Reflectance spectra for Si

Transcription

1 Refletane speta fo Si Notie R and ε i and ε show onsideable stutues in the fom of peas and shouldes. These stutues aise fom the optial tansitions between alene bands to the ondution bands. 16

2 Miosopi Theoy: semilassial appoah We will use a semi-lassial appoah to deie the Hamiltonian desibing the inteation between an extenal eletomagneti field and Bloh eletons inside the semionduto. In this appoah the eletomagneti field is teated lassially while eletons ae desibed by quantum mehanial Bloh waes. This appoah is not igoous as the fully quantum mehanial teatments in whih the eletomagneti waes ae quantized into photons but has the adantage of being simple and easie to undestand and geneates the same esults. We stat with the unpetubed one-eleton Hamiltonian: p H 0 + V. m To desibe the eletomagneti field we intodue a eto potential A,t and a sala potential Φ,t. Beause of gauge inaiane, the hoie of these potentials is not unique. Fo sempliity, we will hoose the Coulomb gauge, in whih Φ 0 and A 0 In this gauge the eleti and magneti fields E,B ae gien by 1 A E and B A t 17

3 Semilassial Appoah The lassial Hamiltonian of a hage Q in the pesene of an extenal magneti field an be obtained fom the fee-patile Hamiltonian by eplaing the momentum P by P-QA/. Coespondingly, we obtain the quantum mehanial Hamiltonian desibing the motion of an eleton -e in an extenal eletomagneti field by eplaing the eleton momentum opeato p in p+ea/: [ p + ea / ] H + V. m The tem [p+ea] /m an be expanded eeping in mind that p is an opeato whih does not ommutate with A as 1 ea p e e e A p + A p + p A + m + m m m m Using the definition of p as the opeato h / i 18

4 Eleton-adiation adiation Inteation. h h p A f A f + A f i i Fom the Coulomb gauge we hae that A 0 and theefoe [e/m]pa [e/m] Ap. Fo the pupose of alulating linea optial popeties we an also neglet the tem whih depends quadatially on the field. Unde this assumption we an appoximate H by H H 0 + e m A p Compaed with the unpetubed Hamiltonian H 0 the exta tem desibes the inteation between the adiation and a Bloh eleton. As a esult, this tem will be efeed to as the eleton-adiation inteation Hamiltonian H er : H er e m A p Note that the fom of H er depends on the hoosen gauge. 19

5 Matix Element Thee ae seeal ways to alulate the dieleti funtion of a semionduto fom H er. We ae onside the simplest appoah. We fist assume that A is wea enough that we an apply time-dependent petubation theoy in the fom of the Femi Golden Rule to alulate the tansition pobability R pe unit olume fo an eleton in the alene band state, with enegy E and waeeto to the ondution band, with oesponding enegy E and waeeto. To do this we need to ealuate the matix element: H e / m A p er. We will now wite the eto potential A as Aê, whee ê is a unit eto paallel to A. In tems of the amplitude of the inident eleti field Eq, the amplitude of A an be witten as E A { exp[ i q t ] +.. } q The integation oe time of the tem [i-t] and the oesponding fatos in the eleton Bloh funtions leads fomally to exp ie t / hexp[ i t]exp ie t / h dt δ E E h This esult means that the eleton in the alene bands absobs the photon enegy and is then exited into the ondution band. 0

6 Matix Element Similaly, the matix element of the omplex onjugate expit gies ise to δ E E + h This means that the poess desibes the tansition fom eleton fom the alene band to the ondution band by an absoption - o emission + of photons. Witing the Blo funtions fo the eletons in the ondution and alene bands, as: [ u, exp[ i ] and u exp[ i [ ], and using the espession fo A we obtain E [ ] * A p, exp, exp 4 u i q e p u i d q Opeating with p on u, expi yields two tems: pu exp i exp i pu, + hu, exp i, The integal of the seond tem multiplied by u*, anishes beause u, and u, ae othogonals.. 1

7 Matix Element We an split the oesponding integal of the fist tem [ + ] * u, exp i q pu, d into two pats by witing R j +, whee lies within one unit ell and R j is a lattie eto. Beause of the peiodiity of the u, and u,, we find [ ] exp exp[ '] * i q R + j u, i q + pu, d' j unit ell The summation of exp[iq- + R j ] oe all the lattie etos R j esults in a delta funtion δq- +. This tem ensues that waeeto is onseed in the absoption poess q+. This is a onsequene of the tanslation symmety of a pefet ystal. Using the waeeto onseation the integal oe the unit ell simplifies to [ ] * * u, exp i q + ' pu, d' u, q pu, d' + unit unit ell ell This expession an be futhe simplified if we assume that q q0 is muh smalle than the size of the Billouin zone, a ondition usually satisfied by isible photons, whose waelengths ae of the ode of 500nm.

