Electrical Standards based on quantum effects: Part II. Beat Jeckelmann

Transcription

1 Electrical Standards based on quantum effects: Part II Beat Jeckelmann

2 Part II: The Quantum Hall Effect Overview Classical Hall effect Two-dimensional electron gas Landau levels Measurement technique Accuracy of the quantized Hall resistance Applications in dc and ac electrical metrology 2

3 Discovery of the Quantum Hall Effect K. Von Klitzing discovers the quantum Hall effect in on 5 February 1980 in Grenoble K. v. Klitzing et al, Phys. Rev. Lett., 45, 494 (1980) 3

4 Classical Hall effect (1879) X X X X BX v X X X X X -e X X X X X F L X X X X X X X X X X Lorentz Force F L q v B Edwin Hall ( ) 4

5 Classical Hall effect (2) V H d F m F e U H n B I e d 3D I n 3D : carrier concentration 2D-system: U H B I ne Hall effect independent of geometrical dimensions! 5

6 Realisation of a 2D electron gas Si MOSFET 6

7 GaAs Heterostructure Layers grown by: Molecular Beam Epitaxy (MBE) or Metal Organic Chemical Beam Epitaxy (MOVCD) 4 mm 7

8 Inversion Layer E E 1 de Broglie wavelength: typically 100 nm (GaAs) E 0 E F Conduction band level spacing at 10 T: Si: 7.9 mev GaAs: 17 mev Graphene: 120 mev z (nm) thermal energy: T = 4 K: kt = 0.36 mev 8

9 Landau levels c cyclotron motion in a strong magnetic field Classical: r c c v r c e B m z Quantum mechanics: Energy values of closed orbits E 1 E0 cn, n 2 0,1, 2, 3... (spin neglected) magnetic length: (8 10 T) r c l 2n 1, l e B 9

10 Landau quantization (2) E Orbital degeneracy N L w 2 2l Number of states in a Landau level State density n B 1 2l 2 eb h Number of flux quanta within the area of the sample B = 0 B z >0 no scattering Filling factor: i n n B s 10

11 Quantum Hall effect? Hall voltage in a 2D-system: U H B I n e s R H B i n B e h, 2 e i i 1, 2, 3... observed when i levels are fully occupied! 11

12 Mobility gap Fermi level mobility gap Disorder and scattering removes orbital degeneracy Localized states do not carry current extended states localized states Plateau forms when E F resides within the localized states B z >0 no scattering B z >0 scattering state density Varenna 2016 / El. Standards II 14

13 QHE in a real device V H I V H + V c 15

14 QHE in a real device (2) E electronic states at Fermi energy E F -L y /2 o +L y /2 No energy gaps for real devices with finite width 16

15 Edge state picture Skipping Orbits source and drain contact are connected by a common edge one-dimensional edge channels carry the current Landauer formalism current = driving force of electronic transport Varenna 2016 / El. Standards II 17

16 Büttiker formalism Current in 1D channels: 5 R 5, T 5 R 6, T 6 6 I h e I R 1 T 1 R 2 T 2 : difference electrochemical potential R 3, T 3 R 4, T 4 R H R i xx 5 3 e 4 3 e i e i e h h 1 2 h i e 0 2 T: transmission R: reflection coefficient T i = 1, R i =0 ; Varenna 2016 / El. Standards II 6 18

17 QHE Model The QHE is a collective phenomenon, it can not be explained by a microscopic model A vanishing longitudinal resistivity indicated the absence of backscattering Under this condition, the QHE is the direct consequence of the transmission of one-dimensional channels. 19

18 QHE in Graphene Graphene: 2D crystal of carbon atoms with charge carriers» massless relativistic particles Geim & Novoselov Nobel Prize in Physics (2010) 20

19 Unconventional quantum Hall effect xy 4 e 2 1 N h 2 Novoselov, et al, Nature 438,

20 E / mev Interest for graphene E required for accuracy of R H at T B / T Potential to develop a quantum Hall resistance standard At T >5 K and B < 5 T Cryogen-free and cheap 22

21 Fine structure constant R K e h 2 0c 2 Test the validity of RK i RH (i) or: additional route to the determination of independent of QED) most accurate value for : measurement of a e + theoretical expansion of a e in a series expansion of (numerical computations: Kinoshita et al.) 23

