THE EFFECTIVE MASS THEORY. Gokhan Ozgur Electrical Engineering SMU

Transcription

1 THE EFFECTIVE MASS THEORY Gokhan Ozgu Electical Engineeing SMU - 003

2 Intoduction In this pesentation, the effective ass theoy EMT fo the electon in the cystal lattice will be intoduced. The dynaics of the electon in fee space and in the lattice will be copaed. The E-k diaga fo diect band gap seiconductos will be studied and the hole concept will be intoduced. Next, the EMT fo single and degeneate bands will be pesented. Finally, soe application aeas of the EMT will be entioned.

3 What is the Effective Mass An electon in cystal ay behave as if it had a ass diffeent fo the fee electon ass 0. Thee ae cystals in which the effective ass of the caies is uch lage o uch salle than 0. The effective ass ay be anisotopic, and it ay even be negative. The ipotant point is that the electon in a peiodic potential is acceleated elative to the lattice in an applied electic o agnetic field as if its ass is equal to an effective ass.

4 Fee Electon Dynaics If the electon is fee then E epesents the kinetic enegy only. It is elated to the wave vecto k and oentu p by 1 k p E h 0 0 Theefoe, the quantu echanical and classical fee paticles exhibit pecisely the sae enegy-oentu elationship, as shown below. E <p>

5 Goup Velocity of a Wavepacket The velocity of the eal paticle is the phase velocity of the wave packet envelope. It is called the goup velocity and its elation to enegy and oentu is obtained fo 1 de 1 de 3 v g dp h dk ψ packet x --- ψx Reψx Hee, E and k ae intepeted as the cente values of enegy and cystal oentu, espectively. Now, what happens when an extenal foce F acts on the wavepacket? F could be any foce othe than the cystalline foce associated with the peiodic potential. The cystalline foce is aleady taken into account in the wavefunction solution. x

6 Electon Dynaics in the Lattice The wok done by the foce on the wavepacket will then be 4 de Fdx Fv Fo that we get the foce expession using 3 1 de 1 de dk 5 F v dt v dk dt g g g dt 6 F dhk dt The acceleation is found taking tie deivative of 3 7 a dv dt d dt de dk d E dk d hk dt g 1 1 h h

7 Effective Mass Expession Finally, we obtain the effective ass equation dvg 8 F dt 9 1 h 1 d E dk The equation 8 is identical to Newton s second law of otion except that the actual paticle ass is eplaced by an effective ass *.

8 Effective Mass Tenso In thee diensional cystals the electon acceleation will not be colinea. Thus, in geneal we have an effective ass tenso. dvg F xˆ xx x + yxf xyˆ + zx F xzˆ dt xx 1 yx 1 zx 1 xy 1 yy 1 zy 1 xz 1 yz 1 zz The cystal and theefoe the k-space can be aligned to the pincipal axis of the syste centeed at a band extea. Since E- k elationship is paabolic at that point, all off-diagonal tes in the tenso will vanish. Fo GaAs, as an exaple, the conduction band effective ass becoes siply a scala e * fo paabolic appoxiation.

9 Measueent of Effective Mass Effective ass is a diectly easuable quantity, which can be obtained fo cycloton esonance expeient. The test ateial is placed in a icowave esonance cavity and cooled down to 4 K. A static agnetic field B and f electic field ε oiented noal to B ae applied acoss the saple, as shown in the figue. The fequency of the obit, called cycloton fequency, is diectly popotional to B and invesely dependent on the effective ass. When B field is adjusted such that cycloton and f fequencies ae equal, then a esonance is B obseved. Then fo B-field stength, diection and f fequency, one can deduce the effective ass coesponding to the given expeient configuation. Fo diffeent B-oientations the effective asses can be easued by this way. f ε

10 E-k Diaga, Velocity and Effective Mass The figue depicts the gaphs fo E, de/dk, and d E/dk fo CB in the fist BZ. At k0, electon has a constant positive value and it ises apidly as k value inceases. Afte expeiencing a singulaity infinite ass the effective ass becoes negative up to the top of the fist BZ. Theefoe: - * is positive nea the bottos of all bands, - * is negative nea the tops of all bands. E v * -π/a k 0 π/a

11 E-k Diagas The E-k cuve is concave at the botto of the CB, so e * is positive. Wheeas, it is convex at the top of the VB, thus e * is negative. This eans that a paticle in that state will be acceleated by the field in the evese diection expected fo a negatively chaged electon. That is, it behaves as if a positive chage and ass. This is the concept of the hole. GaAs e * > 0 e * < 0 E c - E v - L <111> Γ <100> X Fo valance band the degeneate band with salle cuvatue aound k0 is called the heavy-hole band, and the one with lage cuvatue is the light-hole band.

