^{1,a)}, S. Bhattacharya

^{2}, S. Bhowmik

^{3}, R. Benedictus

^{3}and S. Choudhury

^{4}

### Abstract

We study thermoelectric power under strong magnetic field (TPM) in carbon nanotubes(CNTs) and quantum wires (QWs) of nonlinear optical, optoelectronic, and related materials. The corresponding results for QWs of III-V, ternary, and quaternary compounds form a special case of our generalized analysis. The TPM has also been investigated in QWs of II-VI,IV-VI, stressed materials,, , , and bismuth on the basis of the appropriate carrier dispersion laws in the respective cases. It has been found, taking QWs of , , , , , , lattice-matched to InP,, , , , stressed , , , , and bismuth as examples, that the respective TPM in the QWs of the aforementioned materials exhibits increasing quantum steps with the decreasing electron statistics with different numerical values, and the nature of the variations are totally band-structure-dependent. In CNTs, the TPM exhibits periodic oscillations with decreasing amplitudes for increasing electron statistics, and its nature is radically different as compared with the corresponding TPM of QWs since they depend exclusively on the respective band structures emphasizing the different signatures of the two entirely different one-dimensional nanostructured systems in various cases. The well-known expression of the TPM for wide gap materials has been obtained as a special case under certain limiting conditions, and this compatibility is an indirect test for our generalized formalism. In addition, we have suggested the experimental methods of determining the Einstein relation for the diffusivity-mobility ratio and the carrier contribution to the elastic constants for materials having arbitrary dispersion laws.

I. INTRODUCTION

II. THEORETICAL BACKGROUND

A. Investigation of the TPM in (, ) and (, 0) CNTs

B. Investigation of the TPM in QWs of nonlinear optical materials

C. Investigation of TPM for QWs of III-V, ternary, and quaternary materials

D. Investigation of TPM for QWs of II-VImaterials

E. Investigation of TPM for QWs of IV-VI compounds

F. Investigation of TPM for QWs stressed compounds

G. Investigations of TPM for QWs of

H. Investigation of the TPM for the QWs of -type platinum antimonide

I. Investigation of the TPM for the QWs of

J. Investigation of the TPM for the QWs of Bi in accordance with the McClure and Choi, the Hybrid, the Cohen, the Lax, and the parabolic ellipsoidal models

1. The McClure and Choi model

2. The Hybrid model

3. The Cohen model

4. The Lax model

5. The parabolic ellipsoidal model

K. The results of this paper find the following two important applications in the areas of quantum effect devices

III. RESULTS AND DISCUSSION

### Key Topics

- Quantum wells
- 69.0
- Carbon nanotubes
- 27.0
- Band models
- 26.0
- Dispersion relations
- 24.0
- Band structure
- 22.0

## Figures

(a) Plot of the TPM as a function of film thickness for the QWs of in accordance with (*a*) the generalized proposed band model with , (*b*) the generalized band model with , (*c*) the simplified three-band model of Kane, where, and , (*d*) the two-band model of Kane, and (*e*) the parabolic energy bands. (b) Plot of the TPM as a function of film thickness for the QWs of in accordance with (*a*) the generalized proposed band model with , (*b*) the generalized band model with , (*c*) the simplified three-band model of Kane, where and , (*d*) the two-band model of Kane, and (*e*) the parabolic energy bands. (c) Plot of the TPM as a function of electron concentration per unit length for the QWs of in accordance with (*a*) the generalized proposed band model with , (*b*) the generalized band model with , (*c*) the three-band model of Kane, where and , (*d*) the two-band model of Kane, and (*e*) the parabolic energy bands. The curves (*f*) and (*g*) exhibit the corresponding plots for (, ) and (, 0) CNTs, respectively. (d) Plot of the TPM as a function of electron concentration per unit length for the QWs of in accordance with (*a*) the generalized proposed band model with , (*b*) the generalized band model with , (*c*) the three-band model of Kane, where and , (*d*) the two-band model of Kane, and (*e*) the parabolic energy bands. The curves (*f*) and (*g*) exhibit the corresponding plots for (, ) and (, 0) CNTs, respectively.

(a) Plot of the TPM as a function of film thickness for the QWs of in accordance with (*a*) the generalized proposed band model with , (*b*) the generalized band model with , (*c*) the simplified three-band model of Kane, where, and , (*d*) the two-band model of Kane, and (*e*) the parabolic energy bands. (b) Plot of the TPM as a function of film thickness for the QWs of in accordance with (*a*) the generalized proposed band model with , (*b*) the generalized band model with , (*c*) the simplified three-band model of Kane, where and , (*d*) the two-band model of Kane, and (*e*) the parabolic energy bands. (c) Plot of the TPM as a function of electron concentration per unit length for the QWs of in accordance with (*a*) the generalized proposed band model with , (*b*) the generalized band model with , (*c*) the three-band model of Kane, where and , (*d*) the two-band model of Kane, and (*e*) the parabolic energy bands. The curves (*f*) and (*g*) exhibit the corresponding plots for (, ) and (, 0) CNTs, respectively. (d) Plot of the TPM as a function of electron concentration per unit length for the QWs of in accordance with (*a*) the generalized proposed band model with , (*b*) the generalized band model with , (*c*) the three-band model of Kane, where and , (*d*) the two-band model of Kane, and (*e*) the parabolic energy bands. The curves (*f*) and (*g*) exhibit the corresponding plots for (, ) and (, 0) CNTs, respectively.

