Electron-hole Liquid Research Articles

Scaling between the surface and bulk properties of electron-hole liquids

Calculations are presented of the surface structure and surface tension of the electron-hole liquid (EHL) in (111)-stressed Ge. The dramatic changes which the EHL undergoes as the Ge band structure changes with stress make this system ideal for studying the relationship between the surface and bulk properties of liquids. The Cahn-Hilliard scaling rule (1958) kappa sigma approximately L is found to be obeyed under strikingly diverse conditions. A case also occurs where no general relation between the surface tension and the bulk properties can hold.

Two-component electron-hole liquid: A simple model

With the use of a very simple model for a uniaxially stressed semiconductor such as Ge or Si it is shown analytically under which conditions a phase separation of the electron-hole liquid into two distinct liquids of the type predicted by Kirczenow and Singwi would take place.

Properties of the electron-hole liquid in Si: Zero stress to the high-stress limit

We have measured the properties of the electron-hole liquid (EHL) in Si as a function of stress from $\ensuremath{\sigma}=0 \mathrm{to} 163$ kg/${\mathrm{mm}}^{2}$ which is well into the high-stress limit where the valence bands are decoupled. These measurements provide a useful test of many-body theories for EHL in crystals with a simple band structure. To produce these high stresses and to achieve some unique experimental advantages not realized in uniform-stress geometries, we have employed a well-characterized Hertzian stress geometry to strain ultrapure crystals of Si. This geometry creates an electronic potential minimum which confines both electrons and holes inside the crystal volume. At low temperature the $e\ensuremath{-}h$ pairs spatially condense into a small volume (${10}^{\ensuremath{-}6}$ ml) of EHL at the bottom of the well. Because the liquid volume is small the EHL exists in a region of effectively uniform stress. This idea is confirmed by separate uniform-stress experiments which are included as a part of our study. We study the EHL properties by analyzing its photoluminescence with spectral, spatial, and time resolution under conditions in which stress, temperature, and photoexcitation level are varied over wide limits. In our spectral analysis, which incorporates stress-dependent band structure, we determine the Fermi energies and $e\ensuremath{-}h$ pair density as it decreases from 3.5 \ifmmode\times\else\texttimes\fi{} ${10}^{18}$ ${\mathrm{cm}}^{\ensuremath{-}3}$ at zero stress to 3.5 \ifmmode\times\else\texttimes\fi{} ${10}^{17}$ ${\mathrm{cm}}^{\ensuremath{-}3}$ in the high-stress limit. Corresponding measurements of the intrinsic EHL time decay show that the lifetime increases from 0.14 to 3.0 \ensuremath{\mu}s over the same stress range. Combining the density and lifetime measurements, we are able to investigate the density dependence of the enhancement factor ${g}_{\mathrm{eh}}(0)$ and the Auger process which dominates $e\ensuremath{-}h$ recombination. We have measured a phase diagram for excitonic gas-EHL at high stress and find a critical point of ${T}_{c}\ensuremath{\approx}20 \mathrm{K}$, ${n}_{c}\ensuremath{\approx}2.5\ifmmode\times\else\texttimes\fi{}{10}^{17} {\mathrm{cm}}^{\ensuremath{-}3}$. In the high-stress limit the measured EHL binding energy is in the range $\ensuremath{\varphi}=1\ensuremath{-}1.4$ meV. Additional spatial data reveal the presence of a repulsive force between droplets, possibly a "phonon wind," which inhibits large droplet formation. In general, the stress dependence of the ground-state properties shows trends which are in agreement with existing theories. However, some interesting discrepancies between our data and these theories are found.

The ground-state energy of an electron-hole liquid in a strong magnetic field

The behaviour of electron-hole systems such as Bi and Bi1-xSbx in a strong magnetic field has been investigated using simple models. The ground-state energy is numerically calculated using the Hartree-Fock, random-phase and Hubbard approximations. The correlation energy of the system is shown to have the properties inherent in a quasi-one-dimensional and one-component spin system. The effect of electron correlation on ultrasonic attenuation is discussed.

Dynamical properties of the non-equilibrium electron-hole plasma in silicon

The time evolution of electron-hole plasmas above the critical point for electron-hole liquid formation is studied in unstressed and uniaxially stressed Si. It is shown experimentally that the plasma time constant is controlled by the transition of carrier pairs from the plasma to free excitons. For different Si band configurations the dependence of the plasma density on temperature is studied. We are able to explain the experimental time constant and the density-temperature relation by using a stationary non-equilibrium theory for the thermo-diffusion of the plasma.

