Nondegenerate Semiconductors Research Articles

Entropy, Electric Field and Heavily Doped Nanowires

In this paper an attempt is made to study, the entropy in the presence of intense electric field in nanowires (NWs) of heavily doped (HD) III–V and optoelectronic materials on the basis of newly formulated electron dispersion relations within the frame work of k → ⋅ p → formalism. It is found taking HD NWs of InSb, InAs, Hg(1– x) Cd (x)Te and In(1– x) Ga( x) As( y) P(1– y) lattice matched to InP as examples III–V, ternary and quaternary compounds that the entropy increases with increasing electron concentration per unit length and decreasing film thickness in different spiky manners, since the coincidence of Fermi energy with the sub-band energy leads to the step functional dependence of the density state function and this fact is being reflected in the quantized variations of the entropy with the said variables. The entropy increases with increasing electric field and decreasing alloy composition respectively. The numerical values of entropy with all the physical variables are totally band structure dependent for all the cases. The most striking features are that the presence of poles in the dispersion relation of the materials in the absence of band tails creates the complex energy spectra in the corresponding opto-electronic HD NWs and the effective electron mass exists within the band gap which is impossible without the concept of band tailing. The well-known classical result of entropy for non-degenerate bulk semiconductors having parabolic energy bands has been obtained as a special case of our generalized formulation and thus confirming the compatibility test. The content of this paper finds four important applications in the field of quantum effect devices of nanoscience and nanotechnology.

Size effect in thermoelectric power factor of nondegenerate and degenerate low-dimensional semiconductors

Low-dimensional materials have been known to give high thermoelectric (TE) performance by reducing the confinement length of the materials. Recently, we have shown that the TE power factor of low-dimensional semiconductors depends not only on the confinement length, but also on the thermal de Broglie wavelength of electrons or holes [Phys. Rev. Lett. 117, 036602 (2016)], in which the calculation was performed by assuming the semiconductors to be nondegenerate, i.e., we approximated the Fermi energy to lie only within the energy band gap, or in other words, the low doping approximation. Now, in this work, we generalize the previous results considering the degenerate case, in which the Fermi energy can exist in the valence or conduction bands, thus enabling a full consideration of heavy doping. An analytical formula for the TE power factor is derived to describe the size effect in the power factor of the low-dimensional semiconductors. We find that for both nondegenerate and degenerate cases, the TE power factor is enhanced in one- and two-dimensional semiconductors when the confinement length is smaller than the thermal de Broglie wavelength of the semiconductors, with Fermi energy around top (bottom) of valence (conduction) band for the p-type (n-type) semiconductors.

Effect of the Annealing on the Low-Temperature Charge Transport Properties of Heavily Boron-Doped Nanocrystalline Silicon Films for Thermoelectric Applications

Abstract Silicon is the reference material of microelectronics, is readily available, relatively unexpensive, and its use may take profit of a fantastic technology. This may explain why a substantial effort has focused on improving its thermoelectric efficiency, either by top-down nanostructuring or through suitable processing. In this paper we report an analysis of the electronic transport properties of heavily boron-doped nanocrystalline silicon films. High-temperature thermal treatments are confirmed to remarkably increase its thermoelectric power factor. Electrical conductivity and Hall effect measurements were carried out over the temperature range 20–300 K along with Seebeck coefficient measurements. We provide evidence of the occurrence of low-temperature hopping conduction between impurity subbands. Dopant ionization was studied as a function of temperature. Freeze-out temperature was found to correlate with the Seebeck coefficient in agreement with Pisarenko equation. This brings to the conclusion that, while untreated samples are weakly degenerate, the thermal processing reverts them into non-degenerate semiconductors, in spite of the high doping level.

Mechanical and thermoelectric properties of Bi2−xSbxTe3 prepared by using encapsulated melting and hot pressing

Bi2-xSbxTe3 (x = 1.4 - 1.7) solid solutions were synthesized by using encapsulated melting (EM) and were consolidated by using hot pressing (HP). The lattice constants decreased with increasing Sb content, which indicated that solid solutions were successfully synthesized. The relative densities of all the hot-pressed specimens were higher than 97.9 %. X-ray diffraction patterns, pole figures and electron backscattered diffraction spectra of all the hot-pressed specimens indicated randomly oriented textures. Very low values of the orientation factor (F) were obtained, Fmin = 0.008 for Bi0.4Sb1.6Te3 hot-pressed at 673 K and Fmax = 0.115 for Bi0.4Sb1.6Te3 hot-pressed at 723 K, which implied that the microstructures were highly isotropic. A bending strength of 46 MPa and a Vickers hardness of 94 Hv were attained for Bi0.4Sb1.6Te3 hot-pressed at 648 K. However, the mechanical properties were degraded with increasing HP temperature owing to grain growth. An increased HP temperature did not affect the electrical properties significanctly. The carrier concentration increased with increasing Sb content, and the specimen with x = 1.4 showed nondegenerate semiconductor behavior whereas those with x ≥ 1.5 behaved as degenerate semiconductors. All specimens showed p-type conduction, which was confirmed from the positive signs of the Seebeck coefficient and the Hall coefficient. The increased Sb content caused a shift in the peak values of the Seebeck coefficient to higher temperatures and an enhancement of the power factor. The electronic thermal conductivity decreased, but the lattice thermal conductivity increased, with increasing Sb content. The lowest theraml conductivity of 0.83 Wm -1K -1 was obtained at 373 K for Bi0.4Sb1.6Te3. The maximum dimensionless figure of merit, ZTmax = 1.1, was achieved at 323 K for Bi0.4Sb1.6Te3.

