Submit manuscript...
eISSN: 2576-4543

Physics & Astronomy International Journal

Short Communication Volume 7 Issue 1

On the Einstein relation between mobility and diffusion cofficint in an active Nanostructured Materials

Subhamoy Singha Roy

Department of Physics, JIS College of Engineering under Maulana Abul Kalam Azad University of Technology, India

Correspondence: Subhamoy Singha Roy, Department of Physics, JIS College of Engineering under Maulana Abul Kalam Azad University of Technology, India, Tel 8910557519

Received: January 02, 2023 | Published: February 2, 2023

Citation: Roy SS. On the Einstein relation between mobility and diffusion cofficint in an active Nanostructured Materials. Phys Astron Int J. 2023;7(1):7-9. DOI: 10.15406/paij.2023.07.00277

Download PDF

Abstract

It is commonly known that the speed at which contemporary switching semiconductors work much depends on the carrier degeneracy of the band. Furthermore, the relation of the diffusivity to mobility ratio of the carriers in semiconductors (referred to as DMR) is very helpful because it is more accurate than any of the individual relations for diffusivity to mobility ratio, which is thought to be the two most frequently used parameters in carrier transport in semiconductors. With n-InSb and Hg1-xCdxTe as examples of numerical calculations, we will examine DMR under strong magnetic quantization of III-V in line with the three- and two-band models of Kane, respectively, along with parabolic energy bands.

Keywords: Diffusivity to mobility ratio, magnetic quantization, III-V semiconductor

Introduction

There has been a lot of interest in recent years in researching the many physical properties of degenerate semiconductors that follow Kane's dispersion relation and have non-parabolic energy bands. Numerous aspects that result in these semiconductors' unique properties have been discovered to be influenced by the band non-parabolicity. The last few years have seen a significant increase in interest in both new technology applications as well as their ability to elucidate novel phenomena in computational and theoretical nanoscience. The quantization of the wave vector—also known as the quantum size effect (QSE)—allows for 2D electron transport parallel to the surface in QWs, representing new physical features not present in bulk semiconductors.1 This phenomenon is caused by the restriction of carrier motion in the direction normal to the film (let's say the z-direction). To further explore these materials' electrical characteristics, a brand-new structure called QWWs has been put out.2

The charge carriers can only travel in the longitudinal direction in the 1D synthetic material because the electron gas is quantized in two transverse directions.3 The degree of freedom of the free carriers substantially reduces when the quantum well's dimension goes from 1D to 3D, and the density-of-states function switches from a stepped cumulative one to Dirac's delta function.4,5 Because of the improvement in carrier mobility, low-dimensional heterostructures made of diverse materials are being studied extensively.6 These characteristics make such structures appropriate for use in devices such as optical modulators,7 optical switching systems,8 high-speed digital networks,9 high-frequency microwave circuits,10 quantum well lasers,11 and others.

In order to express the performance at the device terminals and the switching speed in terms of carrier concentration, the appropriate Einstein relation must be used. It is well known that the degree of carrier degeneracy of the band has a significant impact on how quickly modern switching semiconductors operate. Furthermore, since it is more accurate than any of the individual relations for diffusivity to mobility ratio, which are thought to be the two most frequently used parameters in carrier transport in semiconductors, the relation of the diffusivity to mobility ratio of the carriers in semiconductors (referred to as DMR) is very helpful. The diffusivity of mobility ratio in non-degenerate semiconductors is k B T e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaam 4AamaaBaaaleaacaWGcbaabeaakiaadsfaaeaacaWGLbaaaaaa@3C1B@  although for degenerate semiconductors, Landsberg demonstrated for the first time12 that under the condition carrier degeneracy, the DMR depends only on the band structure. Additionally, substantial research has been done on the relationship between the DMR and the screening length, its relationship with the noise power, and the several DMR formulations for degenerate semiconductors with varying band structures.13-17

