Charge transport in oxides and metalinsulator

Transcription

1 Charge transport in oxides and metalinsulator transitions M. Gabay School on modern topics in Condensed matter Singapore, 28/01 8/

2 Down the rabbit hole Scaling down impacts critical parameters of a transistor: t=rc (switching time). Id (source drain current), Ioff (leakage current), Nc (carrier concentration), m (mobility), j=nc e m E If Tï Ileak~10A!!

3 Technological hurdles. Scaling down impacts critical parameters of a transistor: t=rc (switching time). Id (source drain current), Ioff (leakage current), Nc (carrier concentration), m (mobility), ì Good for P kv 2 but bad for m~ 1/E 1/3 Bad for t ï Multicore, 3D architecture

4 Avouris, Nanoletters, 10, 4285, 2010 Plus: 100GHz vs 28 GHZ limit for ususal semiconductors, T =10 times that of MOSFET Minus: On/Off ratio of 100 vs 104 to 107 for MOSFET Avouris, Nanoletters, 10, 715, 2010

5 Transition Metal oxide (TMO) SrTiO 3 YBa 2 Cu 3 O 7-d BiFeO 3 La 1-x Sr x MnO 3

6 Large ε r for oxides ï surface carrier doping and «built-in» insulating coating m (mobility) sensitive to the quality of the interface Darrell G. Schlom and Loren N. Pfeiffer, Nature Materials, 9, 881, 2010 C. Cen et al, Nature Materials,7, K Ueno et al, Applied physics Letters 96, , 2010

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9 Doping by chemical substitution or oxygen content introduces disorder

10 Doping 2D materials H n s =e/h*k F l e doping (h or e) n s (cm -2 ) H (cm 2 low T

11 Jy=0 Vx Vy y x Bz I

12 s xy (w c )= w ct s w c t xx (w c =0) s xx (w c )= s w c t xx (w c =0) w c =eb z /m* Hall bar geometryï Jy=0 ïs(w c )=(s 2 xx+s 2 xy)/s xx = s(w c =0)!!

13 Taking into account a realistic Fermi surface leads to Kohler rule for the magneto-resistance DrååP F(w c t) Taking into account multiband structure Also leads to B dependence for the magneto-resistance s 1,2 = r R 1,2 =,

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15 w c t>1 råå(w) Shubnikov-de Haas

16 PRL 105, (2010)

17 1.1. Classical charge transport 1. T >>h D. Phonon scattering 1/T 2. T << h D. Phonon scattering 1/T 5 3. T << T F. e e scattering + Umklapp 1/T 2 4. T << T F. Impurity scattering Const Note: There is no σ(t) dependence in the T=0 limit! (within the classical approximation, for non interacting electrons )

18 n i (impurity concentration) Matthiessen rule d ~ l=1/n i s r ; l=v F t

19 1.2.Semiclassical concept of transport (1960) l min ~ 1 k F Ioffe Regel criterion A.F. Ioffe and A.R. Regel, Prog. Semicond. 4, 237 (1960). 2 ne m ( k 2 F / 2 k F 2 ) e 2 l e h 2 (2D) Abram F. Ioffe minimum metallic conductivity 2 e 2D h kΩ Anatoly R. Regel Nevil Mott ( )

20 A non interacting electron moving in random potential extended localized localized localized E c E critical extended

21 Quantum concept of transport (1979): E.Abrahams A B Interference of electron waves causes localization T.V. Ramakrishnan P.W. Anderson D.C. Licciardello

22 e i Ioffe-Regel criterion for localization: k F l~1 In fact localization at T=0 in 2D even if k F l>>1 G e 2 /h)k F l g R 1 1 m kfl

23 In analogy with optics the amplitude of the wave that travels along path I and scatters on obstacles with transmission coefficients T j evolves according to the sequence E 0 T 1 E 0 exp i k 1 (r 1 -r 0 ) T 2 T 1 E 0 exp i( k 1 (r 1 -r 0 )+ ( k 2 (r 2 -r 1 ) ) A 1 =T 4 T 3 T 2 T 1 E 0 exp i(êj=1,4 k j (r j+1 -r j ) ) (where 5ª 0) the amplitude of the wave that travels along path II and scatters on obstacles with transmission coefficients T j evolves according to the sequence E 0 T 4 E 0 exp- i k 4 (r 4 -r 5 ) T 3 T 4 E 0 exp -i( k 4 (r 4 -r 5 )+ ( k 3 (r 3 -r 2 ) ) A 2 =T 1 T 2 T 3 T 4 E 0 exp -i(êj=1,4 k j (r j -r j+1 ) ) (where 5ª 0) ï <I> rand A 1 +A 2 2

24 L D=mE F /e 0!! m k F l <I> rand 4 T 1 T 2 T 3 T 4 2 <I> rand 2 T 1 T 2 T 3 T 4 2

25 G (L) = e 2 /h)g(l) = s A/L sl d-2 s(t) = e 2 /h)g(l) L 2-d L=Lin(T) Metalï s(t) finite ï g(l)~ L d-2 at large L Quantum ballistic regime Microscopic scale l Quantum diffusive regime Mesoscopic scale L in Classical diffusive regime Macroscopic scale System size

26 R. Landauer X R tot >R 1 +R 2 Add more random potentials ï. R(L)~ exp L/l

27 Scaling theory (gang of four, 1979) Conductance changes when system size is changed. Metal: Insulator: All the states at E F are localized at and below two dimensions! A metal insulator transition at g=g c is continuous (d>2).

28 2D OR NOT 2D D 2 e h ln( T ) ( ~ D 2 p ln(l in /l)) All electrons in 2D become localized at T 0 for ln(1/t ) In a magnetic field, L in L B ~ ï Ds>0

29 R. Scherwitzl, S. Gariglio, M. Gabay, P. Zubko, M. Gibert, and J. M. Triscone, Phys. Rev. Lett. 106, (2011) LaNiO 3 film on SrTiO 3 Delocalization due to a magnetic field

30 Antilocalization due to spin-orbit

31 3 symmetry classes (orthogonal, unitary, symplectic) symplectic class: time reversal, spin rotation spin orbit interaction anti localization Metal insulator transition in 2D critical point in 2D

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34 Spin!

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36 Electron interaction correction to conductivity in the diffusive regime H=0 It is a measure of screening (smaller values more metallic) H>k B T/g* B

37 Quantifying e e interaction in 2D ( ) F i a,s FL constants (harmonics) of the e e interaction