DoS of bulk and quantum structures
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The document discusses the density of states (DoS) for bulk semiconductors and various quantum structures such as quantum wells, wires, and dots. It defines DoS as the number of available energy states per unit energy interval per unit dimension. It then derives expressions for the DoS of bulk semiconductors, quantum wells, quantum wires, and notes that quantum dots have a discrete DoS with delta function peaks.
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G24041050
This document discusses the optimization and simulation of Josephson junctions as switching elements. It begins with an introduction stating the need to optimize Josephson junction parameters and characteristics to increase switching speed. It then provides a brief review of the Josephson effect and properties of Josephson junctions. The rest of the document details a theoretical approach to estimating junction parameters, characteristics, and optimization methods, and using computer simulation to model the dynamic response of Josephson junctions to better understand their switching performance and speed.
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Semiconductor nanodevices
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Quantum Electronics Lecture 7
This document is a lecture on the quantum Hall effect. It discusses how the resistance of a quantum well system becomes quantized under low temperature and high magnetic field conditions, known as the quantum Hall effect. It also provides calculations of the quantum Hall resistance, showing that it is constant and equal to h/q^2, where h is Planck's constant and q is the elementary charge. The significance is that the resistance remains constant as long as the Fermi level lies between localized states in the Landau bands formed under the magnetic field.
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This document discusses semiconductor materials and devices. It covers the classification of materials based on their band structure, including direct and indirect bandgap semiconductors. It also discusses the classification of semiconductors as n-type or p-type based on the position of the Fermi level. Finally, it derives expressions for the effective mass of electrons and holes in semiconductors.
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The document summarizes a lecture on the quantum Hall effect. It defines the quantum Hall effect as a phenomenon where the resistance of a quantum well system is quantized under low temperature and high magnetic field conditions. It then provides calculations to show that the quantum Hall resistance is quantized and equal to h/q^2, where h is Planck's constant and q is the elementary charge. Finally, it discusses how the quantization occurs due to the formation of discrete energy levels called Landau levels in the presence of a magnetic field.
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#Prerequisites:
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DoS of bulk and quantum structures
- 1. Course: Quantum Electronics Arpan Deyasi Topic: Density of states of Bulk and Quantum Structures 1 Arpan Deyasi, RCCIIT 1/17/2021
- 2. 1/17/2021 2Arpan Deyasi, RCCIIT What is DoS? Number of available energy states per unit energy interval per unit dimension in real space EC EC + dEC EV EV + dEV dEV dEC E k
- 3. 1/17/2021 Arpan Deyasi, RCCIIT 3 What do we mean by ‘dimension’? For ‘bulk’, it is ‘volume’ For ‘quantum well’, it is ‘area/surface’ For ‘quantum wire’, it is ‘line/length’ For ‘quantum dot’, it is a ‘point/dot’