Theoretical background

The DMR is expressed as a result of the basic analysis as  

d μ = 1 e n 0 ( n 0 E F ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaam izaaqaaiabeY7aTbaacqGH9aqpdaWcaaqaaiaaigdaaeaacaWGLbaa amaalaaabaGaamOBamaaBaaaleaacaaIWaaabeaaaOqaamaabmaaba WaaSGaaeaacqGHciITcaWGUbWaaSbaaSqaaiaaicdaaeqaaaGcbaGa eyOaIyRaamyramaaBaaaleaacaWGgbaabeaaaaaakiaawIcacaGLPa aaaaaaaa@47CE@   (1)

According to the band Kane model, the following is the equation for the electron concentration in magnetic quantization in III-V semiconductors

n 0 = eB ( 2 m * ) π 2 2 n=0 n max [ Φ(n, E F )+ π 2 k B 2 T 2 6 Φ (n, E F ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad6gadaWgaa WcbaGaaGimaaqabaGccqGH9aqpdaWcaaqaaiaadwgacaWGcbWaaOaa aeaadaqadaqaaiaaikdacaWGTbWaaWbaaSqabeaacaGGQaaaaaGcca GLOaGaayzkaaaaleqaaaGcbaGaeqiWda3aaWbaaSqabeaacaaIYaaa aOGaeS4dHG2aaWbaaSqabeaacaaIYaaaaaaakmaaqahabaWaamWaae aacqqHMoGrcaGGOaGaamOBaiaacYcacaWGfbWaaSbaaSqaaiaadAea aeqaaOGaaiykaiabgUcaRmaalaaabaGaeqiWda3aaWbaaSqabeaaca aIYaaaaOGaam4AamaaDaaaleaacaWGcbaabaGaaGOmaaaakiaadsfa daahaaWcbeqaaiaaikdaaaaakeaacaaI2aaaaiqbfA6agzaagaGaai ikaiaad6gacaGGSaGaamyramaaBaaaleaacaWGgbaabeaakiaacMca aiaawUfacaGLDbaaaSqaaiaad6gacqGH9aqpcaaIWaaabaGaamOBam aaBaaameaaciGGTbGaaiyyaiaacIhaaeqaaaqdcqGHris5aaaa@6654@   (2)

Using equations (1) and (2), we get

d μ = 1 e n=0 n max [ Φ(n, E F )+ π 2 k B 2 T 2 6 Φ (n, E F ) ] [ n=0 n max [ Φ (n, E F )+ π 2 k B 2 T 2 6 Φ'''(n, E F ) ] ] 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaam izaaqaaiabeY7aTbaacqGH9aqpdaWcaaqaaiaaigdaaeaacaWGLbaa amaaqahabaWaamWaaeaacqqHMoGrcaGGOaGaamOBaiaacYcacaWGfb WaaSbaaSqaaiaadAeaaeqaaOGaaiykaiabgUcaRmaalaaabaGaeqiW da3aaWbaaSqabeaacaaIYaaaaOGaam4AamaaDaaaleaacaWGcbaaba GaaGOmaaaakiaadsfadaahaaWcbeqaaiaaikdaaaaakeaacaaI2aaa aiqbfA6agzaagaGaaiikaiaad6gacaGGSaGaamyramaaBaaaleaaca WGgbaabeaakiaacMcaaiaawUfacaGLDbaaaSqaaiaad6gacqGH9aqp caaIWaaabaGaamOBamaaBaaameaaciGGTbGaaiyyaiaacIhaaeqaaa qdcqGHris5aOWaamWaaeaadaaeWbqaamaadmaabaGafuOPdyKbauaa caGGOaGaamOBaiaacYcacaWGfbWaaSbaaSqaaiaadAeaaeqaaOGaai ykaiabgUcaRmaalaaabaGaeqiWda3aaWbaaSqabeaacaaIYaaaaOGa am4AamaaDaaaleaacaWGcbaabaGaaGOmaaaakiaadsfadaahaaWcbe qaaiaaikdaaaaakeaacaaI2aaaaiabfA6agjaacEcacaGGNaGaai4j aiaacIcacaWGUbGaaiilaiaadweadaWgaaWcbaGaamOraaqabaGcca GGPaaacaGLBbGaayzxaaaaleaacaWGUbGaeyypa0JaaGimaaqaaiaa d6gadaWgaaadbaGaciyBaiaacggacaGG4baabeaaa0GaeyyeIuoaaO Gaay5waiaaw2faamaaCaaaleqabaGaeyOeI0IaaGymaaaaaaa@8424@   (3)