(a) Plot of the TPM as a function of film thickness for the QWs of in accordance with (*a*) the three-band model of Kane, (*b*) the two-band model of Kane, and (*c*) the parabolic energy bands. (b) Plot of the TPM as a function of film thickness for the QWs of in accordance with (*a*) the three-band model of Kane, (*b*) the two-band model of Kane, and (*c*) the parabolic energy bands. (c) Plot of the TPM as a function of film thickness for the QWs of in accordance with (*a*) the three-band model of Kane, (*b*) the two-band model of Kane, and (*c*) the parabolic energy bands. (d) Plot of the TPM as a function of film thickness for the QWs of in accordance with (*a*) the three-band model of Kane, (*b*) the two-band model of Kane, and (*c*) the parabolic energy bands. (e) Plot of the TPM as a function of film thickness for the QWs of lattice matched to InP in accordance with (*a*) the three-band model of Kane, (*b*) the two-band model of Kane, and (*c*) the parabolic energy bands.

(a) Plot of the TPM as a function of film thickness for the QWs of in accordance with (*a*) the three-band model of Kane, (*b*) the two-band model of Kane, and (*c*) the parabolic energy bands. (b) Plot of the TPM as a function of film thickness for the QWs of in accordance with (*a*) the three-band model of Kane, (*b*) the two-band model of Kane, and (*c*) the parabolic energy bands. (c) Plot of the TPM as a function of film thickness for the QWs of in accordance with (*a*) the three-band model of Kane, (*b*) the two-band model of Kane, and (*c*) the parabolic energy bands. (d) Plot of the TPM as a function of film thickness for the QWs of in accordance with (*a*) the three-band model of Kane, (*b*) the two-band model of Kane, and (*c*) the parabolic energy bands. (e) Plot of the TPM as a function of film thickness for the QWs of lattice matched to InP in accordance with (*a*) the three-band model of Kane, (*b*) the two-band model of Kane, and (*c*) the parabolic energy bands.

(a) Plot of the TPM as a function of electron concentration per unit length for the QWs of in accordance with (*a*) the three-band model of Kane, (*b*) the two-band model of Kane, and (*c*) the parabolic energy bands. The curves (*d*) and (*e*) are the corresponding plots for (, ) and (, 0) CNTs. (b) Plot of the TPM as a function of electron concentration per unit length for the QWs of in accordance with (*a*) the three-band model of Kane, (*b*) the two-band model of Kane, and (*c*) the parabolic energy bands. The curves (*d*) and (*e*) exhibit the corresponding plots for (, ) and (, 0) CNTs, respectively. (c) Plot of the TPM as a function of electron concentration per unit length for the QWs of in accordance with (*a*) the three-band model of Kane, (*b*) the two-band model of Kane, and (*c*) the parabolic energy bands. The curves (*d*) and (*e*) exhibit the corresponding plots for (, ) and (, 0) CNTs, respectively. (d) Plot of the TPM as a function of electron concentration per unit length for the QWs of in accordance with (*a*) the three-band model of Kane, (*b*) the two-band model of Kane, and (*c*) the parabolic energy bands. The curves (*d*) and (*e*) exhibit the corresponding plots for (, ) and (, 0) CNTs, respectively. (e) Plot of the TPM as a function of electron concentration per unit length for the QWs of lattice matched to InP in accordance with (*a*) the three-band model of Kane, (*b*) the two-band model of Kane, and (*c*) the parabolic energy bands. The curves (*d*) and (*e*) exhibit the corresponding plots for (, ) and (, 0) CNTs, respectively.