Two-component electron-hole liquid in Ge and Si crystals under tensile stress

The properties of the two-component electron-hole liquid (TCEHL) in Ge and Si under tensile stress at $T=0$ are calculated. It is predicted that the TCEHL should not phase separate in Ge under $〈110〉$ tensile stress. In Ge under $〈111〉$ tensile stress the TCEHL should not phase separate for low concentrations of hot electrons but should phase separate for high concentrations of hot electrons. In Si under $〈001〉$ tensile stress the TCEHL should not phase separate for low concentrations of hot electrons, but for high concentrations of hot electrons the theory does not make a unique prediction. In Si under $〈110〉$ tensile stress the qualitative predictions for high and low hot-electron concentrations are similar to those for $〈001〉$ tensile stress for low and high hot-electron concentrations, respectively. The main differences between the cases of tensile and compressive stress are due to the differences in the structure of the conduction band. This makes experiments on the TCEHL under tensile stress a practical means of obtaining data which approximate those which would be obtained at high hot-electron concentrations under compressive stress, i.e., in a part of the phase diagram inaccessible to direct experimental study. This complementarity between compressive and tensile stress should be quite precise in Si but only qualitatively valid in Ge because of pronounced valence-band effects.

Exchange-correlation energy and the phase seperation of the electron- hole liquid in stressed semiconductors

Corrections to the exchange-correlation energy ( E xc ) used by Kirczenow and Singwi to predict the phase seperation of the electron-hole liquid in <111>-stressed Ge are examined. Andryushin, Gel'fond and Silin (AGS) claimed that the corrections destroy this phase seperation. Combescot, Singwi and Vignale have claimed the opposite. The claims of AGS are shown to be unfounded. They stem from an inaccurate hole kinetic energy approximation. Kulakovskii and Kukushkin's finding that the corrections oppose the phase separation in Si conflicts with theory. Possible sources of the discrepancy are discussed. The effect of valence band structure on E XC is considered.

A calculation of valence band masses, exciton and acceptor energies and the ground state properties of the electron-hole liquid in cubic SiC

The Luttinger parameters of cubic SiC are determined by using a modified Lawaetz approach, including the hole-phonon coupling for small spin—orbit interaction: γ ∗ 1 = 2.817, γ ∗ 2 = 0.508, γ ∗ 3 = 0.860, κ ∗ = -0.41. Revised ground state calculations of the electron-hole liquid, including the nonparabolic valence band dispersion, for conflicting theories of the electron-phonon interaction, yield a good agreement between the ϵ ∗ 0-approximation and experiment. The calculated ground state binding energy of the free exciton and of the point-charge acceptor are 26.7 and 179 meV respectively. These calculations yield a revised value of the band gap energy E g = 2.416 eV.

Radiative and phonon-induced corrections to the ground state energy of electron-hole drops in indirect band gap semiconductors and a new two-component fluid

The ground state energy of electron-hole drops in optically excited indirect band gap semiconductors is examined. The radiative and the phonon-induced contributions are calculated in the Hartree-Folk approximation for the first time. Apart from the condensation of electron-hole liquid for weak electron-phonon interaction, there is a possibility of condensation of electron-electron and hole-hole pairs to form a two-component fluid of the electron-phonon interaction is strong. If there are strong eight-particle interactions between pairs, the biexcitons will condense rather than the electron-hole pairs.

Spatially separated electron-hole system in high magnetic fields

The excitonic phase in the system of spatially separated electrons and holes in high magnetic fields proves to be absolutely unstable because of the negative sign of isothermal compressibility. Instead of this phase, electron-hole liquid drops may exist in the system.

Density-functional theory of the electron-hole liquid in doped Ge

Density-functional theory of the electron-hole liquid in doped Ge

Far infrared and millimeter wave studies of impurity scattering within electron-hole liquid in semiconductors

Despite the experimental evidence in far-infrared that impurities are ionized within the electron-hole liquid in silicon and germanium, combination of luminescence data with millimeter wave cyclotron resonance and far-infrared magneto-optics tells us that the impurity-assisted intervalley electron scattering within EHL in silicon can apparently be interpreted on the basis of neutral impurity scattering. Justification of simulating the impurity center within EHL as “neutral” is grounded for the electron scattering problem, making use of the atomic scattering theory in contrast with a simple ionized impurity model calculation with proper screening.