For the two-band model of Kane, equation (3) gets transformed as:

d μ = k B T e n=0 n max [ 1 a { ( 1+ 3αb 2 ) F 1/2 ( η )+ 3α k B T 4 F 1/2 ( η ) } ] n=0 n max [ 1 a { ( 1+ 3αb 2 ) F 3/2 ( η )+ 3α k B T 4 F 1/2 ( η ) } ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaam izaaqaaiabeY7aTbaacqGH9aqpdaWcaaqaaiaadUgadaWgaaWcbaGa amOqaaqabaGccaWGubaabaGaamyzaaaadaWcaaqaamaaqahabaWaam WaaeaadaWcaaqaaiaaigdaaeaadaGcaaqaaiaadggaaSqabaaaaOWa aiWaaeaadaqadaqaaiaaigdacqGHRaWkdaWcaaqaaiaaiodacqaHXo qycaWGIbaabaGaaGOmaaaaaiaawIcacaGLPaaacaWGgbWaaSbaaSqa amaalyaabaGaeyOeI0IaaGymaaqaaiaaikdaaaaabeaakiaacIcacu aH3oaAgaqbaiaacMcacqGHRaWkdaWcaaqaaiaaiodacqaHXoqycaWG RbWaaSbaaSqaaiaadkeaaeqaaOGaamivaaqaaiaaisdaaaGaamOram aaBaaaleaadaWcgaqaaiaaigdaaeaacaaIYaaaaaqabaGccaGGOaGa fq4TdGMbauaacaGGPaaacaGL7bGaayzFaaaacaGLBbGaayzxaaaale aacaWGUbGaeyypa0JaaGimaaqaaiaad6gadaWgaaadbaGaciyBaiaa cggacaGG4baabeaaa0GaeyyeIuoaaOqaamaaqahabaWaamWaaeaada WcaaqaaiaaigdaaeaadaGcaaqaaiaadggaaSqabaaaaOWaaiWaaeaa daqadaqaaiaaigdacqGHRaWkdaWcaaqaaiaaiodacqaHXoqycaWGIb aabaGaaGOmaaaaaiaawIcacaGLPaaacaWGgbWaaSbaaSqaamaalyaa baGaeyOeI0IaaG4maaqaaiaaikdaaaaabeaakiaacIcacuaH3oaAga qbaiaacMcacqGHRaWkdaWcaaqaaiaaiodacqaHXoqycaWGRbWaaSba aSqaaiaadkeaaeqaaOGaamivaaqaaiaaisdaaaGaamOramaaBaaale aadaWcgaqaaiabgkHiTiaaigdaaeaacaaIYaaaaaqabaGccaGGOaGa fq4TdGMbauaacaGGPaaacaGL7bGaayzFaaaacaGLBbGaayzxaaaale aacaWGUbGaeyypa0JaaGimaaqaaiaad6gadaWgaaadbaGaciyBaiaa cggacaGG4baabeaaa0GaeyyeIuoaaaaaaa@9301@   (4)

where we have applied the formula

D Dη F j ( η )= F j1 ( η ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaam iraaqaaiaadseacqaH3oaAaaGaamOramaaBaaaleaacaWGQbaabeaa kmaabmaabaGaeq4TdGgacaGLOaGaayzkaaGaeyypa0JaamOramaaBa aaleaacaWGQbGaeyOeI0IaaGymaaqabaGcdaqadaqaaiabeE7aObGa ayjkaiaawMcaaaaa@48A1@

