Inverse Compressibility Research Articles

Phase separation and enhanced charge-spin coupling near magnetic transitions

The generic changes of the electronic compressibility in systems which show magnetic instabilities are studied within a mean-field approach. It is shown that, when going into the ordered phase, the inverse compressibility is reduced by an amount comparable to its original value, making charge instabilities also possible. We discuss, within this framework, the tendency towards phase separation of the double-exchange systems, the pyrochlores, and the magnetic transitions of the Hubbard model.

Transverse electric field in a mesoscopic Luttinger-liquid ring

The electric field within a ring containing repulsively interacting electrons is estimated using the spinless Luttinger-liquid model extended to allow transverse excitations due to the radial centrifugal force. The radial voltage drop across a ring, threaded by the Aharonov-Bohm flux, consists of a flux-periodic part that is independent of the interaction, and a flux-independent part that is proportional to the square root of the inverse compressibility of the electron gas. The voltage drop obtained from perturbation theory is consistent with Bernoulli's equation.

Statistics of Hartree-Fock levels in small disordered systems

We study the statistics of quasiparticle and quasihole levels in small interacting disordered systems within the Hartree-Fock approximation. The distribution of the inverse compressibility, given according to Koopmans' theorem by the distance between the two levels across the Fermi energy, evolves from a Wigner distribution in the noninteracting limit to a shifted Gaussian for strong interactions. On the other hand, the nature of the distribution of spacings between neighboring levels on the same side of the Fermi energy (corresponding to energy differences between excited states of the system with one missing or one extra electron) is not affected by the interaction and follows Wigner-Dyson statistics. These results are derived analytically by isolating and solving the appropriate Hartree-Fock equations for the two levels. They are substantiated by numerical simulations of the full set of Hartree-Fock equations for a disordered quantum dot with Coulomb interactions. We find enhanced fluctuations of the inverse compressibility compared to the prediction of the random matrix theory, possibly due to the localization of the wave functions around the edge of the dot. The distribution of the inverse compressibility calculated from the discrete second derivative with respect to the number of particles of the Hartree-Fock ground state energy deviates from the distribution of the level spacing across the Fermi energy. The two distributions have similar shapes but are shifted with respect to each other. The deviation increases with the strength of the interaction thus indicating the breakdown of Koopmans' theorem in the strongly interacting limit.

Fluctuation of inverse compressibility for electronicsystems with random capacitive matrices

This article is concerned with the statistics of the addition spectra of certain many-body systems of identical particles. In the first part, the pertinent system consists of N identical particles distributed among K<N independent subsystems, such that the energy of each subsystem is a quadratic function of the number of particles residing on it with random coefficients. On a large scale, the ground-state energy E(N) of the whole system grows quadratically with N, but in general there is no simple relation such as EN = aN+bN 2. The deviation of E(N) from exact quadratic behaviour implies that its second difference (the inverse compressibility) XN ≡E(N+1)−2E(N)+E(N−1) is a fluctuating quantity. Regarding the numbers XN as values assumed by a certain random variable X, we obtain a closed-form expression for its distribution F (X). Its main feature is that the corresponding density P (X)=dF (X)/d X has a maximum at the point X=0. As K→∞ the density is Poissonian, namely, P(X)→e−X This result serves as a starting point for the second part, in which coupling between subsystems is included. More generally, a classical model is suggested in order to study fluctuations of Coulomb blockade peak spacings in large two-dimensional semiconductor quantum dots. It is based on the electrostatics of several electron islands among which there are random inductive and capacitive couplings. Each island can accommodate electrons on quantum orbitals whose energy depends also on an external magnetic field. In contrast to a single-island quantum dot, where the spacing distribution between conductance peaks is close to Gaussian, here the distribution has a peak at small spacing value. The fluctuations are mainly due to charging effects. The model can explain the occasional occurrence of couples or even triples of closely spaced Coulomb blockade peaks, as well as the qualitative behaviour of peak positions with the applied magnetic field.