Simple considerations on the surface tension and the critical temperature of the electron-hole liquids

The surface tension $\ensuremath{\sigma}$ of an electron-hole liquid (EHL) and its temperature dependence are related to its bulk properties, following the work of Kohn and Yaniv. The relation $\ensuremath{\sigma}=\ensuremath{\alpha}{(\mathrm{AC})}^{\frac{1}{2}}$, where $A$ is related to the bulk compressibility $\ensuremath{\chi}$ and $C$ is the Lifshitz parameter, is found to be obeyed in the EHL at zero temperature with $\ensuremath{\alpha}$ a constant of the order of 3. The same approach is used to justify the relation $\ensuremath{\sigma}\ensuremath{\chi}\ensuremath{\simeq}{a}_{B},{a}_{B}$ being the excitonic Bohr radius. We are also able to derive the temperature dependence of the surface tension and to estimate the liquid-gas temperature of the EHL, the latter being in good agreement with experiment.

Two-component electron-hole liquid in〈110〉-stressed Ge and Si: A test case for theory and experiment

Calculations of the properties of the two-component electron-hole liquid (TCEHL) in $〈110〉$-stressed Ge and Si and $T=0$ have been carried out. It is predicted that the TCEHL in $〈110〉$-stressed Ge should not phase separate at all while in $〈110〉$-stressed Si there should be no phase separation under experimentally accessible conditions. These model-independent predictions are in marked contrast with the case of $〈111〉$-stressed Ge where a phase separation of the TCEHL was predicted by Kirczenow anad Singwi and has recently been observed experimentally by Bajaj and Wong. The differences between the behavior of the TCEHL in $〈110〉$- and in $〈111〉$-stressed Ge are shown to be due primarily to the different structure of the conduction band in the two cases. This is also true of the difference between the cases of $〈100〉$- and $〈110〉$-stressed Si. Detailed experimental studies of the TCEHL in $〈110〉$-stressed Si and especially in $〈110〉$-stressed Ge would now be very useful for testing both the range of validity of the theory and the reliability of the recently developed experimental techniques.

Electron-hole liquid in GaS

Photoluminescence spectra of “pure” GaS are recorded under high excitation levels and low temperatures between 1.9 and 10K. This work reports a new and broad band which dominates the spectrum at high excitation intensities and temperatures near 1.9K. Its dependence on the excitation level is highly superlinear. When temperature is raised from 1.9 to 10K the spectrum observed of low excitation levels is recovered. We interpret the band here observed as due to radiative recombination from electron-hole droplets. From best fitting between experimental spectrum and calculated theoretical curves the critical density and the condensation energy are estimated as, n c = 4.5 × 10 20 cm −3 and φ = 9.0 meV.

Electron Scattering by Impurity Centres in Electron-Hole Liquid

The pseudo-neutral nature of the impurity centre embedded in electron-hole liquid in a semiconductor is discussed from the viewpoint of electron scattering. Such a remarkable difference in the scattering cross-section by a factor of \(\gtrsim 130\) as observed between donor and acceptor within EHL in silicon is interpreted by the direct analogy with the isolated neutral impurity scattering model rather than by the simplified ionised centre model with proper screening.

Luminescence from an Electron‐Hole Liquid in the Indirect Compound Ga(As1−xPx)

AbstractThe radiative recombination from an electron‐hole liquid (EHL) is investigated in highly excited pure Ga(As1−xPx) with indirect band structure. The experimental results reveal an arsenic‐induced radiative no‐phonon (NP) transition from the liquid phase. The comparison of the experimental and calculated lineshape of the EHL luminescence band provides the fundamental parameters of the EHL obtained for a sample with x = 0.88. The equilibrium carrier density and the work function at T = 2 K are determined to be neq = 9 × 1018 cm−3 and Φ = (20 ± 1) me V, respectively. The influence of the arsenic atoms on the energetic properties of the EHL is qualitatively discussed.

Effect of the electron-phonon interaction on the subband structure of inversion layers in compound semiconductors