For α→0 as for wide gap semiconductors, we get

d μ = k B T e n=0 n max F 1/2 ( η ) n=0 n max F 3/2 (η) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaam izaaqaaiabeY7aTbaacqGH9aqpdaWcaaqaaiaadUgadaWgaaWcbaGa amOqaaqabaGccaWGubaabaGaamyzaaaadaWcaaqaamaaqahabaGaam OramaaBaaaleaadaWcgaqaaiabgkHiTiaaigdaaeaacaaIYaaaaaqa baGcdaqadaqaaiabeE7aObGaayjkaiaawMcaaaWcbaGaamOBaiabg2 da9iaaicdaaeaacaWGUbWaaSbaaWqaaiGac2gacaGGHbGaaiiEaaqa baaaniabggHiLdaakeaadaaeWbqaaiaadAeadaWgaaWcbaWaaSGbae aacqGHsislcaaIZaaabaGaaGOmaaaaaeqaaOGaaiikaiabeE7aOjaa cMcaaSqaaiaad6gacqGH9aqpcaaIWaaabaGaamOBamaaBaaameaaci GGTbGaaiyyaiaacIhaaeqaaaqdcqGHris5aaaaaaa@5EEE@   (5)

where

η [ E F ( n+1/2 ) ω 0 ] k B T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeE7aOjabgg Mi6oaalaaabaWaamWaaeaacaWGfbWaaSbaaSqaaiaadAeaaeqaaOGa eyOeI0YaaeWaaeaacaWGUbGaey4kaSYaaSGbaeaacaaIXaaabaGaaG OmaaaaaiaawIcacaGLPaaacqWIpecAcqaHjpWDdaWgaaWcbaGaaGim aaqabaaakiaawUfacaGLDbaaaeaacaWGRbWaaSbaaSqaaiaadkeaae qaaOGaamivaaaaaaa@4C21@

Under the condition of non-degeneracy, we know

F j ( η ) e η MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAeadaWgaa WcbaGaamOAaaqabaGcdaqadaqaaiabeE7aObGaayjkaiaawMcaaiab gIKi7kaadwgadaahaaWcbeqaaiabeE7aObaaaaa@41F4@

and equations (4) and (5) lead to well known result as

d μ = k B T e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaam izaaqaaiabeY7aTbaacqGH9aqpdaWcaaqaaiaadUgadaWgaaWcbaGa amOqaaqabaGccaWGubaabaGaamyzaaaaaaa@3FA0@   (6)

as it should be for non-degenerate semiconductors.

Experimental advice for determining ratio of d μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaam izaaqaaiabeY7aTbaaaaa@3B0B@  indirectly in degenerate semiconductors with arbitrary band structures and strong magnetic quantization.

It is widely known that only the dispersion rules can explain the thermoelectric power when there is substantial magnetic quantization.18 The amount of thermoelectric power is expressed as

g= 1 eT n 0 + ( EE F )R( E )[ ( f 0 E ) ] dE MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEgacqGH9a qpdaWcaaqaaiaaigdaaeaacaWGLbGaamivaiaad6gadaWgaaWcbaGa aGimaaqabaaaaOWaa8qmaeaadaqadaqaaiaadweacqGHsislcaWGfb WaaSraaSqaaiaadAeaaeqaaaGccaGLOaGaayzkaaGaamOuamaabmaa baGaamyraaGaayjkaiaawMcaamaadmaabaWaaeWaaeaacqGHsislda WcaaqaaiabgkGi2kaadAgadaWgaaWcbaGaaGimaaqabaaakeaacqGH ciITcaWGfbaaaaGaayjkaiaawMcaaaGaay5waiaaw2faaaWcbaGaey OeI0IaeyOhIukabaGaey4kaSIaeyOhIukaniabgUIiYdGccaWGKbGa amyraaaa@5974@   (7)

where R( E ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkfadaqada qaaiaadweaaiaawIcacaGLPaaaaaa@3B86@  is the total number of states and f 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadAgadaWgaa WcbaGaaGimaaqabaaaaa@3A2D@ is the Fermi –Dirac function. Following Tsidilkovski18 the equation (7) can be expressed by

g= π 2 3e n 0 k B 2 T n 0 E F MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadEgacqGH9a qpdaWcaaqaaiabec8aWnaaCaaaleqabaGaaGOmaaaaaOqaaiaaioda caWGLbGaamOBamaaBaaaleaacaaIWaaabeaaaaGccaWGRbWaa0baaS qaaiaadkeaaeaacaaIYaaaaOGaamivamaalaaabaGaeyOaIyRaamOB amaaBaaaleaacaaIWaaabeaaaOqaaiabgkGi2kaadweadaWgaaWcba GaamOraaqabaaaaaaa@4A9B@   (8)