22 Fine structure constant R K h e 2 0c 2 a e : Anomalous magnetic moment of e µ Mu : Ground state hyperfine splitting G 90 : Gyro-magnetic moment of the proton h/m: Neutron diffraction; cold atoms.. R K : QHE QED theory necessary without QED 24

23 Metrological application Ideal systems: T = 0 K, I = 0 A No dissipation: R xx = 0 R H i h i e 2 R i K R K is a universal quantity Localization theory, Edge state model Real experiment: T > 0.3 K, I = 40 µa Non-ideal samples Dissipation: R xx > 0 R H i, R 0?? xx Is R H (i) a universal quantity? Independent of device material, mobility, carrier density, plateau index, contact properties...? Few quantitative theoretical models available empirical approach, h i e precision measurements 2 25

24 QHE devices for metrology Carrier concentration B i n s e R H i m -2 < n s < m -2 n s > m -2 : 2nd subband fills up Mobility: > 10 T -1 to have clear separation of the LL up to plateau 4 Hall bar defined by photolithography and wet etching techniques Alloyed AuGeNi contacts 26

25 R H (kw QHE in a real device R H h i e 2 i = DEG B z I R xx (kw B (T)

26 Temperature dependence Thermal activation: 1 K T 10 K electrons thermally activated to the nearest extended states xx T 0 xx e kt E F E LL xy T xy (T) ie2 h xy (T) s xx (T) Cage et al. 1984: 1.2 K < T < 4.2 K, 2 GaAs samples, i = < s <

27 Current dependence GaAs T = 0.3 K i = 2 W = 400 µm Breakdown 29

28 Measurement technique To make use of the QHE for metrological applications, a measurement technique, capable of transferring the QHR to room temperature resistance standards must be available. Most accurate resistance bridge technique: Cryogenic Current Comparator (CCC) 30

29 The cryogenic current comparator (CCC): Principles Meissner effect: Harvey 1972 SQUID Harvey 1972 n 1I1 n2i 2 31

30 The CCC bridge: SQUID: N P I P N S I S (1 d) Detector: with d N t N S R L R L R H U m R S I S R P I P R P R S N P 1 N S 1 d 1 1 U m U 32

31 Ratio accuracy: W 1 N(1 w 1 ) W 2 N(1 w 2 ) U SQUID (w 1 w 2 ) Windings in a binary series: 1, 1, 2, 4, 8, 10, 16, 32, 32, 64, 100, 128, 256, 512, 1000, 1097, 2065,

32 Ratio accuracy 34

33 Performance V n-rms (nv/sqrt(hz)) Detector Johnson 300 K SQUID noise Thermal noise Hz Transfer function I CC s 4 A turn SQUID V nth 4 k B T R B 0.01 Johnson 1.2 K k 100k R (W) R H (2) : 100 Ω N P = 2065, N S = 16, I P = 50 µa V n-rms : 7 nv/hz u A : 1 nw/w in 2 min 35

34 Universality of the quantum Hall effect Width dependence Contact resistance Device mobility Plateau index Device material: MOSFET-GaAs - Graphene 36

35 Geometry of the QHE device Some theories predict a width dependence of the QHR Jeanneret et al., µm Value of : 2 ( ) ( ) 10 3 OFMET 200 µm 50 µm EPFL 20 µm 10 µm Deviation on 500 µm wide samples: i = 2 < ppb i = 4 < ppb No influence No size effect observed within the measurement uncertainty 37

36 Effect of the contact resistance Rc M. Büttiker, 1992:...It is likely, therefore, that in the future, contacts will play an essential role in assessing the accuracy of the QHE. On a perfectly quantised plateau: R P V I c 1 P1 P2 g I B z P 1 P 2 = i C Real sample R c > 0 Transmission 1 Reflection 0 Bad contacts electron gas depletion in the contact region non-ohmic behaviour of metal semiconductor interface 38

37 Contact resistance (2) g : depleted region in the contact arm 12 h/2e vg R H e h 2 1 g h/12e 2 4 B (T) 6 h/6e

38 R H /R H (ppb) R H /R H (ppb) Contact resistance (3) i = 2 80 W Deviation of R H related to finite V xx i = 4 6 W R c /R H (i) 40

39 Scattering parameters Electron mobility: measure of the electron velocity Jeckelmann et al. 97 R H independent of the device mobility or the fabrication process to 2 parts in