Ohm’s Law for a Bipolar Semiconductor: The Role of Carrier Concentration and Energy Nonequilibria

The effective linear electrical conductivity of a nondegenerate bipolar semiconductor, sandwiched between two metals, is investigated taking into account both its nonequilibrium charge carriers (both electrons and holes) and nonequilibrium temperature. We stress that even in the linear perturbative approximation both carrier concentration and energy nonequilbria arise automatically when an electrical current flows. The expression for the effective electrical conductivity is obtained and shown to depend on the electron and hole electrical conductivity, the thermal conductivity, the bandgap, charge carriers lifetimes, and both bulk and surface recombination rates. The effective electrical conductivity is equal to the classical result, i.e., the sum of the electron and hole electrical conductivities, only if the surface recombination rate at the interface is sufficiently strong or the charge carrier lifetime is sufficiently small. In this article, partial cases are considered, specifically, semiconductors with small and large thermal conductivities, semiconductors with monopolar electron and monopolar holes, strong and weak surface recombination rates, and small and large charge carrier lifetimes. Expressions for the effective electrical conductivity are obtained in all partial cases.

Quantum Effects in the Thermoelectric Power Factor of Low-Dimensional Semiconductors.

We theoretically investigate the interplay between the confinement length L and the thermal de Broglie wavelength Λ to optimize the thermoelectric power factor of semiconducting materials. An analytical formula for the power factor is derived based on the one-band model assuming nondegenerate semiconductors to describe quantum effects on the power factor of the low-dimensional semiconductors. The power factor is enhanced for one- and two-dimensional semiconductors when L is smaller than Λ of the semiconductors. In this case, the low-dimensional semiconductors having L smaller than their Λ will give a better thermoelectric performance compared to their bulk counterpart. On the other hand, when L is larger than Λ, bulk semiconductors may give a higher power factor compared to the lower dimensional ones.

A note on the electrochemical nature of the thermoelectric power

While thermoelectric transport theory is well established and widely applied, there remains some degree of confusion on the proper thermodynamic definition of the Seebeck coefficient (or thermoelectric power) which is a measure of the strength of the mutual interaction between electric charge transport and heat transport. Indeed, as one considers a thermoelectric system, it is not always clear whether the Seebeck coefficient is to be related to the gradient of the system's chemical potential or to the gradient of its electrochemical potential. This pedagogical article aims to shed light on this confusion and clarify the thermodynamic definition of the thermoelectric coupling. First, we recall how the Seebeck coefficient is experimentally determined. We then turn to the analysis of the relationship between the thermoelectric power and the relevant potentials in the thermoelectric system: As the definitions of the chemical and electrochemical potentials are clarified, we show that, with a proper consideration of each potential, one may derive the Seebeck coefficient of a non-degenerate semiconductor without the need to introduce a contact potential as seen sometimes in the literature. Furthermore, we demonstrate that the phenomenological expression of the electrical current resulting from thermoelectric effects may be directly obtained from the drift-diffusion equation.

Linear Electrical Conductivity of a Bipolar Semiconductor: Heating and Recombination

The linear electrical conductivity of a nondegenerate bipolar semiconductor, with metal contacts at both sides, is investigated for small values of the thermal conductivity in a general case, i.e., when both nonequilibrium charge carriers (electrons and holes) and nonequilibrium temperature are present. It must be emphasized that both concentration and energy nonequilibria arise automatically when an electric current flows, even in a linear approximation with respect to perturbation. The expression for the electrical conductivity is obtained. This expression depends on electrical conductivities of electrons and holes, the thermal conductivity, the bandgap, the lifetime of charge carriers, and the surface recombination rate at the contacts of a semiconductor with a metal.