Using equations (1) and (2), we get

d μ = π 2 k B 2 T 3g e 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaam izaaqaaiabeY7aTbaacqGH9aqpdaWcaaqaaiabec8aWnaaCaaaleqa baGaaGOmaaaakiaadUgadaqhaaWcbaGaamOqaaqaaiaaikdaaaGcca WGubaabaGaaG4maiaadEgacaWGLbWaaWbaaSqabeaacaaIYaaaaaaa aaa@45D0@   (9)

Therefore, by knowing the associated thermo-electric power, which is a quantifiable experimental quantity, we may experimentally estimate DMR for any degenerate semiconductors under strong magnetic quantization having arbitrary dispersion laws.

Discussion

For n-InSb, we have plotted DMR vs 1/B according to (Figure 1) the three- and two-band models of Kane, as well as the parabolic band model, using the appropriate equation. According to figure 1, all types of III-V semiconductors exhibit DMR oscillations because of the SdH effect. Figure 1 shows that DMR oscillates as well, despite the fact that the nature of the oscillation is very different from that seen in figure 1. We have displayed DMR vs. electron concentration for the same in Figure 2. Our experimental recommendation excludes any band parameter for constant temperature that changes inversely with thermoelectric power g in equation (9) used to determine DMR.19,20 For each model, the only necessary value of g is the experimental value for that model. Since g declines in an oscillatory manner with increasing doping under magnetic quantization, we may infer from equation (9) that the DMR will likewise grow in an oscillating manner, as shown by Figure 2. This claim is a veiled examination of our theoretical investigation. A method for examining the band structure of degenerate semiconductors under magnetic quantization is provided by the experimental value of g.

Figure 1 The plots of the DMR versus the inverse magnetic field ( 1/B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalyaabaGaaG ymaaqaaiaadkeaaaaaaa@39F4@ ) for n-InSb in accordance with (a) the parabolic band model, (b) the three-band model of Kane and (c) the two-band model of Kane respectively. ( n o = 10 23 m 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad6gadaWgaa WcbaGaam4BaaqabaGccqGH9aqpcaaIXaGaaGimamaaCaaaleqabaGa aGOmaiaaiodaaaGccaWGTbWaaWbaaSqabeaacqGHsislcaaIZaaaaa aa@416D@ , m * =0.0145 m o MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2gadaahaa WcbeqaaiaacQcaaaGccqGH9aqpcaaIWaGaaiOlaiaaicdacaaIXaGa aGinaiaaiwdacaWGTbWaaSbaaSqaaiaad+gaaeqaaaaa@41A9@ and α=4.167 (eV) 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiabeg7aHjabg2 da9iaaisdacaGGUaGaaGymaiaaiAdacaaI3aGaaiikaiaadwgacaWG wbGaaiykamaaCaaaleqabaGaeyOeI0IaaGymaaaaaaa@43A0@ ).

Figure 2 The plots of the DMR versus the normalized concentration ( k o MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadUgadaWgaa WcbaGaam4Baaqabaaaaa@3A6C@ ) ( k o = n o / ( 10 22 m 3 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadUgadaWgaa WcbaGaam4BaaqabaGccqGH9aqpdaWcgaqaaiaad6gadaWgaaWcbaGa am4BaaqabaaakeaacaGGOaGaaGymaiaaicdadaahaaWcbeqaaiaaik dacaaIYaaaaOGaamyBamaaCaaaleqabaGaeyOeI0IaaG4maaaakiaa cMcaaaaaaa@44FF@ , the electron concentration n o MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad6gadaWgaa WcbaGaam4Baaqabaaaaa@3A6F@ has been expressed in m 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaad2gadaahaa WcbeqaaiabgkHiTiaaiodaaaaaaa@3B25@ ) for n-InSb in accordance with (a) parabolic model, (b) three-band model of Kane and (c) two-band model of Kane ( B=2Tesla MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=PjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadkeacqGH9a qpcaaIYaGaamivaiaadwgacaWGZbGaamiBaiaadggaaaa@3F77@ ).

