The BTE with a High B-field

Transcription

1 ECE 656: Electronic Transport in Semiconductors Fall 2017 The BTE with a High B-field Mark Lundstrom Electrical and Computer Engineering Purdue University West Lafayette, IN USA 10/11/17

2 Outline 1) Introduction 2) Landau levels 3) Shubnikov-deHaas oscillations 4) Quantum Hall Effect 2

3 High magnetic fields ω c = qb m * ω c τ m >> 1 µ n B z >> 1 Interesting things happen when the B-field is large. D. F. Holcomb, Quantum electrical transport in samples of limited dimensions, American Journal of Physics, 67, pp , April

4 Reference Longitudinal magnetoresistance quantized Hall voltage zero longitudinal resistance, R xx V x Shubnikov-deHaas (SdH) oscillations V H 4 M.E. Cage, R.F. Dziuba, and B.F. Field, A Test of the Quantum Hall Effect as a Resistance Standard, IEEE Trans. Instrumentation and Measurement, Vol. IM-34, pp , 1985

5 Some numbers silicon µ n = 1000 cm 2 V-s B z = 2,000 Gauss B z = 0.2 Tesla µ n B z 0.02 << 1 Hall effect measurements with typical laboratory-sized magnets are in the low B-field regime. Except for very high mobility sample such as modulation doped films.) Birck Nanotechnology Center: National High Magnetic Field Lab (Florida State Univ.): 1-8 T 45 T 5

6 Large magnetic fields InAlAs/InGaAs T L = 300K InAlAs/InGaAs T L = 77K µ n 10,000 cm 2 V-s µ n 100,000 cm 2 V-s B z = 2,000 Gauss B z = 2,000 Gauss B z = 0.2 Tesla µ n B z 0.2 < 1 B z = 0.2 Tesla µ n B z 2 > 1 6 Shubnikov-deHaas (SdH) oscillations and quantum Hall effect

7 Modulation doping E F E C E C E F E V wide band gap doped n + small band gap nominally undoped E V R. Dingle, et al, Appl. Phys. Lett., 33, 665,

8 Outline 1) Introduction 2) Landau levels 3) Shubnikov-DeHaas oscillations 4) Quantum Hall Effect 8

9 Schrödinger equation with a B-field ε 1 + i! + q" A 2m * B = A ( ) 2 ( ) +U y Ψ x, y ( ) = EΨ( x, y) See S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge, 1995, pp

10 Cyclotron frequency B = B z ẑ y d( k ) dt = q υ B υ B = B z ẑ k y θ k x k x dk x dt dk y dt = qυ y B z = k d cosθ dt = +qυ x B z = k d sinθ dt d 2 cosθ = qυb z dt 2 k cosθ ( t) = cosθ ( 0)e iω ct = qυ sinθb z = +qυ cosθb z 2 cosθ = ω 2 c cosθ 10

11 B = B z ẑ y Cyclotron frequency d( k ) dt = q υ B υ B = B z ẑ k y θ k x k x harmonic oscillator: ω c = qυb z k Quantum mechanically: E n = n ω c Landau levels 11

12 B = B z ẑ y Cyclotron frequency E n = n ω c ω c = qυb z k i) parabolic energy bands: υ x υ = k m * ii) graphene: ω c = q B z m * E = υ F k ω c = q B z E υ F 2 ( ) 12

13 Effect on DOS D 2 D m * π 2 ( E) E n = n ω c ω c = q B z m * ε 1 E 0 E 1 E 2 E D 2 D ( E, B ) = D δ E ε n + 1 z ω c n=0 13

14 Degeneracy of Landau levels D 2 D m * π 2 ( E) D 0 ω c E n = n ω c ω c = q B z m * ε 1 E 0 E 1 E 2 E D 0 = ω c m* π = 2qB z 2 h 14

15 Broadening D 2 D ( E) ΔE D 0 ω c E n = n ω c ω c = q B z m * m * π 2 D 0 = 2qB z h ε 1 E 0 E 1 E 2 E ΔEΔt = ΔE τ to observe Landau levels: ω c >> ΔE ω c τ >> 1 15

