ssp_22 - 6 Continued 6-28 6.3 Experimental methods Crystal...

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Unformatted text preview: 6 Continued 6-28 6.3 Experimental methods Crystal electron in a magnetic field Fig. 6.19 Orbit of an electron in a solid ender the influence of an applied magnetic field B. The orbit is perpendicular to B and confined to surfaces of constant energy in k space (see eq. (6.3.1)). On closed orbits the period is proportional to the derivative of the cross-sectional area. A with respect to E (see eq. (6.3.2)). In metals electrons move on the Fermi surface. In semiconductors they move at surfaces of constant energy near the minimum of the conduction band. Lorentz force: v= F = -e(v x B) 1 grad k E h & F = hk ⇒ dk e = 2 B × grad k E (k ) dt h Orbit of electron: ⊥B ⊥ gradkE, i.e. on surface E = const. 6-29 (6.3.1) Fig. 6.20 Possible orbits of electrons in a solid in a magnetic field. For electronlike orbit the direction of motion is as for a free electron (b). For holelike orbits the filled electron states lie outside the Fermi surface (cf. Fig.5.23b)Therefore the direction of gradkE is such that the movement is opossite to that of a free electron, i.e. in the direction of a positive particle (c). In addition, there are also open orbits (a). Period on closed loop: ( 6 . 3 . 1) ⇒ T = ∫ dt dk e = 2 B ( grad k E ) ⊥ dt h 1 42 43 = dE / dk ⊥ ωc = 2 T = eB h 2π 2πeB =2 T h ( dA / dE ) ωc: cyclotron frequency 6-30 ∫ h 2 dk ⋅ dk ⊥ dA = dE eB dE (6.3.2) Cyclotron frequency ωc for free electrons h2k 2 Free electrton: E = 2m , v= k h m 1 grad k E = k ⇒ ⊥ = h m v⊥ h Fermi surface: sphere 1 dE = v⊥ h dk ⊥ ⇒ ωc = ⇒ e B m ∫ dk dk ⊥ 1 = dE v⊥h ∫ dk = 1 2π k ⊥ 2π m = h v⊥ h2 E.g.: B = 1T ⇒ T = 2π/ωc = 3.6 x 10-11 s (6.3.3) Lorentz force 144244 = centrifuga l444 3 144 2 force 4 3 Elementary derivation: − evB − mω 2 v ↑ =ωr Fig. 6.21 Elementary derivation of the cyclotron frequency as a balance of Lorentz and centrifugal force. Real space: r: radius of the spherical or spiral (for v|| ≠ 0) orbit. Electron in solid: ωc = e B mc eq. (6.3.2) ⇒ h 2 dA mc = 2π dE 6-31 cyclotron mass (6.3.4) Energy of electrons in magnetic field (Landau levels) Motion on spherical orbit: composed of two harmonic oscillations. ⇒ discrete energy of the harmonic oscillator in plane ⊥ B. (Landau levels of the free electron gas) hh 14243 123 2 1 E = const. + n + ωc + k z2 . 2 2m (6.35) harmonic oscillator translational in variance in z −direction (Spin degeneration not accounted for) Allowed states in k-space ″condence″ onto tubes (″Landau tubes″) Fig. 6.22 Change of the density D(E) of state of a free electron gas in a magnetic field. According to eq. (6.3.5) D(E) is composed of discreate energy levels at (n + 1/ 2)hωc , i.e. δ-functions, and the density of states of a 1dimensional electron gas perpendicular to the direction of the applied field. In 1-dimension D(E)∝ 1 / E (see practical course). Since ωc = ωc(B) D(EF) oscillates with increasing magnetic field. ωc = ωc(B) ⇒ Oscillations of D(E) with B. 6-32 Fig 6.23 (a) The dots indicate the allowed states of a free electron gas in k-space. According to eq. (6.3.5). The electrons ″condence″ on tubes (cross-sections shown as dashed lines) in the presence of an applied magnetic field. (b) The radius of the tubes increases with increasing quantum number n. (c) Three-dimentional sketch of the ″Landau tubes″. within the Fermi spheres. Even in extremely strong fields the number of tubes within the Fermi surface is much larger than shown (c) since ωc = 1.1 x 10-4 eV for B = 1T and EF is of the order of 5eV. (d)The Landau tubes are only parallel to B if B is parallel to an axis of main symetry but not for the general case shown here. B = 1T ⇒ ω c = 1.1 × 10−4 eV ⇒ 300 K : kT = 0.025eV ⇒ electrons can leave Landau tubes. 6-33 De Haas - van Alphen effect Periodic oscillations of the magnetic susceptibility χmag as a function of 1/B χmag ∝ D(EF) 1 2πe 1 ∆ = h AF B (see Chpt. 7) (see practical course) ⇒ Cross sections of Fermi surface. Condition: kT < hω c , i.e. low temperature. 6-34 Cyclotron resonance in semiconductors Fig. 6.24 Classical description of cyclotron resonance. The electrons move on surfaces of content energy in a static B-field, e.g., on the ellipsoidal surfaces, near the conduction band minimum (see Fig.6.4). The electrons absorbs energy from an applied alternating high frequency field (additional force by @Ã field) if ωHF = ωcotherwise acceleration and decceleration compensate on average. Resonance absorbtion in HF alternating field if: ω HF = ω c = e B ⇒ mc mc * * m c ⇒ m nt , m nl , m * p Requirement: ωτ >> 1 (many periods between collisions) (see practical course) ⇒ strong B-field, low temperature (conduction electrons generated by light), pure crystals. Clear adsorbtion maximum from ″external orbits″: number of states per frequency interval particularly high. Resonance also for holes: opposite direction distinguished by circularly polarized HF-field. 6-35 of orbits ...
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This note was uploaded on 01/19/2010 for the course MATERIALS M504 taught by Professor Adelung during the Spring '02 term at Uni Kiel.

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