Acknowledgments

None.

Conflicts of interest

None.

References

  1. LL Chang, H Esaki, CA Chang, et al.Shubnikov—de Haas Oscillations in a Semiconductor Superlattice. Phys Rev Lett.1977;38:1489.

K Lee, MS Shur, JJ Drunnond. 1/f noise in modulation-doped field effect transistors. IEEE Trans. Electron Devices. 1983;30:207.

  1. H Sakaki. J Vac Sci Technol. 1981;19:198.
  2. WJ Stoopol, LD Jackel, EL Hu, et al. A. Fetter. Phys Rev Lett. 1982;49:951.

VK Arora, M Prasad. Phys Status Solidi B. 1983;117:127.
VK Arora.Quantum well wires: electrical and optical properties. J Phys C. 1985;28:3011.

  1. Y Arakawa, H Sakaki. Multidimensional quantum well laser and temperature dependence of its threshold current. Appl Phys Lett. 1982;40:939.
  2. T Tsuboi. Energy Level Discreteness of CuCl Electrons Confined to a Three-Dimensional Potential Well. Phys Status Solidi B, 1988;146.
  3. NT Linch. Festkorperprobleme. 1985;23:227.
  4. I Suemune, LA Coldren. IEEE J Quant Electron. 1988;24:1178.
  5. DAB Miller, DS Chemla, TC Damen, et al. IEEE J Quant Electron. 1985;1462.
  6. D Kasemet, CS Hong, NB Patel. Very narrow graded‐barrier single quantum well lasers grown by metalorganic chemical vapor deposition. Appl Phys Lett. 1982;41:912.

K Woodbridge, P Blood, ED Fletcher. Short wavelength (visible) GaAs quantum well lasers grown by molecular beam epitaxy. Appl Phys Lett. 1984;45:16.
S Tarucha, H Okamoto. GaAlAs buried multiquantum well lasers fabricated by diffusion‐induced disordering. Appl Phys Lett. 1984;45:1.
H Hieblum, DC Thomas, CM Knoedler. Tunneling hot‐electron transfer amplifier: A hot‐electron GaAs device with current gain. Appl Phys Lett. 1985;47:1105.

  1. O Aina, M Mattingly, FY Juan. High‐quality InAlAs grown by organometallic vapor phase epitaxy. Appl Phys Lett. 1987;50:43.
  2. DR Scifres, C Lindstrom, RD Burnham, et al. Electron. Lett. 1983;19:170.
  3. PT Landsberg. In Recombination inSemiconductor Compounds, Cambridge University Press, U.K, 1991.
  4. V K Arora. High-field distribution and mobility in semiconductors. J Phys C. 1985;18:3011–3016.
  5. S Singha Roy. Electrical Engineering. 2005;87.
  6. S Singha Roy. A simple Theoretical Analysis of Quantum Transport of 3-Dimension Photoelectric Effect. Physics of Semiconductor device. 2003;2:932.
  7. S Singha Roy. Nanosensors, Biosensors, and Info-Tech Sensors and Systems 2010. SPIE,7646,76461Q. 2010.
  8. S Singha Roy. Thesis “On some Electronic and Optical Properties of Non-Linear Optical and Optoelectronic Materials” Jadavpur University, India, 2005.
  9. IM. Tsidilkovski. Band Structure of Semiconductors, Pergamon Press, Oxford, U.K, 1982.
  10. S Singha Roy. Quantum Sensing and Nano Electronics and Photonics XIV. Proc. SPIE 10111. 2017;14:1011133.
  11. LL Chang, H Esaki, CA Chang, et al.Shubnikov—de Haas Oscillations in a Semiconductor Superlattice. Phys Rev Lett.1977;38:1489.
K Lee, MS Shur, JJ Drunnond. 1/f noise in modulation-doped field effect transistors. IEEE Trans. Electron Devices. 1983;30:207.
Creative Commons Attribution License

©2023 Roy. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work non-commercially.