Classical Size Effects in Films of n-PbTe

Based on kinetic Boltzmann equation the boundary problem of calculating the conductivity and Seebeck coefficient for a film with a rectangular cross section is solved. Mirror- diffuse mechanism of reflection of the charge carriers from the surfaces of the film is considered. Calculations were performed for different thicknesses nondegenerate semiconductor n-PbTe.
 A comparison of theoretical calculations with obtained experimental data for vapor-phase condensates based on PbTe is made.

The high-frequency conductivity of a thin semiconductor wire in the longitudinal magnetic field

The problem of high-frequency conductivity of a thin semiconductor cylindrical wire at arbitrary temperature in the longitudinal magnetic field is solved by the kynetic method. The charge carrier reflection from inner wire surface is assumed to be diffuse. The ratio between the mean free path of charge carriers and the wire radius is supposed to be arbitrary. The results obtained for limiting cases of degenerate and nondegenerate semiconductor were subjected to comparative analysis.

Energy flux in semiconductors: Interaction of thermal and concentration nonequilibriums

The energy flux in bipolar semiconductors is investigated taking into account the influence of recombination on it. The general expression of an energy flux in a nondegenerate semiconductor is obtained in a linear approximation with respect to perturbation taking into account recombination (the presence of nonequilibrium charge carriers in the semiconductor) and thermal electrical currents of electrons and holes. The energy flux density has been calculated in two different cases, the case of weak recombination and the case of strong recombination, for a one-dimension case.

Elucidating the electron transport in semiconductors via Monte Carlo simulations: an inquiry-driven learning path for engineering undergraduates

Within the context of higher education for science or engineering undergraduates, we present an inquiry-driven learning path aimed at developing a more meaningful conceptual understanding of the electron dynamics in semiconductors in the presence of applied electric fields. The electron transport in a nondegenerate n-type indium phosphide bulk semiconductor is modelled using a multivalley Monte Carlo approach. The main characteristics of the electron dynamics are explored under different values of the driving electric field, lattice temperature and impurity density. Simulation results are presented by following a question-driven path of exploration, starting from the validation of the model and moving up to reasoned inquiries about the observed characteristics of electron dynamics. Our inquiry-driven learning path, based on numerical simulations, represents a viable example of how to integrate a traditional lecture-based teaching approach with effective learning strategies, providing science or engineering undergraduates with practical opportunities to enhance their comprehension of the physics governing the electron dynamics in semiconductors. Finally, we present a general discussion about the advantages and disadvantages of using an inquiry-based teaching approach within a learning environment based on semiconductor simulations.

Electron-acoustic phonon field induced tunnel scattering

Theory of electron-acoustic single phonon scattering has been reconsidered. It is assumed that the non-degenerate semiconductor has a spherical parabolic band structure. In the basis of the reconsideration there is a phenomenon of the tilting of semiconductor bands by the perturbing potential of an electric field. In this case, electron eigenfunctions are not plane waves or Bloch functions. In low-field regime, the expressions for electron intraband transition probability and scattering time are obtained under elastic collision approximation. Dependencies of scattering time on electron energy and uniform electric field are analyzed. The results of corresponding numerical computations for n-Si at 300 K are presented. It is established that there is no fracture on the curve of electron scattering time dependence on the electron energy.

Piezoelectric scattering limited mobility as controlled by the transverse component of the phonon wave vector in quantum layers at low temperatures

Theory of the momentum relaxation time of the carriers and of their ohmic mobility in a quantum well in a non-degenerate compound semiconductor has been developed for interaction with piezoelectric phonons at low temperatures. The calculations have been made without taking recourse either to the Kawaji's traditional model of interaction with only the in-plane component of the phonon wave vector or to the equipartition approximation for the phonon population. The numerical results obtained for inversion layers in GaAs and CdS describe how significantly does the consideration of the transverse component of the phonon wave vector in the light of the Ridley's momentum conservation approximation or of the true phonon distribution bring in significant changes in the transport characteristics compared to what the traditional approximations predict.

Characterization of Lorenz number with Seebeck coefficient measurement

In analyzing zT improvements due to lattice thermal conductivity (κL) reduction, electrical conductivity (σ) and total thermal conductivity (κTotal) are often used to estimate the electronic component of the thermal conductivity (κE) and in turn κL from κL = ∼ κTotal − LσT. The Wiedemann-Franz law, κE = LσT, where L is Lorenz number, is widely used to estimate κE from σ measurements. It is a common practice to treat L as a universal factor with 2.44 × 10−8 WΩK−2 (degenerate limit). However, significant deviations from the degenerate limit (approximately 40% or more for Kane bands) are known to occur for non-degenerate semiconductors where L converges to 1.5 × 10−8 WΩK−2 for acoustic phonon scattering. The decrease in L is correlated with an increase in thermopower (absolute value of Seebeck coefficient (S)). Thus, a first order correction to the degenerate limit of L can be based on the measured thermopower, |S|, independent of temperature or doping. We propose the equation: L=1.5+exp−|S|116 (where L is in 10−8 WΩK−2 and S in μV/K) as a satisfactory approximation for L. This equation is accurate within 5% for single parabolic band/acoustic phonon scattering assumption and within 20% for PbSe, PbS, PbTe, Si0.8Ge0.2 where more complexity is introduced, such as non-parabolic Kane bands, multiple bands, and/or alternate scattering mechanisms. The use of this equation for L rather than a constant value (when detailed band structure and scattering mechanism is not known) will significantly improve the estimation of lattice thermal conductivity.

