ECE 874: Physical Electronics Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University ayresv@msu.edu

Lecture 16, 23 Oct 12 VM Ayres, ECE874, F12

Effective mass: How: practical discussion: VM Ayres, ECE874, F12

Reminder: how you got the E-k curves: Kronig-Penney model allowed energy levels, Chp. 03: LHS RHS Graphical solution for number and values of energy levels E1, E2,…in eV. a = width of well, b = width of barrier, a + b = Block periodicity aBl VM Ayres, ECE874, F12

k = 0 k = ± p a + b VM Ayres, ECE874, F12

(b) VM Ayres, ECE874, F12

(b) VM Ayres, ECE874, F12

Matlab can do numerical derivatives Get: the E-k curves. Matlab can do numerical derivatives Note that the effective mass m* isn’t a single number. Note also that a + b = aBl varies depending on what direction you move in, so there are more curves than are on this single ± direction chart. VM Ayres, ECE874, F12

Which band has the sharpest curvature d2E/dk2? Get: the E-k curves. Region of biggest change of tangent = greatest curvature: the parabolas shown. Example problem: Which band has the sharpest curvature d2E/dk2? Which band has the lightest effective mass? Which band has the heaviest effective mass? Where in k-space, for both? VM Ayres, ECE874, F12

Which band has the sharpest curvature d2E/dk2? Band 4 Get: the E-k curves. Region of biggest change of tangent = greatest curvature: the parabolas shown. Example problem: Which band has the sharpest curvature d2E/dk2? Band 4 Which band has the lightest effective mass? Which band has the heaviest effective mass? Band 1: broadest = least curvature divide by smallest number = heaviest m* Where in k-space, for both? At k= 0 called the G point VM Ayres, ECE874, F12

VM Ayres, ECE874, F12

Where in k-space, for both? VM Ayres, ECE874, F12

Where in k-space, for both? m*A at G  k = 0 m*B at about ½ way between G and X in [100] direction: k = 0 VM Ayres, ECE874, F12

k = p/aBl = p/aLC at end of Zone 1 a + b = aBl aBl for [100] = aLC k = p/aBl = p/aLC at end of Zone 1 This is X for [100] VM Ayres, ECE874, F12

Where in k-space, for both? m*A at G  k = 0 m*B at about ½ way between G and X in [100] direction at k = p/2 aLC k = 0 VM Ayres, ECE874, F12

Assume T = 300K and it doesn’t change Ec = Egap = constant at a given T Hint: compare the answers for b = 0 and b ≠ 0 in (a) VM Ayres, ECE874, F12

Pick correct curve: VM Ayres, ECE874, F12

Pick conduction or valence bands:: E – Ec (eV) VM Ayres, ECE874, F12

Pick conduction minima. Where in k-space are they? E – EV (eV) <111> L G X <100> VM Ayres, ECE874, F12

Pick conduction minima. Where in k-space are they? G at k = 0 L at k = p/aBl for <111> Could work out the aBl distance between atomic cores in a <111> direction if needed. Not needed to finish answering the question. E – EV (eV) <111> L G X <100> VM Ayres, ECE874, F12

Note that the effective mass m* isn’t a single number. Go back to here: Note that the effective mass m* isn’t a single number. Note also that a + b = aBl varies depending on what direction you move in, so there are more curves than are on this single ± direction chart. VM Ayres, ECE874, F12

VM Ayres, ECE874, F12

VM Ayres, ECE874, F12

VM Ayres, ECE874, F12

VM Ayres, ECE874, F12

(From practical to fundamental!) VM Ayres, ECE874, F12

In 3 D: VM Ayres, ECE874, F12

Write this in 2D: all three parts. Integrate a -> v -> r. Vector r (t) is the direction. The final answer contains time t. VM Ayres, ECE874, F12

Then a = dv/dt for dvx/dt and dvy/dt Start with [m*ij] Then F = qE Then a = dv/dt for dvx/dt and dvy/dt Integrate with respect to time, 2x’s, to get x(t) and y(t). VM Ayres, ECE874, F12

k = 0 VM Ayres, ECE874, F12

1D: Any one of these parabolas could be modelled as: Region of biggest change of tangent = greatest curvature: the parabolas shown. 1D: Any one of these parabolas could be modelled as: VM Ayres, ECE874, F12

For any of these parabolas: Region of biggest change of tangent = greatest curvature: the parabolas shown. 3D: <111> + <100> E – EV (eV) <111> L G X <100> For any of these parabolas: There’s a major axis but also two minor ones VM Ayres, ECE874, F12

E – EV (eV) <111> L G X <100> Same: truncate 1/2 VM Ayres, ECE874, F12

k = 0 VM Ayres, ECE874, F12