is adopted. When plotted against ${\ensuremath{\gamma}}_{0}^{\ensuremath{-}1}$, the internal energy shows parabolic variations in a periodic way until it approaches the average kinetic energy for large ${\ensuremath{\gamma}}_{0}^{\ensuremath{-}1}$. The ideal susceptibility shows zigzag changes characteristic of absolute zero when it is plotted against ${\ensuremath{\gamma}}_{0}^{\ensuremath{-}1}$. It is zero at ${\ensuremath{\gamma}}_{0}^{\ensuremath{-}1}=2m+1$, where it jumps from $\ensuremath{-}2m$ to $2(m+1)$, $m$ being an integer. The Coulomb interaction suppresses the zigzag variations for small ${r}_{s}$ and inverts the oscillating pattern at ${r}_{s}$ \ensuremath{\sim}0.7. The oscillations thereafter may be considered characteristic of a liquid state rather than a gaseous state as they are somewhat different from those for small ${r}_{s}$. The susceptibility is convergent for all ${r}_{s}$.

DOI
10.1103/physrevb.27.4743

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A many-body theory is given to show that the magnetic properties of a two-dimensional electron system are dependent on the filling factor \ensuremathγ0\ensuremath−1{\ensuremath{\gamma}}_{0}^{\ensuremath{-}1}\ensuremathγ0\ensuremath−1​ defined as the ratio of the ideal Fermi energy against the field energy. Explicit results are obtained for absolute zero and large values of this factor. The internal energy is evaluated explicitly in consideration of the ideal, first-order exchange and ring contributions. The result converges for all rs{r}_{s}rs​ because for the ring contribution no small-momentum-transfer assumption is made and for low densities a rigorous analytical continuation is adopted. When plotted against \ensuremathγ0\ensuremath−1{\ensuremath{\gamma}}_{0}^{\ensuremath{-}1}\ensuremathγ0\ensuremath−1​, the internal energy shows parabolic variations in a periodic way until it approaches the average kinetic energy for large \ensuremathγ0\ensuremath−1{\ensuremath{\gamma}}_{0}^{\ensuremath{-}1}\ensuremathγ0\ensuremath−1​. The ideal susceptibility shows zigzag changes characteristic of absolute zero when it is plotted against \ensuremathγ0\ensuremath−1{\ensuremath{\gamma}}_{0}^{\ensuremath{-}1}\ensuremathγ0\ensuremath−1​. It is zero at \ensuremathγ0\ensuremath−1=2m+1{\ensuremath{\gamma}}_{0}^{\ensuremath{-}1}=2m+1\ensuremathγ0\ensuremath−1​=2m+1, where it jumps from \ensuremath−2m\ensuremath{-}2m\ensuremath−2m to 2(m+1)2(m+1)2(m+1), mmm being an integer. The Coulomb interaction suppresses the zigzag variations for small rs{r}_{s}rs​ and inverts the oscillating pattern at rs{r}_{s}rs​ \ensuremath{\sim}0.7. The oscillations thereafter may be considered characteristic of a liquid state rather than a gaseous state as they are somewhat different from those for small rs{r}_{s}rs​. The susceptibility is convergent for all rs{r}_{s}rs​.
We present an effective theory describing the low-energy properties of an interacting two-dimensional electron gas at large noninteger filling factors \ensuremath{\nu}\ensuremath{\gg}1. Assuming that the interaction is sufficiently weak, ${\mathit{r}}_{\mathit{s}}$1, we integrate out all the fast degrees of freedom, and derive the effective Hamiltonian acting in the Fock space of the partially filled Landau level only. This theory enables us to find two energy scales controling the electron dynamics at energies less than \ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$. The first energy scale, (\ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$/\ensuremath{\nu})ln(\ensuremath{\nu}${\mathit{r}}_{\mathit{s}}$), appears in the one-electron spectral density as the width of a pseudogap. The second scale, rs{\mathit{r}}_{\mathit{s}}rs​\ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$, is parametrically larger, it characterizes the exchange-enhanced spin splitting and the thermodynamic density of states.