16 Example If B = 1T, how many states are there in each LL? D 0 = 2qB z h = cm -2 If n S = 5 x cm -2, then 10.4 LL s are occupied. How high would the mobility need to be to observe these LL s? µ B > 1 µ > 10,000 cm 2 /V-s ( B = 1 T) modulation-doped semiconductors 16

17 Outline 1) Introduction 2) Landau levels 3) Shubnikov-deHaas oscillations 4) Quantum Hall Effect 17

18 SdH oscillations Longitudinal magnetoresistance V x Shubnikov-deHaas (SdH) oscillations V H M.E. Cage, R.F. Dziuba, and B.F. Field, A Test of the Quantum Hall Effect as a Resistance Standard, IEEE Trans. Instrumentation and Measurement, Vol. IM-34, pp ,

19 LL filling D 2 D ( E) ω c ν = n S 2qB h E n = n ω c ω c = q B z m * m * π 2 D 0 = 2qB z h ε 1 E 0 E 1 E 2 E 4 E F E For a given sheet carrier density, n s, some (fractional) number of LL s are occupied. 19

20 SdH oscillations As the B-field is varied, the longitudinal resistance will oscillate as the number of filled LL s changes. The period of the oscillation corresponds to the change in filling factor from one integer to the next. ν 1 ν 2 = 1 n S 2qB 1 h n S 2qB 2 h = 1 n S = 2q h 1 1 B 1 1 B 2 20

21 Outline 1) Introduction 2) Landau levels 3) Shubnikov-deHaas oscillations 4) Quantum Hall Effect 21

22 SdH oscillations quantized Hall voltage Zero longitudinal resistance M.E. Cage, R.F. Dziuba, and B.F. Field, A Test of the Quantum Hall Effect as a Resistance Standard, IEEE Trans. Instrumentation and Measurement, Vol. IM-34, pp ,

23 Edge states ν edge states ν edge states Assume that the Fermi level lies between Landau levels in the bulk, then only the edge states are occupied (edge states play the role of modes). 23

24 QHE: quantized Hall Voltage T = 1 because +v and v channels are spatially isolated. (from J.H. Davies, The Physics of Low-Dimensional Semiconductors, Cambridge, 1998, Fig. 6.18, p. 240) V 1 = 0 and V 2 = +V V 3 = V 4 = V 1 = 0 I = 2q2 h νv V H = V 3 V 5 = 1 h ν 2q I 2 V 6 = V 5 = V 2 = +V quantized Hall resistance 24

25 SdH oscillations quantized Hall voltage V H = 1 h ν 2q I 2 Shubnikov-deHaas (sdh) oscillations M.E. Cage, R.F. Dziuba, and B.F. Field, A Test of the Quantum Hall Effect as a Resistance Standard, IEEE Trans. Instrumentation and Measurement, Vol. IM-34, pp ,

26 QHE: zero longitudinal magnetoresistance (from J.H. Davies, The Physics of Low-Dimensional Semiconductors, Cambridge, 1998, Fig. 6.18, p. 240) V 1 = 0 and V 2 = +V V 3 = V 4 = V 1 = 0 I = 2q2 h νv V x = V 4 V 3 = 0 V 6 = V 5 = V 2 = +V zero longitudinal resistance 26

27 SdH oscillations vanishing longitudinal resistance V H = 1 h ν 2q I 2 Shubnikov-deHaas (sdh) oscillations M.E. Cage, R.F. Dziuba, and B.F. Field, A Test of the Quantum Hall Effect as a Resistance Standard, IEEE Trans. Instrumentation and Measurement, Vol. IM-34, pp ,

28 For a more complete discussion See Chapter 4 in Electronic Transport in Mesoscopic Systems, Supriyo Datta, Cambridge, J.H. Davies, The Physics of Low-Dimensional Semiconductors, Cambridge, D. F. Holcomb, Quantum electrical transport in samples of limited dimensions, American J. Physics, 64, pp ,

29 For a more complete discussion Chapter 17 in the following book provides a good, general discussion of transport in higg magnetic fields. J. Singh, The Physics of Semiconductors and Their Heterostructures, McGraw Hill, New York,