# A1_507A - Phys 507A Solid State Physics I Assignment 1...

This preview shows pages 1–3. Sign up to view the full content.

Phys 507A - Solid State Physics I Assignment 1: Electrons in crystals. Due Oct. 2nd 1. Periodic potential in 1d. Consider an electron subject to a 1d periodic potential, U ( x ) = X n = -∞ v ( x - na ) , (1) where v ( x - na ) represents the potential barrier against an electron tunnelling between the ions on opposite sides of the point na . For simplicity we assume that v ( x ) = v ( - x ) (inversion symmetry), and v ( x ) = 0 for | x | ≥ a/ 2. But other than this the potential U ( x ) is quite general. Remarkably, the band structure of the 1d solid can be expressed quite simply in terms of the transmittance and reflectance of an electron hitting the barrier v ( x ). Denote the energy of the incident electron to be = ¯ h 2 K 2 / 2 m . The variable K = 2 m / ¯ h parametrizes the energy of the electron. Consider an electron incident from the left on the potential barrier v ( x ); since v ( x ) = 0 for x a/ 2, in these regions the wave function ψ l ( x ) will have the form ψ l ( x ) = e iKx + r e - iKx , x ≤ - a 2 = t e iKx , x a 2 . (2) Here t and r are the transmission and reflection coefficients, respectively. Their actual dependence on K is given by the form of the barrier v ( x ). However, one can deduce the properties of the band structure of the periodic potential U by appealing only to very general properties of t and r . Because v ( x ) is even, ψ r ( x ) = ψ l ( - x ) is also a solution to the Schr¨ oedinger equation with energy . From Eq. (2) it follows that ψ r ( x ) = t e - iKx , x ≤ - a 2 , = e - iKx + r e iKx , x a 2 . (3) Since ψ l and ψ r are two independent solutions to the single-barrier Schr¨ oedinger equa- tion with the same energy, any other solution with that energy will be a linear combi- nation of these two; in addition, since the crystal potential is identical to v ( x ) in the region | x | ≤ a/ 2, any solution to the crystal Schr¨oedinger equation with energy in the | x | ≤ a/ 2 region must be given by ψ ( x ) = l ( x ) + r ( x ) . (4)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Now Bloch’s theorem asserts that ψ can be chosen to satisfy ψ ( x + a ) = e ika ψ ( x ) , (5) for a suitable k (different than K !). Differentiating the above equation we also find that ψ 0 ( x + a ) = e ika ψ 0 ( x ) . (6) (a) By imposing the conditions (5) and (6) at x = - a/ 2, and using Eqs. (2)–(4), show that the energy of the Bloch electron is related to its wavevector k by: cos ( ka ) = t 2 - r 2 2 t e iKa + 1 2 t e - iKa , = ¯ h 2 K 2 2 m . (7) Verify that this gives the right answer in the free electron case ( v = 0). We write the complex number t in terms of its magnitude and phase: t = | t | e . (8) The real number δ is known as the phase shift, since it represents the change in phase of the transmitted wave relative to the incident one. Electron conservation requires that the probability of transmission plus the probability of reflection be unity, | t | 2 + | r | 2 = 1 . (9) This, and some other useful information, can be proved as follows. Let φ 1 and φ 2 be
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern