ssp_17 - 5-525 Continued5-535.3 Crystal electrons in...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 5-525 Continued5-535.3 Crystal electrons in external fieldsFree electron ))((),(tkxkietxω−⋅∝Ψfixed momentum kph=infinity extended plane waveLocation within ∆x(1 dim) ⇒momentum uncertainty h∆k[ ]∫∆+∆−−∝Ψ2/2/)()(),(kkkktkkxidkekatxω(5.3.1)wave packet ∆p∆x= h∆k∆x ≥(1/2)hGroup velocity:kv∂∂=ωPhase velocity:kkEkkc)(1)(h==ω)(kωω =dispertion ⇒broadening of packetFig. 5.30 Real space representation of the wave packet describing the motion of a spatiallylocalized free electron at times t= 0, t, 2t...(Re{ }ψ: ; ψ: ).The centerof the wave packet, i.e., in the particle picture the electron itself, moves with the groupvelocity k∂∂=/ωυ. The halfwidth of the envelope increases with time. As the wave packetspreads, the wavelenght of the oscillations of Re{ }ψbecomes smaller at the front and largerat the rear.5-54Electron in crystalBloch wavesif localized packets of Bloch waves)(1)(kEgradkgradkkh==ων(5.3.2)k: from band where electron stemms fromv: expected value of the velocity operatorsnot instantaneous velocity of an electron in V(r)Without external field or in filled band: v, -vpairwiseExternal field and partially filled band: assymetryNarrow band ⇒gradkE(k) smaller ⇒smaller v(k).Fig. 5.31 Expected or average velocityof an electron in a crystal as function ofthe wave number kat given E(k) relation(1-dim).5-55Semiclassical model:External field: treated classicallyInternal fielddue to V(r): q.m. treatmentElectron in crystal: Bloch wave packetCenter of wave packet obays classical equation of motionExternal force F⇒energy gain dE(k) =F•vdt(correspondence principle)kdkEgradkdEvk43421h)()(=E(k) from q.m.⇒Fdtkd=heq. of motion of a crystal electron (5.3.3)Time-dependent Schrödinger equation ⇒eq. (5.3.3) valid forcondition in Fig.5.32.Fig. 5.32 A rigoroustreatment by means ofthe time-dependentSchrödinger eq. showsthat eq. (5.3.3) holds forwave packets of Blochstates if the externalelectromagnetic field isnot too large comparedwith the atomic fieldsand is slowly varying onthe atomic length andtime scale.5-565.3.1 Effective mass and holes[ ]∑∂∂∂∂∂=∂∂==jijiiikitkkkEkEdtdkEgraddtddtdv211)(1hhh∑∂∂∂=jjjiFkkE221hsince Fdtkd=hClassical equation of motion Fmdtvd1=analogously Fmdtvd=*1withjiijkkEm∂∂∂=221*1h(5.3.4)↑curvature ofE(k)m*ij = m*ji⇒can be transformed to principle axis:...
View Full Document

Page1 / 15

ssp_17 - 5-525 Continued5-535.3 Crystal electrons in...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online