(a) Plot of the TPM as a function of electron concentration per unit length for the QWs of in accordance with (*a*) the three-band model of Kane, (*b*) the two-band model of Kane, and (*c*) the parabolic energy bands. The curves (*d*) and (*e*) are the corresponding plots for (, ) and (, 0) CNTs. (b) Plot of the TPM as a function of electron concentration per unit length for the QWs of in accordance with (*a*) the three-band model of Kane, (*b*) the two-band model of Kane, and (*c*) the parabolic energy bands. The curves (*d*) and (*e*) exhibit the corresponding plots for (, ) and (, 0) CNTs, respectively. (c) Plot of the TPM as a function of electron concentration per unit length for the QWs of in accordance with (*a*) the three-band model of Kane, (*b*) the two-band model of Kane, and (*c*) the parabolic energy bands. The curves (*d*) and (*e*) exhibit the corresponding plots for (, ) and (, 0) CNTs, respectively. (d) Plot of the TPM as a function of electron concentration per unit length for the QWs of in accordance with (*a*) the three-band model of Kane, (*b*) the two-band model of Kane, and (*c*) the parabolic energy bands. The curves (*d*) and (*e*) exhibit the corresponding plots for (, ) and (, 0) CNTs, respectively. (e) Plot of the TPM as a function of electron concentration per unit length for the QWs of lattice matched to InP in accordance with (*a*) the three-band model of Kane, (*b*) the two-band model of Kane, and (*c*) the parabolic energy bands. The curves (*d*) and (*e*) exhibit the corresponding plots for (, ) and (, 0) CNTs, respectively.

(a) Plot of the TPM as a function of film thickness for the QWs of CdS in accordance with the Hopfield model where (*a*) and (*b*) . The curves (*c*), (*d*), and (*e*) exhibit the TPM for the QWs of , PbSnTe, and , respectively. (b) Plot of the TPM as a function of electron concentration per unit length for the QWs of CdS in accordance with the Hopfield model where (*a*) and (*b*) . The curves (*c*), (*d*), and (*e*) exhibit the TPM for QWs of , PbSnTe, and . The curves (*f*) and (*g*) exhibit the corresponding plots for (, ) and (, 0) CNTs, respectively.

(a) Plot of the TPM as a function of film thickness for the QWs of CdS in accordance with the Hopfield model where (*a*) and (*b*) . The curves (*c*), (*d*), and (*e*) exhibit the TPM for the QWs of , PbSnTe, and , respectively. (b) Plot of the TPM as a function of electron concentration per unit length for the QWs of CdS in accordance with the Hopfield model where (*a*) and (*b*) . The curves (*c*), (*d*), and (*e*) exhibit the TPM for QWs of , PbSnTe, and . The curves (*f*) and (*g*) exhibit the corresponding plots for (, ) and (, 0) CNTs, respectively.

(a) Plot of the TPM as a function of film thickness for the QWs of stressed InSb in accordance with (*a*) the absence of stress and (*b*) the presence of stress. The curves (*c*), (*d*), and (*e*) exhibit the TPM for QWs of , , and , respectively. (b) Plot of the TPM as a function of electron concentration per unit length for the QWs of stressed InSb in accordance with (*a*) the absence of stress and (*b*) the presence of stress. The curves (*c*), (*d*), and (*e*) exhibit the TPM in QWs of , , and , respectively. The curves (*f*) and (*g*) are the corresponding plots for (, ) and (, 0) CNTs, respectively.

(a) Plot of the TPM as a function of film thickness for the QWs of stressed InSb in accordance with (*a*) the absence of stress and (*b*) the presence of stress. The curves (*c*), (*d*), and (*e*) exhibit the TPM for QWs of , , and , respectively. (b) Plot of the TPM as a function of electron concentration per unit length for the QWs of stressed InSb in accordance with (*a*) the absence of stress and (*b*) the presence of stress. The curves (*c*), (*d*), and (*e*) exhibit the TPM in QWs of , , and , respectively. The curves (*f*) and (*g*) are the corresponding plots for (, ) and (, 0) CNTs, respectively.

(a) Plots of the TPM as a function of film thickness for QWs of Bi in accordance with (*a*) the McClure and Choi model, (*b*) the Hybrid model, (*c*) the Cohen model, (*d*) the Lax model, and (*e*) the parabolic ellipsoidal model. (b) Plots of the TPM as a function of for QWs of Bi in accordance with (*a*) the McClure and Choi model, (*b*) the Hybrid model, (*c*) the Cohen model, (*d*) the Lax model, and (*e*) the parabolic ellipsoidal. The curves (*f*) and (*g*) exhibit the TPM for the (, ) and (, 0) CNTs, respectively.

(a) Plots of the TPM as a function of film thickness for QWs of Bi in accordance with (*a*) the McClure and Choi model, (*b*) the Hybrid model, (*c*) the Cohen model, (*d*) the Lax model, and (*e*) the parabolic ellipsoidal model. (b) Plots of the TPM as a function of for QWs of Bi in accordance with (*a*) the McClure and Choi model, (*b*) the Hybrid model, (*c*) the Cohen model, (*d*) the Lax model, and (*e*) the parabolic ellipsoidal. The curves (*f*) and (*g*) exhibit the TPM for the (, ) and (, 0) CNTs, respectively.

## Tables

The numerical values of the energy band constants of the materials as used in this article.

The numerical values of the energy band constants of the materials as used in this article.

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