Alexander Shashkin Institute of Solid State Physics, Chernogolovka, Moscow District 142432, Russia Sergey Kravchenko Physics Department, Northeastern University, Boston, Massachusetts 02115, U.S.A. Two-dimensional (2D) electron systems are realized when the electrons are free to move in a plane but their motion perpendicular to the plane is quantized in a confining potential well. Quantum phase transitions realized experimentally in such systems so far include metal-insulator transitions in perpendicular magnetic fields, metal-insulator transition in zero magnetic field, and a possible transition to a Wigner crystal. The first transition is governed by the externally controlled electron density or magnetic field, while the other two are governed by the electron density. At low electron densities in 2D systems, the strongly-interacting limit is reached because the kinetic energy is overwhelmed by the energy of electron-electron interactions. The interaction strength is characterized by the ratio between the Coulomb energy and the Fermi energy, r∗s = Eee/EF . Assuming that the effective electron mass is equal to the band mass, the interaction parameter r∗s in the single-valley case reduces to the Wigner-Seitz radius, rs = 1/(πns)1/2aB, and therefore increases as the electron density, ns, decreases (here aB is the Bohr radius in the semiconductor). Possible candidates for the ground state of the system include a Wigner crystal characterized by spatial and spin ordering [1], a ferromagnetic Fermi liquid with spontaneous spin ordering [2], a paramagnetic Fermi liquid [3], etc. In the strongly-interacting limit (rs 1), no analytical theory has been developed to date. According to numerical simulations [4], Wigner crystallization is expected in a very dilute regime, when rs reaches approximately 35. Refined numerical simulations [5] have predicted that prior to the crystallization, in the range of the interaction parameter 25 ≤ rs ≤ 35, the ground state of the system is a strongly correlated ferromagnetic Fermi liquid. At higher electron densities, rs ∼ 1, the electron liquid is expected to be paramagnetic, with the effective mass, m, and Lande´ g factor renormalized by interactions. Apart from the ferromagnetic Fermi liquid, other intermediate phases between the Wigner crystal and the paramagnetic Fermi liquid may also exist.
We present a quantitative study of the quasiparticle properties of a three-dimensional electron Fermi liquid. Our approach is based on the theory of the many-body local field factors which we use to include vertex corrections associated with charge and spin fluctuations. Extensive use is made of the results of recent quantum Monte Carlo calculations. Several models for the wave-vector dependence of the spin-antisymmetric many-body local field factor ${G}_{\ensuremath{-}}$, for which no numerical results are currently available, are discussed and compared. Both the real and imaginary parts of the self-energy as well as the quasiparticle renormalization constant and the enhancement of the effective mass are calculated in the experimentally accessible range of electron densities. The results obtained by means of the on-shell approximation are critically compared with those given by a self-consistent solution of the Dyson equation. An ultraviolet catastrophe in the Coulomb-hole part of the self-energy is identified and a satisfactory resolution of this impasse is presented. The same many-body local field factors are also used to obtain the Landau interaction function. This allows us to obtain both the spin susceptibility and the proper compressibility of the system.
We investigate the properties of the one-electron Green's function in an interacting two-dimensional electron system in a strong magnetic field, which describes an electron tunneling into such a system. From finite-size diagonalization, we find that its spectral weight is suppressed near zero energy, reaches a maximum at an energy of about 0.2e2{\mathit{e}}^{2}e2/\ensuremath{\epsilon}lc{\mathit{l}}_{\mathit{c}}lc​, and decays exponentially at higher energies. We propose a theoretical model to account for the low-energy behavior. For the case of Coulomb interactions between the electrons, at even-denominator filling factors such as \ensuremath{\nu}=1/2, we predict that the spectral weight varies as ${\mathit{e}}_{0}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\omega}}}$/\ensuremath{\Vert}\ensuremath{\omega}\ensuremath{\Vert}, for \ensuremath{\omega}\ensuremath{\rightarrow}0.
The formalism introduced in a previous paper is used for discussing the Coulomb interaction of many electrons moving in two space-dimensions in the presence of a strong magnetic field. The matrix element of the Coulomb interaction is evaluated in the new basis, whose states are invariant under discrete translations (up to a gauge transformation). This paper is devoted to the case of low filling factor, thus we limit ourselves to the lowest Landau level and to spins all oriented along the magnetic field. For the case of filling factor νf = 1/u we give an Ansatz on the state of many electrons which provides a good approximated solution of the Hartree–Fock equation. For general filling factor νf = u′/u a trial state is given which converges very rapidly to a solution of the self-consistent equation. We generalize the Hartree–Fock equation by considering some correlation: all quantum states are allowed for the u′ electrons with the same translation quantum numbers. Numerical results are given for the mean energy and the energy bands, for some values of the filling factor (νf = 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 4/5). Our results agree numerically with the Charge Density Wave approach. The boundary conditions are shown to be very important: only large systems (degeneracy of Landau level over 200) are not affected by the boundaries. Therefore results obtained on small scale systems are somewhat unreliable. The relevance of the results for the Fractional Quantum Hall Effect is briefly discussed.
The correlation energy per particle (expressed in rydbergs) of a two-dimensional electron gas is calculated by summing the ring diagrams. For high densities, this is found to be of the form C+Drslnrs+O(rs)C+D{r}_{s}\mathrm{ln}{r}_{s}+O({r}_{s})C+Drs​lnrs​+O(rs​), where rs{r}_{s}rs​ is a dimensionless parameter such that ${r}_{s}^{\ensuremath{-}2}$ is proportional to the number of electrons per unit area. The value of DDD is -0.172 and that of CCC is -0.38 \ifmmode\pm\else\textpm\fi{} 0.04, the latter involves a numerical evaluation. By the same methods, we have also calculated the difference in total energy of the nonmagnetic and ferromagnetic states and we find for rs>2.3{r}_{s}>2.3rs​>2.3 the system may be ferromagnetic. We have also obtained some exact relationships between the pair correlation function at zero separation and the asymptotic behavior of the structure factor and the momentum distribution function. These are of interest in relation to scattering experiments.