Recently electron inversion layers have been fabricated on a number of compound (principally III–V) semiconductors. Because of the polar nature of these materials, the effect of the electron-optical phonon interaction is expected to be significant. This has certainly been true in recent calculations of the ground state energy and equilibrium density of electron-hole liquids in a number of compound semiconductors [1]. We have calculated the subband structure of inversion layers in polar semiconductors. The starting point of our many-body calculation is the Hartree subband structure determined using a variational method described in ref. [2]. Using the Hartree wavefunctions as basis functions we have calculated the self-energy of the ground and first excited subbands to lowest order in the effective interaction. This interaction is the sum of Coulomb (direct and image) and electron-phonon terms which are screened by a dielectric function that includes both e-e and e-ph effects. The screening by inversion layer electrons has been treated in the plasmon-pole approximation. We have obtained the electron self-energy, the quasi-particle energy, and the subband separations as a function of inversion layer concentration for parameters corresponding to GaAs. The coupling of electrons to bulk phonons is considered. Our results are compared to those obtained using the simpler ϵ 0 ∗ approximation [1]. Here, explicit phonon terms are neglected, but the bare band mass is replaced by the polaron mass, and the high frequency dielectric constant (ϵ ∞) is replaced by the static dielectric constant (ϵ 0). This approximation has been applied to the electron-hole liquid for situations in which the optical phonon frequency is larger than the electron plasma frequency. The idea behind the ϵ 0 ∗ approximation is to account in a very rough way for vertex corrections to the electron-phonon interaction. A principal criticism of our dynamic RPA calculations is the neglect of vertex corrections in the effective interaction. In three-dimensional metallic systems (high density), Migdal's theorem applies and vertex corrections are small. However, it is not clear that we may neglect such corrections in inversion layers. This point will be discussed further.

“Neutrality” Question of Impurities within Electron-Hole Liquid in Semiconductors

From the correspondence between our past cyclotron resonance work and the time resolved EHD luminescence result by Wagner-Schmid-Sauer for impurity-assisted intervalley scattering in silicon, a rather surprising conclusion is drawn that the impurities can remain “neutral” in the electron-hole liquid.

Quantum oscillations and phonon-wind effects in the magnetic field luminescence lifetime and total intensity of the electron-hole liquid in Ge

Magnetic field studies of the lifetime $\ensuremath{\tau}$ and the total luminescence intensity ${I}_{\mathrm{tot}}$ of the electron-hole liquid (EHL) in Ge up to 18 T are reported. At very low fields some highly anomalous increases of up to 20% in ${I}_{\mathrm{tot}}$ from 0 to 0.1 T and decreases of up to 50% from 0.1-0.5 T are observed. Most unexpectedly the size of these variations is found to be dependent on the crystallographic orientation of the excited crystal face and on the magnitude of the exciting laser intensity. The increases are attributed to magnetic field inhibition of the escape of Auger carriers from electron-hole drops and the decreases to magnetic field confinement of the carriers at the crystal surface, where nonradiative recombination mechanisms dominate. The striking angular dependence of the 0 to 0.1 T increase is a consequence of a variation in the size distribution of the EHD's with the orientation of the excited crystal face. This is shown to be consistent with, and to arise from, the strong anisotropy in the phonon wind which is the driving force behind the motion of the carriers and drops away from the point of excitation. Consequently the effective zero-field quantum efficiency is also not a constant but varies between 25% and 33% depending on the excitation conditions. The variation of the EHL density with field is determined from normalized plots of $\frac{{I}_{\mathrm{tot}}}{\ensuremath{\tau}}$ for $\stackrel{\ensuremath{\rightarrow}}{H}\ensuremath{\parallel}〈001〉 \mathrm{and} 〈111〉$ from 0 to 18 T. For $\stackrel{\ensuremath{\rightarrow}}{H}\ensuremath{\parallel}〈001〉$ a much higher density increase with field is found than for $\stackrel{\ensuremath{\rightarrow}}{H}\ensuremath{\parallel}〈111〉$. This is explained by the higher electron and hole densities of states for this direction. Evidence is presented that the dominant Auger mechanism is a phonon-assisted three-particle process. The uncorrelated coefficients of Auger recombination ${C}_{1}$ and radiative recombination ${B}_{1}$ are estimated to be ${C}_{1}=1.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}32}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ ${\mathrm{cm}}^{6}$ and ${B}_{1}=3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ ${\mathrm{cm}}^{3}$. Finally, detailed studies of Landau oscillations in ${I}_{\mathrm{tot}}$ and $\ensuremath{\tau}$ are described. By employing sensitive field-modulation techniques, high-field hole oscillations are observed for the first time. Using the hole-density-of-states evaluations of Hensel and Suzuki, these hole oscillations are used to provide an independent, precise determination of the high-field EHL density. Values in good agreement with those determined from the $\frac{{I}_{\mathrm{tot}}}{\ensuremath{\tau}}$ ratio are found. Analysis of the electron oscillations which occur at much lower field ($Hl3$ T) provides evidence for a renormalization of the transverse electron mass of \ensuremath{\sim} 10%. For $\stackrel{\ensuremath{\rightarrow}}{H}\ensuremath{\parallel}〈110〉$ from the electron oscillations the EHL density is found to decrease to $\frac{1}{2{n}_{0}}$ at 7 T, before increasing towards higher field, in contrast to the behavior for the other two directions.