8 Eleti dipole appoximation Fo small q the waefuntion u +q an be expanded into a Taylo seies in q: u u + q u..., + q,, + When q is small enough that all the q-dependent tems an be negleted, the matix element is gien by * e p u, e p u, d' unit ell This appoximation is now as the eleti dipole appoximation. Notie that the eleti dipole appoximation is equialent to expanding the tem expiq into a Taylo seies: 1+iq +... and neteting all the q-dependent tems. In this ase we hae that and the tansitions ae said to be etial o diet. If the eleti dipole matix element is zeo, the optial tansition is detemined by the q tem in the top equation and the matix element is * e p q u, + q e p u, d' unit ell gies ise to eleti quadupole and magneti dipole tansitions. 3

9 Eleti Dipole Tansition Pobability Fom now we shall estit ouseles to eleti dipole tansitions and theefoe and the momentum matix element is not stongly dependent on so we shall eplae it by the onstant P. The equation an by semplified H e / m A p e / m A whee A ontains the tem exp-it absoption poess and his omplex onjugate emission not onside in the following disussion. The eleti dipole tansition pobability R fo photon absoption pe unit time is: R π / h H δ E E h is thus gien by. er P, er e E R π / h P δ E E h m, The powe lost by the field due to absoption in unit olume of the medium is simply the tansition pobability pe unit olume multiply by the enegy in eah photon: Poweloss Rh 4

10 5 Dieleti Funtions Dieleti Funtions This powe lost by the field an also be expessed in tems of α and ε i of the medium by noting that the ate of deease in the enegy of the inident beam pe unit olume is gien by di/dt, whee I is the intensity of the inident beam: we obtain By using the KKR we an obtain the espession fo the eal pat: whee ε α h R n I I n dt dx dx di dt di i δ π ε h E E P m e i + P m m e 4 1 π ε h E E h n n i i ε κ α κ ε

11 Joint Density of States The dispesion in ε i omes fom the summation oe the delta funtion of the enegy onseation. The matix elements P between a gien ouple of alene and ondution bands ae shown to be not stongly dependent fom, exept nea speial etos whee P anishes beause of symmety. Neleting suh situation and taing P as a onstant, we find that the ontibution to the dieleti funtion fom a pai of bands is popotional to 1/ and to the quantity: 1 D j dδ E E h 3 4π BZ whih is alled joint density of states beause it gies the density of pai states one oupied and the othe one empty, sepaated by an enegy. The integation an be pefomed by using the popeties of the δ funtion. We now that b a g x δ [ f x ] dx x0 g x o df dx 1 x x0 in whih x 0 epesents a zeo of the funtion fx ontained in the inteal a,b. In 3D we hae 1 ds D whee E is the abbeiation fo E -E, and j 3 4π E S is the onstant enegy sufae defined by E 6 onst.

12 Van Hoe singulaities The joint of density fo inteband tansitions as a funtion of E shows stong aiations in the neighbouhood of paiula alues of E whih ae alled itial point enegies. Fom the espession of D j we see that singulaities in the joint density of states ae expeted when E E 0 o moe geneally when E E 0 This point ae now as itial points and ou in geneal at high simmety points of the Billouin zone and the oesponding singulaities in the density of states ae nown as Van Hoe singulaities. The analyti behaiou of Dj nea a singulaity may be found by expanding E -E in a Taylo seies about the ytial point. In the expansion the linea tems ae zeo beause of the aboe ondition. Limiting the expansion to quadati tems and denoting the wae etos along the pinipal axes with the oigin at the ytial point by x, y, z, E h y + x E Eo ε + + x ε y ε mx m y with m x,m y,m z ae positie quantities and ε x, ε y, ε z equal to +1 and -1. z m z z 7

13 Citial Points We obtain fou type of singulaities, depending on the signs of ε x, ε y, ε z. The itial points ae alled: h x y z E E Eo + ε x + ε y + ε z mx m y mz M 0 when all oeffiients of the quadati expansion ae positie minimum; M 1 when two oeffiients of the quadati expansion ae positie and one negatie saddle point; M when two oeffiients of the quadati expansion ae negatie and one positie saddle point; M 3 when all oeffients of the quadati expansion ae negatie maximum; whee the subsipts attahed to M indiate the numbe of negatie oeffiients in the expansion of enegy diffeenes. The analyti behaiou of the joint density of states nea itial points an be obtain using the espession of Dj and the enegy expansion. We an notie that thee ae shap disontinuities at the itial points. 8