40 Step ratio measurements R H independent of the plateau index to 3 parts in i R H (i) 2 R H 2 i 1,3, 4,6,8 1 ( ) Jeckelmann et al

41 QHR comparison GaAs - MOSFET Jeckelmann et al., METAS, 1996 R H MOSFET GaAs R H

42 QHR comparison GaAs - Graphene GaAsgraphene ( ) A. Tzalenchuk et al., Nat. Nanotech. (2010) High precision measurements in epitaxial graphene on SiC at NPL T. J. B. M. Janssen et al., New J. of Phys. and Metrologia (2012) 44

43 QHR comparison GaAs Graphene (2) GaAsgraphene ( ) F. Lafont et al. LNE, 2014: graphene grown by CVD on SiC 45

44 Universality: Summary The quantum Hall resistance is a universal quantity independent of: Device width Device material: MOSFET- GaAs - Graphene Device mobility Plateau index..to a level of < R H h i, Rxx 0 2 i e CCEM Technical Guidelines: F. Delahaye and B. Jeckelmann, Metrologia 40, , (2003) 46

45 International QHR Key-comparison (r Lab -r BIPM )/r Lab BNM-LCIE METAS PTB NPL NIST PTB R H (2) / 100 W 10 kw / 100 W 100 W / 1 W Key Comparison BIPM.EM-K

46 (R-R nom )/R nom (nww Deviation from fit (nww Application: DC Resistance Standard (data METAS) Ω resistance standard year Deviation from fit < 2 nω/ω over a period of 10 years 48

47 QHE arrays Connection of several Hall bars (Delahaye 1993) R 2 1 AB R H c I B z A A Rc B B single connection c R R c H I B z A A R c R c B B double connection c R R c H

48 QHE array (LNE) 129 i = 2 10 mm 50

49 AC-QHE: Applications SI realisation of the Farad: Calculable capacitor complicated experiment Representation of the Farad: DC QHE New route: AC measurements of the QHR 51

50 First AC measurements of the QHE i = 2 T = 1.3 K I = 40 µa f = 800 Hz Narrow bumpy plateau (PTB, NPL, NRC, BIPM) Frequency dependence: R H ( i, )/ R H = = 1 to / khz Measurement problems: AC Losses Delahaye et al. 2000, Shurr et al. 2001, Overney et al Delahaye 94 52

51 Ratio Bridge: Principle U id it ib iq U jc Z Z T B 0 U(1 ) R R B T 1 RH, I Rc 1, I 53

52 Calculable Resistor: Quadrifilar Resistor (QR) Transmission line equations R// f R, L, C, G, R c R dc 2 c (1 ) = /khz 2 54

53 AC measurements of the QHE (with grounded back gate) Linear frequency dependence of the Hall resistance observed B C B C Capacitive losses scale with the surface 55

54 Model for ac Losses z V H B y x Hp Lc i i H Hc Lp i l d V G Z H V i H V i i H H R H i R 1 l H R 1 l H VH Overney et al.,

55 Adjusting capacitive losses (2 back gates) V H V G 57

56 Double shielding technique Meet the defining condition: ALL currents which have passed the Hall-potential line are collected and measured. Adjust the high-shield potential su so that dr H /di = 0. B.P. Kibble, J. Schurr, Metrologia 45, L25-L27 (2008). 58

57 QHR as impedance standard R H /R H / P741-63a " " P741-2B " P741-2C P741-2J P741-E -20 frequency of application f / khz = khz -1 capacitive effects < khz -1 for different devices better than artefacts J. Schurr, J. Kucera, K. Pierz, and B.P. Kibble, Metrologia 48, (2011). 59

58 Realization of the Farad 10 nf10 nf 10 nf RC = 1 quadrature bridge frequency QHR R K QHR = he / 2 10:1 bridge 1 nf 10:1 bridge 100 pf 10:1 bridge 10:1 calibration - relative uncertainty of 10 pf: (k = 1) (cryogenic quantum effect without calculable artefacts) - quantum standard of capacitance, analogous to R DC 10 pf J. Schurr, V. Bürkel, B. P. Kibble, Metrologia 46, ,

59 Conclusions R H is a universal quantity QHR improved electrical calibrations in National Metrology Institutes considerably QHR plays an important role in the determination of the fine structure constant QHR can be used as quantum standard for impedance 61

60 Thank you very much for your attention