Fluctuations of the inverse compressibility in disordered electron systems

We consider interacting electrons in one- and two-dimensional disordered systems and observe within the Hartree-Fock approximation that relative fluctuations of the inverse compressibility in general decrease with the hopping strength, signaling metal-insulator transitions in the systems. Particularly, in one dimension the perturbative approach reveals two regimes where the behaviors of the relative fluctuations with the system size are different: In the localized regime the relative fluctuations are independent of the size while in the opposite limit corresponding to the diffusive regime the relative fluctuations scale linearly with the system size. To check the validity of the Hartree-Fock approximation, we also obtain exact results for small systems via the Lanczos method, which suggests that for intermediate values of the hopping strength the relative fluctuations are overestimated in the Hartree-Fock approximation.

On the extended states in the quantum Hall regime and in zero magnetic field

For a two-dimensional electron system in a strong perpendicular magnetic field B, there are extended states at each Landau level. We show that if the inverse compressibility is negative, then the extended states float downward in energy when B decreases. We set up a condition for the case where all extended states are below the Fermi energy in the low B limit. The condition may explain why a conducting state has been observed in a high mobility Si-Mosfet but so far not in high mobility n-GaAs.

Statistics of addition spectra of independent quantum systems

Motivated by recent experiments on large quantum dots, we consider the energy spectrum in a system consisting of $N$ particles distributed among $K<N$ independent sub-systems, such that the energy of each sub-system is a quadratic function of the number of particles residing on it. On a large scale, the ground state energy E(N) of such a system grows quadratically with $N$, but in general there is no simple relation such as $E(N)=a N +b N^{2}$. The deviation of E(N) from exact quadratic behavior implies that its second difference (the inverse compressibility) $\chi_{N} \equiv E(N+1)-2 E(N)+E(N-1)$ is a fluctuating quantity. Regarding the numbers $\chi_{N}$ as values assumed by a certain random variable $\chi$, we obtain a closed-form expression for its distribution $F(\chi)$. Its main feature is that the corresponding density $P(\chi)=\frac{dF(\chi)} {d\chi}$ has a maximum at the point $\chi=0$. As $K \to \infty$ the density is Poissonian, namely, $P(\chi) \to e^{-\chi}$.

Acoustic measurements of condensation/evaporation and crystal growth

Using a volume-modulation technique developed to study thermoacoustics, the complex compressibility of a liquid in coexistence with its vapor is measured. Such a system exhibits a distinctive behavior. At low frequencies, the inverse complex compressibility vanishes because the pressure remains at the equilibrium vapor pressure as the total volume is changed slowly. The volume change is accompanied by an exchange of molecules between the two phases. At higher frequencies, however, this exchange is too slow, and the pressure response approaches that which would be expected in the absence of the liquid phase. The presence of the liquid acts effectively as a high-pass filter whose time constant depends on the rate of condensation/evaporation. The application of this technique to the measurement of condensation coefficients of liquids will be discussed. The same technique can also be applied to solid/vapor systems where the results provide a measure of the crystal growth rate. A unique feature of this technique is that the crystal growth rate can be monitored continuously as a function of temperature. [Work supported by Ohio University Research and Sponsored Programs.]

Effect of finite quantum-well width on the compressibility of a two-dimensional electron gas

Double quantum-well structures have been used to measure the inverse compressibility (${\mathrm{\ensuremath{\kappa}}}^{\mathrm{\ensuremath{-}}1}$) of a two-dimensional electron gas as a function of the confining quantum-well width. The lower layer is contacted independently and carrier density changes within this layer are used to measure ${\mathrm{\ensuremath{\kappa}}}^{\mathrm{\ensuremath{-}}1}$ of the upper layer. As the well width increases we observe a decrease in ${\mathrm{\ensuremath{\kappa}}}^{\mathrm{\ensuremath{-}}1}$ which may be attributed to the effect of the Hartree term. The inclusion of Hartree effects produces the correct trend of ${\mathrm{\ensuremath{\kappa}}}^{\mathrm{\ensuremath{-}}1}$ with well width, although discrepancies still exist between the model and experiment.