- 4. 1/17/2021 4Arpan Deyasi, RCCIIT Energy band diagram is drawn in E-k plane ‘k’ is wave-vector, not a physical quantity No of electrons is measured by magnitude of current So we must know the density of electrons in real space instead of k-space
- 5. 1/17/2021 Arpan Deyasi, RCCIIT 5 Fermi sphere
- 6. 1/17/2021 Arpan Deyasi, RCCIIT 6 Fermi surface
- 7. 1/17/2021 7Arpan Deyasi, RCCIIT DoS for bulk semiconductor Let’s start with Bloch theorem ( , , ) ( , , )x y zx y z x L y L z Lψ ψ= + + + Consider a 3D semiconductor with dimensions Lx, Ly, Lz
- 8. 1/17/2021 8Arpan Deyasi, RCCIIT DoS for bulk semiconductor For validity of wave function 2 2 2 x x x y y y z z z k L n k L n k L n π π π = = = Volume in k-space 3 (2 ) x y z x y z x y z n n n k k k L L L π =
- 9. 1/17/2021 9Arpan Deyasi, RCCIIT DoS for bulk semiconductor Let 1x y zn n n= = = x y zL L L L= = = Volume of unit cell in k-space 3 3 (2 ) kV L π =
- 10. 1/17/2021 10Arpan Deyasi, RCCIIT DoS for bulk semiconductor Volume of Fermi sphere in k-space 34 3 FV kπ= Volume of semiconductor in real space 3 RV L=
- 11. 1/17/2021 11Arpan Deyasi, RCCIIT DoS for bulk semiconductor number of energy states in real space 1 1 F k R N V V V = × × 3 3 3 3 4 1 3 8 L N k L π π = × × 3 2 ( ) 6 k N N k π = =
- 12. 1/17/2021 12Arpan Deyasi, RCCIIT DoS for bulk semiconductor Introducing Pauli’s exclusion principle 3 3 2 2 ( ) 2 6 3 k k N k π π =× = 2 2 ( ) k N k k π ∂ = ∂
- 13. 1/17/2021 13Arpan Deyasi, RCCIIT DoS for bulk semiconductor From parabolic dispersion relation 2 2 * 2 k E m = 2 * E k k m ∂ = ∂ 2 * * 2 * 2E m E E k m m ∂ = = ∂
- 14. 1/17/2021 14Arpan Deyasi, RCCIIT DoS for bulk semiconductor * 1 2 k m E E ∂ = ∂ N N k E k E ∂ ∂ ∂ = × ∂ ∂ ∂ 2 * 2 1 2 N k m E Eπ ∂ = × ∂
- 15. 1/17/2021 15Arpan Deyasi, RCCIIT DoS for bulk semiconductor * 2 2 2 * 1 2 N m E m E Eπ ∂ = × ∂ 3/2* 2 2 1 2 2 N m E E π ∂ = × ∂ For a particular material N E E ∂ ∝ ∂ E ρ(E)
- 16. 1/17/2021 16Arpan Deyasi, RCCIIT DoS for Quantum Well Area in k-space 2 2 (2 ) kA L π = Area of Fermi sphere in k-space Area of semiconductor in real space 2 FA kπ= 2 RA L=
- 17. 1/17/2021 17Arpan Deyasi, RCCIIT DoS for Quantum Well number of energy states in real space 1 1 F k R N A A A = × × 2 2 2 2 1 4 L N k L π π = × × 2 ( ) 4 k N N k π = =
- 18. 1/17/2021 18Arpan Deyasi, RCCIIT DoS for Quantum Well Introducing Pauli’s exclusion principle 2 2 ( ) 2 4 2 k k N k π π =× = ( ) k N k k π ∂ = ∂
- 19. 1/17/2021 19Arpan Deyasi, RCCIIT DoS for Quantum Well From parabolic dispersion relation * 1 2 k m E E ∂ = ∂ N N k E k E ∂ ∂ ∂ = × ∂ ∂ ∂ * 1 2 N k m E Eπ ∂ = × ∂
- 20. 1/17/2021 20Arpan Deyasi, RCCIIT DoS for Quantum Well * * 2 1 2 N m E m E Eπ ∂ = × ∂ * 2 N m E π ∂ = ∂ DoS is independent of energy?
- 21. 1/17/2021 21Arpan Deyasi, RCCIIT DoS for Quantum Well The result is obtained for a particular sub-band Considering all the sub-bands * 2 1 ( ) n i i N m E E E π = ∂ = Θ − ∂ ∑ E ρ(E) E1 E2 E3 Ei-1 Ei Ei+1
- 22. 1/17/2021 22Arpan Deyasi, RCCIIT DoS for Quantum Wire Length in k-space Length of Fermi sphere in k-space Length of semiconductor in real space 2 kL L π = 2FL k= RL L=
- 23. 1/17/2021 23Arpan Deyasi, RCCIIT DoS for Quantum Wire number of energy states in real space 1 1 F k R N L L L = × × 1 2 2 L N k Lπ = × × k N π =
- 24. 1/17/2021 24Arpan Deyasi, RCCIIT DoS for Quantum Wire Introducing Pauli’s exclusion principle 2 k N π = 2 ( )N k k π ∂ = ∂
- 25. 1/17/2021 25Arpan Deyasi, RCCIIT DoS for Quantum Wire From parabolic dispersion relation * 1 2 k m E E ∂ = ∂ N N k E k E ∂ ∂ ∂ = × ∂ ∂ ∂ * 2 1 2 N m E Eπ ∂ = × ∂
- 26. 1/17/2021 26Arpan Deyasi, RCCIIT DoS for Quantum Wire 2 * 1N m E Eπ ∂ = ∂ E ρ(E) E1 E2 E3 E4 E5
- 27. 1/17/2021 27Arpan Deyasi, RCCIIT DoS for Quantum Dot E4E3E2E1 ρ(E) E E ρ(E) E1 E2 E3 E4