Parametric modeling of energy filtering by energy barriers in thermoelectric nanocomposites

We present a parametric modeling of the thermoelectric transport coefficients based on a model previously used to interpret experimental measurements on the conductivity, σ, and Seebeck coefficient, S, in highly Boron-doped polycrystalline Si, where a very significant thermoelectric power factor (TPF) enhancement was observed. We have derived analytical formalism for the transport coefficients in the presence of an energy barrier assuming thermionic emission over the barrier for (i) non-degenerate and (ii) degenerate one-band semiconductor. Simple generic parametric equations are found that are in agreement with the exact Boltzmann transport formalism in a wide range of parameters. Moreover, we explore the effect of energy barriers in 1-d composite semiconductors in the presence of two phases: (a) the bulk-like phase and (b) the barrier phase. It is pointed out that significant TPF enhancement can be achieved in the composite structure of two phases with different thermal conductivities. The TPF enhancement is estimated as a function of temperature, the Fermi energy position, the type of scattering, and the barrier height. The derived modeling provides guidance for experiments and device design.

Is Cu3SbSe3 a promising thermoelectric material?

Cu3SbSe3 characterized by ultralow thermal conductivity is a wide-gap, nondegenerate semiconductor with a large effective mass and deformation potential, yielding zTmax = 0.25.

The Seebeck coefficient and phonon drag in silicon

We present a theory of the phonon-drag Seebeck coefficient in nondegenerate semiconductors, and apply it to silicon for temperatures 30 < T < 300 K. Our calculation uses only parameters from the literature, and previous calculations of the phonon lifetime. We find excellent agreement with the measurements of Geballe and Hull [Phys. Rev. 98, 940 (1955)]. The phonon-drag term dominates at low temperature, and shows an important dependence on the dimensions of the experimental sample.

Vacancy-donor pairs and their formation in irradiated n-Si

The present experiments using fast electron and gamma-irradiation are aimed at checking the model of defect formation in oxygen-lean n-Si(FZ) in a quantitative way. Electrical measurements are taken over a wide temperature range of 20 to 300 K. Analysis of equations of charge balance making use of the statistics of charge carriers in non-degenerate semiconductors demonstrates that group-V impurity atoms strongly interact with intrinsic point defects. As a result, the concentration of shallow donor states is markedly decreased. This loss of shallow donors, −δND, is accompanied with an increase in the concentration of radiation-produced deep acceptors, +δNArad, being equal in magnitude but opposite in sign. Such behavior correlates quantitatively with the formation model of donor-vacancy pairs put forward earlier by Watkins and Corbett, what has been proved on the basis of electrical data for the first time. The formation kinetics of these complexes is discussed. Defects of interstitial type in irradiated material appear to be electrically neutral in n-Si. However, their production in the course of electron- and gamma-irradiation is believed to be responsible for drastic changes in the mobility of charge carriers at cryogenic temperatures.

Thermoelectric properties of Mg2Si0.7Ge0.3Bi m prepared using a solid-state reaction

Mg2Si0.7Ge0.3Bi m (m = 0 − 0.03) solid solutions were synthesized using a solid-state reaction and consolidated by hot pressing. All specimens showed n-type conduction, and the carrier concentration increased from 4.0 × 1017 to 2.1 × 1020 cm−3 with increasing Bi-doping content; the electrical conductivity thereby increased from 7.3 × 102 to 7.0 × 104 Sm−1 at room temperature. The electrical conductivity of the undoped specimen increased with increasing temperature, and the specimen behaved similar to a non-degenerate semiconductor. The absolute value of the Seebeck coefficient of the undoped specimen was very high at low temperature, but it decreased with increasing temperature. Bi-doped Mg2Si0.7Ge0.3 showed a Seebeck coefficient ranging from −235 to −197 μVK−1 at 823 K. The power factor significantly increased with Bi doping and increased with increasing temperature; it were also improved by around 10 times at 823 K. The lowest thermal conductivity was 2.2 Wm−1K−1 for Mg2Si0.7Ge0.3Bi0.02 at 823 K, and the maximum ZT value of 0.79 was obtained for Mg2Si0.7Ge0.3Bi0.02 at 823 K.