Compressibility, capacitance, and ground-state energy fluctuations in a weakly interacting quantum dot

We study the effect of electron-electron (e-e) interactions on compressibility, capacitance, and inverse compressibility of electrons in a quantum dot or a small metallic grain. The calculation is performed in the random-phase approximation. As expected, the ensemble-averaged compressibility and capacitance decreases as a function of the interaction strength, while the mean inverse compressibility increases. Fluctuations of the compressibility are found to be strongly suppressed by the e-e interactions. Fluctuations of the capacitance and inverse compressibility also turn out to be much smaller than their averages. The analytical calculations are compared with the results of a numerical calculation for the inverse compressibility of a disordered tight-binding model. Excellent agreement for weak-interaction values is found. Implications for the interpretation of current experimental data are discussed.

Compressibility studies of two-dimensional electron gases

Double quantum well structures have been used to study the compressibility of two-dimensional electron gases. By using independent contacts to the two layers we are able to measure accurately the carrier density changes of the lower layer (which acts as a detector) in response to altering the top layer density (whose compressibility we are probing). The carrier density changes are related to the chemical potential changes in the top layer which is proportional to the inverse compressibility. Comparison with a model which solves the Poisson and Schrödinger equations in a self-consistent manner provides good qualitative agreement with the experimental results, with the compressibility changing from positive at high carrier densities to negative at low densities when the energy changes due to interactions dominate the chemical potential changes. Samples with different well widths have also been studied, these show that the compressibility varies with well width. When analysing this data we find the Hartree contribution is significant and cannot be ignored.

Even-odd correlations in capacitance fluctuations of quantum dots.

We investigate effects of short range interactions on the addition spectra of quantum dots using a disordered Hubbard model. A correlation function \cS(q) is defined on the inverse compressibility versus filling data, and computed numerically for small lattices. Two regimes of interaction strength are identified: the even/odd fluctuations regime typical of Fermi liquid ground states, and a regime of structureless $\cS(q)$ at strong interactions. We propose to understand the latter regime in terms of magnetically correlated localized spins.

Thermodynamics of a Charged Fermion Layer at High rs Values.

The chemical potential $\ensuremath{\mu}$ of a hole layer is mapped as a function of temperature and density $p$. We present the first investigation of the crossover from the strongly interacting degenerate regime to the classical ideal gas limit. The low temperature inverse compressibility, ${\ensuremath{\kappa}}^{\ensuremath{-}1}\ensuremath{\equiv}{p}^{2}\ensuremath{\partial}\ensuremath{\mu}/\ensuremath{\partial}p$, is negative and temperature independent. At $T\ensuremath{\approx}17\mathrm{K}$ temperature dependence commences abruptly: ${\ensuremath{\kappa}}^{\ensuremath{-}1}$ turns less negative with rising $T$ and eventually reverses sign. An ideal classical gas is attained at temperatures comparable to the interaction energy, more than an order of magnitude larger than the Fermi energy.

Exchange, correlation and compressibility in quantum wires: Anisotropic confinement model

We calculate the exchange energy, the correlation energy and the compressibility for a quantum wire using a three-sum-rule approach for the local-field correction. An anisotropic confinement model with δ-confinement in one direction and an oscillator confinement in the other direction is used in the calculation. We perform a systematic study versus the electron density and the width parameter of the wire and we present analytical and numerical results. In the low density regime the inverse compressibility is strongly negative and this might be observable via capacitance measurements. We predict a spin instability at low density, depending on the confinement parameter.

The thermodynamics of a charged fermion layer at high rs values

We report the measurement of the chemical potential of a two-dimensional hole layer in wide temperature and density ranges. Due to their heavy mass, the holes are in a regime where the ratio of Coulomb to Fermi energies r s is very high, thus facilitating the first detailed study of the full temperature-dependent crossover from the strongly interacting degenerate regime to the classical ideal gas limit. The measured low-temperature chemical potential and inverse compressibility are negative and uninfluenced by temperature. At T ≈ 17K they simultaneously and abruptly become temperature-dependent for all carrier concentrations. As the temperature is further raised, the inverse compressibility becomes less negative and reverses sign. An ideal classical gas behavior is attained only at temperatures of the order of the interaction energy, more than an order of magnitude larger than the Fermi energy.

Interacting electrons in disordered potentials: The inverse compressibility

The inverse compressibility, i.e., the change in the chemical potential as the number of particles in the sample is changed, is studied for a small quantum dot. It is found that the inverse compressibility behaves differently for different values of disorder and electron-electron interactions. For weak interactions or strong disorder one may understand this behavior in the framework of a random matrix theory.