12 Paabolic Appoxiations of Bands Thus, fo paabolic bands, the electon will ove uch like a fee paticle with *, which is elated to the cuvatue of the band. Fo nonpaabolic bands, * is not constant and the local slope and cuvatue of E k elationship ust be used to obtain the velocity and acceleation of the paticle with enegy E. The shape of the botto of the CB and the top of the VB can be appoxiated by paabolas, which esults in constant effective asses. 1a E E c + h k e E electon E c E v 1b E E v h k h E hole

13 Definition of Effective Mass, Using k p theoy, Slide no 4-5, by Jin Wang The enegy eigenvalues 13a O 13b The effective ass can be defined fo 13c 13d

14 The EMT fo a Single Band If the enegy dispesion elation fo a single band n nea k 0 assuing 0 is given by 1 E k E 0 14 n n + h kαk β α, β fo the Hailtonian H 0 with a peiodic potential V p 15 H 0 + V 16 H 0ψ nk E n k ψ nk 0 then the solution fo the Schödinge equation with a petubation U such as an ipuity o quantu-well potential 17 [ H + U ] ψ Eψ 0 is obtainable by solving the following α β

15 The EMT fo a Single Band cont. fo the envelope function F and the enegy E. The wave function is appoxiated by [ ] 0 1 β β, α α α β F E E F U x i x i n + h 18 The peiodic potential deteines the enegy bands and the effective asses, 1/* αβ, and the EM equation 18 contains only the exta petubation U, since the effective asses aleady take into account the peiodic potential. The petubation potential U can also be a quantu-potential in a seiconducto heteostuctue, such as GaAs/AlGaAs. 0 nk u F ψ 19

16 The EMT fo Degeneate Bands Following the discussion on k p ethod fo degeneate bands, like the heavy-hole, light-hole and split-off bands, the dispesion elation is given by LK α β H jj a k j E j 0δ jj + D jj kαk a k k k β j E a j j 1 j 1 α, β which satisfies the following p 1 Hψ nk E n k ψ nk H + V + H so 0 In equation 0 B α β h D + jj δ jj δα β 0 γ p α jγ p 0 β γ j E + 0 p β jγ E γ p α γ j

17 The EMT fo Degeneate Bands cont. Equation is siila to 13b, the single band case whee j j' single band index n. It is genealized to include the degeneate bands. Then, the solution ψ fo the seiconductos in the pesence of a petubation potential U fo the following 3 [ H + U ] ψ Eψ is given by 4 ψ 6 j 1 F j u jo whee F j satisfies

18 The EMT fo Degeneate Bands cont. δ 0δ 6 1 β, α β α α β j j j jj jj jj j EF F U x i x i D E Recall that the wavefunction ψ nk that satisfies 1 was k k n i n u e c \ ψ j jo j n u a u k k

19 Applications of EMT- Conductivity Unde the influence of electic field ε the acceleation of an electon in the lattice and the velocity gained by the electon in tie τ is obtained by ε e 8 a e 9 v τ N being the nube of conduction electons pe unit volue, the cuent density is found to be Ne τ 30 j Nev 31 j ε Following the Oh s law, the conductivity is calculated as Ne τ 3 σ ε

20 Applications of EMT Density of States The expessions fo the conduction and valance band densities of states nea the band edges in the seiconducto ae n n E 33a gc E 3 π h p p E 33b gv E 3 π h v E c E whee n * and p * ae the electon n and hole p density of states effective asses. As an exaple, fo GaAs the conduction band effective ass becoes siply a scala e * fo paabolic appoxiation. Theefoe, fo GaAs it will be n * e *. The density of states effective asses takes into account all band contibutions in CB and VB.

21 Effective Mass Values fo Soe Mateials Hee ae the effective ass values fo soe ateials: Effective Mass AlAs GaAs GaP InP InAs InSb e * / hh * / lh * / so * / n * / p * /