Lecture 8. Band theory con.nued

Transcription

1 Lecture 8 Bnd theory con.nued

2 Recp: Solved Schrodinger qu.on for free electrons, for electrons bound in poten.l box, nd bound by proton. Discrete energy levels rouse. The Schrodinger qu.on pplied to periodic 1D lce resulted in the form.on of energy bnds nd forbidden gps. How cn this be understood from bsic physics or science perspec.ve?

3 One wy of looking t the bnds is by pushing N individul toms together. The discrete energy sttes of the toms re degenerte (sme energy). Upon squeezing the new forces brek tht degenercy nd the single energy vlue spreds into bnds of energies. Poten.ls interfere. Degenercies re broken. nergy levels spred out into bnds

4 Recp: Solved Schrodinger qu.on for free electrons, for electrons bound in poten.l box, nd bound by proton. Discrete energy levels rouse. The Schrodinger qu.on pplied to periodic 1D lce resulted in the form.on of energy bnds nd forbidden gps. The mthem.cl solu.on to llowed nd forbidden regions showed sin( q) m m cos( k) = P + cos( q) P = Vob q = 3 ( q) Decresing the sepr.on distnce, P drops from infinite to discrete vlues nd energy levels become energy bnds nd finlly overlp. m P = Vob = = n lectron m = m V b > in box P o Bnd structure like in figure Right side q(=1 nm) = m P V b = o cos( k) = cos( q) = k m lectron in box Tht s the mth. How cn this be understood from physics or science point of view?

5 For free electrons: = m P V b = o cos( k) = cos( q) = k m But there is nother set of solu.ons: m cos( k) = cos( q + n ) q = k n = = =, ± 1, ±,... k n n m So this turns into (k+3g) (k+g) (k+g) (k) -4-4 k (/) = n m ( k G) G = n =, ± 1, ±,... G

6 For free electrons: = m P V b = o cos( k) = cos( q) = k m But there is nother set of solu.ons: m cos( k) = cos( q + n ) q = k n = = =, ± 1, ±,... k n n m So this turns into (k+3g) (k+g) (k+g) (k) -4-4 k (/) = n m ( k G) G = n =, ± 1, ±,... G The region to + is the 1 st Brillouin zone. All re equivlent: need to look only t 1 st.

7 nd 1 st (k+g) Brillouin zone (k+g) (k) = n m ( k G) G = n =, ± 1, ±,... The region to + is the 1 st Brillouin zone. All re equivlent: need to look only t 1 st. So wht hppens t the cross- overs? G

8 Zimn model Recll X- rys (or electrons) sc\ering from lce (or the nrrow slit experiment) sin(θ ) n λ = n = sin( θ) k = n k Plne wves with wve vector k trvel bck nd forth through the lce. 1 ikx ikx cos( kx) ψ ± ( k) = ( e ± e ) = i sin( kx) For most k, the X- ry would pss through the lyer, but for certin k- vectors ll sc\er events t the lce sites dd together or subtrct. Interference is lrge. k = n The poten.l V(x) vries long the probbility distribu.on of ψ ± (k). Integr.ng the probbility nd V(x) cross one lce unit results on men poten.l energy of the wve (electron) 1 cos k 1 1+ cos(k) 1 1+ cos(k) ± 1 k = ± k V± x dx = V± x dx = V± x dx = V± x dx = V ± ± k k k cos( ) ψ ( ) ( ) ( ) ( ) ( ) sin 1 cos( ) 1 cos( ) cos(k) V ( x) dx = ± V pot V ± = ±V pot

9 Summing up the kine.c energy of the free electron nd the verge poten.l energy: This is for the specil cses when k nd the lce constnt line up! tot = kin + pot = k + V± m = k m ± V pot k = n (k±g) (k) (k±g) -4-4 k (/) G

10 Summing up the kine.c energy of the free electron nd the verge poten.l energy: This is for the specil cses when k nd the lce constnt line up! tot = kin + pot = k + V± m = k m k = n The energy levels no longer overlp but seprte by V pot. nergy gps open up. This occurs t the edges of the Brillouin zones. ± V pot. Allowed bnd 5 Allowed bnd 4 sepr.on from ±V pot Allowed bnd nd 1 st K (/) Brillouin zone Allowed bnd Allowed bnd 1

11 Summing up the kine.c energy of the free electron nd the verge poten.l energy: This is for the specil cses when k nd the lce constnt line up!. Allowed bnd 5 F Allowed bnd 4 Fill bnds with electrons up to the Fermi level F nd 1 st K (/) Brillouin zone Allowed bnd 3 Allowed bnd Allowed bnd 1

12 . Allowed bnd 5 F Allowed bnd 4 Fill bnds with electrons up to the Fermi level F nd 1 st K (/) Brillouin zone Allowed bnd 3 Allowed bnd Allowed bnd 1 The highest energy bnd tht is filled with electrons (no. 3 here) is the vlence bnd, the next higher one is the conduc.on bnd. The gp between these two is the Bnd- gp typiclly referred to. As shown here, electrons t the F Fermi energy cn pick up exceedingly smll mounts of ddi.onl energy nd get excited into empty slightly higher levels in the sme bnd. This is metl. In semiconductors, the vlence bnd is filled. The Fermi energy ( F ) lies in the Bnd- gp. Here, electrons hve to pick up n energy of t lest the size of the Bnd- gp to be excited into slightly higher level. This energy is comprble to therml energies (t Room temperture). (see next slide) In insultors the Bnd- gp is much lrger. Therml energies re not sufficient.

13 . Allowed bnd 5 F Allowed bnd 4 Fill bnds with electrons up to the Fermi level F (here it is shown in the Bnd gp): semiconductor! nd 1 st K (/) Brillouin zone Allowed bnd 3 Allowed bnd Allowed bnd 1 In rel mterils, the mxim nd minim of djcent bnds do not hve to line up on top of ech other. They my be shifed (sidewys) in momentum (k- spce). In the cse of semiconductors, such offset gps re clled indirect gps. Photons cn be used to excite electrons into higher energy levels nd cross gps. However, the electron will NOT chnge momentum (k- vlue). In the cse of indirect gps photon of energy gp (the bnd gp energy width) is not sufficient to trnsfer n electron into the higher bnd. The electron must pick up momentum (k) t the sme.me from seprte source. This could be phonons (het from the lce).

14 At the center (k=) the dispersion ( vs. k) hs zero slope nd cn be pproximted s prbol. The curvture is determined by h- br nd the mss of the electron. In the lce the electron is not quite free. This lters the curvture t k= slightly nd cn be interpreted s chnge in mss to n effec.ve mss m *. ffec.ve msses re importnt for the understnding of semiconductors. tot = n ± m * k The ctul defini.on is = k ffec.ve msses lrger thn the ctul mss of the free electron (m * >m e ) mke the prbol more shllow. m * K (/)