Bulk diffusivity of lattice gases close to criticality

We consider lattice gases where particles jump at random times constrained by hard-core exclusion (simple exclusion process with speed change). The conventional theory of critical slowing down predicts that close to a critical point the bulk diffusivity vanishes as the inverse compressibility. We confirm this claim by proving a strictly positive lower bound for the conductivity.

Fluctuation phenomena in structurally symmetric polymer blends

Polymer reference interaction site model theory with the new molecular closures is employed to numerically and analytically study structurally and interaction potential symmetric binary blends. Both the compressibility and free energy routes to the thermodynamics are studied and the issue of thermodynamic consistency is addressed. A variety of non-Flory–Huggins effects, or ‘‘fluctuation phenomena,’’ are found. These include nonuniversal renormalization of the critical temperature and effective chi-parameter from their mean field values, composition-dependent chi-parameters, and nonlinear dependence of the inverse osmotic compressibility on inverse temperature. All these fluctuation effects depend on degree of polymerization, N, chain length asymmetry, polymer density, range and precise form of the attractive tail potentials, chain stiffness, and proximity to the phase boundary. Some of the fluctuation effects are intrinsic, i.e., survive in the long chain N→∞ limit, while others are finite size effects which arise from chain-connectivity-induced coupled local density and long wavelength concentration fluctuations. Due to the multiple sources of the fluctuation effects, even asymptotic finite size effects can appear ‘‘intrinsic’’ over extended ranges of N. Comparison with lattice Monte Carlo simulations of Deutsch and Binder shows good agreement with the theoretical predictions. All the fluctuation effects can be understood in simple terms by examining the enthalpy of mixing and local interchain correlations. The key physical process is thermally driven local interchain rearrangements corresponding to the formation of diffuse interfaces and clusters or droplets. Analytic results are derived using the Gaussian thread model, which provides a simple physical understanding of the origin of the numerically determined fluctuation effects. In the long chain limit the predictions for the thread blend are shown to be exactly thermodynamically consistent which is a unique circumstance for liquid state theories. The relation of the blend fluctuation stabilization process to the corresponding diblock copolymer problem is briefly discussed.

Thermodynamics of a liquid-metal model from high-temperature series expansions

A lattice model for liquid metals proposed by Nara, Ogawa and Matsubara(1979) is studied using high-temperature series expansions. The grand-canonical potentials are expanded exactly to order beta 9 and the susceptibilities are expanded to order beta 11 on two-, three-, and four-dimensional hypercubic lattices. It is conjectured that, for fixed values of the fermion hopping energy and fermion chemical potential, the system goes critical only at particular values of the lattice-gas chemical potential. The critical values of the lattice-gas chemical potential and the critical temperature are calculated self-consistently and the critical exponent of the inverse isothermal compressibility is estimated using Pade approximants. It is concluded that the critical behaviour of the model is not affected by the presence of the fermions. This conclusion is consistent with the exact solution on a one-dimensional lattice by Thompson, Matsubara and Yang(1994).

Critical exponents and phase diagram of polymer blend solutions: polystyrene/ poly(methyl methacrylate)/d6-benzene system

Critical behaviour of blend solutions of polystyrene ( M w = 3.55 × 10 5)/poly(methyl methacrylate) ( M w = 3.27 × 10 5) in d 6-benzene was investigated by static light scattering. The phase diagram (spinodal, binodal and cloud points) was determined at a fixed total polymer concentration. The system had the lower critical solution temperature. The inverse isothermal osmotic compressibility ( ∂π ∂c 1 T ) and correlation length ζ of concentration fluctuations were determined as functions of the reduced temperature ϵ= | T s T−1 | near the stability limit, where T is the absolute temperature and T s is the spinodal temperature. Both ( ∂π ∂c 1 T ) and ξ were described by single exponents over the investigated temperature range. The critical exponents γ and ν for ( ∂π ∂c 1 T ) and ζ were 1.23 and 0.63, respectively. They were very close to the three-dimensional Ising exponents and obviously different from the mean-field or Fisher's renormalized Ising exponents, as theoretically predicted by Broseta et al.