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
Outline 1) Introduction 2) Landau levels 3) Shubnikov-deHaas oscillations 4) Quantum Hall Effect 2
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. 278-297, April 1999. 3
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. 301-303, 1985
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
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
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, 1978. 7
Outline 1) Introduction 2) Landau levels 3) Shubnikov-DeHaas oscillations 4) Quantum Hall Effect 8
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. 29-27. 9
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
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 + 1 2 ω c Landau levels 11
B = B z ẑ y Cyclotron frequency E n = n + 1 2 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
Effect on DOS D 2 D m * π 2 ( E) E n = n + 1 2 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 0 1 2 ω c n=0 13
Degeneracy of Landau levels D 2 D m * π 2 ( E) D 0 ω c E n = n + 1 2 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
Broadening D 2 D ( E) ΔE D 0 ω c E n = n + 1 2 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
Example If B = 1T, how many states are there in each LL? D 0 = 2qB z h = 4.8 10 10 cm -2 If n S = 5 x 10 11 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
Outline 1) Introduction 2) Landau levels 3) Shubnikov-deHaas oscillations 4) Quantum Hall Effect 17
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. 301-303, 18 1985
LL filling D 2 D ( E) ω c ν = n S 2qB h E n = n + 1 2 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
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
Outline 1) Introduction 2) Landau levels 3) Shubnikov-deHaas oscillations 4) Quantum Hall Effect 21
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. 301-303, 22 1985
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
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
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. 301-303, 25 1985
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
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. 301-303, 27 1985
For a more complete discussion See Chapter 4 in Electronic Transport in Mesoscopic Systems, Supriyo Datta, Cambridge, 1995. J.H. Davies, The Physics of Low-Dimensional Semiconductors, Cambridge, 1998. D. F. Holcomb, Quantum electrical transport in samples of limited dimensions, American J. Physics, 64, pp. 278-297, 1999. 28
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, 1993. 29