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ECE216-Lecture-03-Semiconductor-Physics

ECE216-Lecture-03-Semiconductor-Physics - ECE 216 DEVICE...

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ECE 216 DEVICE PHYSICS FOR INTEGRATED CIRCUITS Lecture 03 Chapter 2 – SEMICONDUCTOR PHYSICS Professor Hisham Z. Massoud Department of Electrical and Computer Engineering Fitzpatrick Building, Room 3521 Duke University, Durham, NC 27708–0291 [email protected] https://courses.duke.edu/webapps/portal/frameset.jsp ECE 216 Chapter 2 – Semiconductor Physics Lecture 03.1

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TOPICS 4. Electrons in a Crystal (continued) 4.5. Wave Packets 4.5. The Electron and Hole Effective Masses 5. Density of Electron Energy States 6. The Effective Density of States 6.1. The Fermi-Dirac Distribution Function f FD ( E ) 6.2. The Electron Concentration 6.3. The Hole Concentration ECE 216 Chapter 2 – Semiconductor Physics Lecture 03.2
4. ELECTRONS IN A CRYSTAL 4.5. Wave Packets The wave-vector ~ k is restricted in real crystals by boundary conditions to discrete values. Each discrete value of ~ k has a corresponding wave-function solution Ψ k . The solution Ψ of the Schr¨ odinger equation (SE) is the superpo- sition of all wave-functions Ψ k . If each wave-function in the one-dimensional problem (i.e. ~ k is k x ) is the form A k exp i ( ω t - k x x ) , then the solution of SE is in the form Ψ ( x, t ) = X k A k exp i ( ω t - k x x ) . This sum is known as a wave packet where ω is the de-Broglie-wave angular frequency. ECE 216 Chapter 2 – Semiconductor Physics Lecture 03.3

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4. ELECTRONS IN A CRYSTAL Each wave travels with a velocity v k given by ~v k = ¯ h m ~ k , and during a time Δ t , each wave travels a distance ~ r = ~v k Δ t = (¯ h ~ k/m ) Δ t . Each wave in a wave packet has a different value of ~ k and thus travels a distance that is, in general, different from that traveled by other waves in the packet. ECE 216 Chapter 2 – Semiconductor Physics Lecture 03.4
4. ELECTRONS IN A CRYSTAL When an electron is left alone, it remains in a state of uniform motion in a straight line in real space, with velocity ~v g . The electron is represented then by the same location in ~ k -space, independent of time unless : 1. The electron is scattered, thus changing its energy and wave-vector (i.e. disappearing from one point in ~ k -space and appearing at an- other). 2. The electron experiences an electric field. Then the electron energy changes. The progression of the electron through ~ k -space and real space is determined by the rate of change of energy. 3. The electron experiences a magnetic field. Such a field alone requires that the direction of motion changes without affecting the electron energy. Thus the electron moves over a constant-energy surface in ~ k -space in response to a magnetic field alone. ECE 216 Chapter 2 – Semiconductor Physics Lecture 03.5

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4. ELECTRONS IN A CRYSTAL The wave packet travels with a velocity known as the group velocity which is the velocity of an electron in a real crystal. It is given by ~v g = d ω d ~ k = 1 ¯ h dE d ~ k = 1 ¯ h -→ k E , which indicates that the electron is moving in a direction perpendicular to a constant-energy contour.
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ECE216-Lecture-03-Semiconductor-Physics - ECE 216 DEVICE...

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