Metal-insulator transition in heavily doped semiconductors: disorder, magnetic moments and electron-electron interactions

Transcription

1 (TH) Metal-insulator transition in heavily doped semiconductors: disorder, magnetic moments and electron-electron interactions Hilbert v. Löhneysen Institute of Technology (KIT) Physikalisches Institut and Institut für Festkörperphysik Pohang, September 7, 2010

2 (TH) Outline Electronic structure of phosphorus-doped silicon The heavily doped semiconductor Si:P an amorphous metal Transport: electron-electron interactions in the metallic and insulating states Localized magnetic moments in the insulating phase Localized magnetic moments and Kondo effect in the metallic phase Role of on-site Hubbard interaction at half-filling: uncompensated Si:P vs. compensated Si:(P,B) Scaling properties of the metal-insulator transition

3 (TH) Acknowledgments T. Trappmannn, C. Sürgers, M. Schöck STM/STS M. Lakner, X. Liu, P. Ziegler specific heat, thermoelectric power H. G. Schlager magnetic susceptibility S. Waffenschmidt, C. Pfleiderer conductivity under uniaxial stress M. Gaymann, H. P. Geserich optical conductivity M. Hering, M. Scheffler, M. Dressel ac conductivity W. Zulehner, Wacker Siltronic samples P. Wölfle, A. Rosch, A. Langenfeld, A. Mildenberger theory

4 (TH) Two scenarios for quantum phase transitions Transition at finite temperature driven to absolute zero by control parameter δ No finite-temperature transition, critical line of zero-temperature transitions terminating in QCP T ordered phase (?) d Ginzburg-Landau theory for (magnetic) quantum phase transition in metals: but experiments differ strongly! QCP Metal-insulator transition in the presence of disorder and interactions: what is the order parameter? d

5 (TH) Metal-insulator transitions Classical percolation s dc Band transitions (Bloch, Peierls) Localization transitions disorder (Anderson) e-e interactions (Mott-Hubbard) T > 0 QCP T = 0 d Strict definition insulator: s dc 0 for T 0 T metal: s dc > 0 metal s dc ~ d - d c µ insulator QCP d

6 (TH) Anderson transition

7 (TH) Mott-Hubbard transition

8 (TH) Metal-insulator transition in heavily doped semiconductors

9 (TH) Isolated donor in silicon Hydrogen-like orbital E I= 45 mev (cf. E I = 13.6 ev for H) Modifications: E i = E i o (m*/m o ) (1/ε 2 ) ε 12, m* = 0.3 m o valley degeneracy: 6 conduction-band minima along <100> crystalline electric field of neighboring Si (tetrahedral symmetry) valley-orbit splitting 10 mev effective mass is anisotropic

10 (TH) Electronic densitity of states of n-doped semiconductors N << N c N N c bandwidth of impurity band E F = ħ 2 2 k F * ~ 8 mev 2m At the metal insulator transition: impurity band separated from conduction band? N >> N c

11 Far-infrared reflectance of Si:P Universität (TH) A. Gaymann, H. P. Geserich, HvL PRL 71, 3681 (1993), PRB 52, (1994) Reflectance Optical conductivity

12 (TH) The (2x1) surface structure of Si(111) after cleavage at room temperature T. Trappmann et al., Appl. Phys. A 68, 167 (1999) dangling-bond states at Si-1 predominantly occupied, at Si-2 predominantly empty

13 (TH) P-doped Si (111) (2x1) surface T. Trappmann et al., EPL 38, 177 (1997)

14 (TH) P-doped Si (111) (2x1) surface T. Trappmann et al., EPL 38, 177 (1997)

15 (TH) V V Analysis of STM data V V T. Trappmann et al., EPL 38, 177 (1997) At negative bias along D : an additional Si atom

16 (TH) Surface band structure of Si:P and Si:B T. Trappmann et al., EPL 38, 177 (1997) n-doped Si:P p-doped Si:B n near a positively charged donor near a negatively charged acceptor

17 (TH) Distribution of nearest-neighbor P atoms in heavily doped Si:P T. Trappmann et al., EPL 38, 177 (1997)

18 (TH) Conductivity of uncompensated Si:P 0== 8 0d=0phi6l, L

19 Conductivity of Si:P well in the metallic regime (N 1.2 N c ) Universität (TH) HvL, M. Welsch, PRB 44, 9045 (1989) Similar experiments: Rosenbaum et al.

20 (TH) Electron-electron interaction in disordered systems Corrections to the conductivity Altshuler, Aronov exchange Hartree gµ B B > k B T screened el-el interaction (Thomas-Fermi) Si:P with six conduction-band minima: With decreasing N N C : F σ should increase m should become more negative

21 (TH) Conductivity of metallic Si:P (N N c ) weak localization l in ~ T p e-e interaction, two terms: Hartree (-) and exchange (+) for N sufficiently above N c (N > 1.1 N c ): ~ magnetic field affects Hartree and exchange:

22 Variable-range hopping between localized states Constant DOS near E F E F Compromise between distance and energy difference variable range hopping (VRH) σ (T) = σ o exp [-(T M /T) 1/4 ] Mott VRH (d = 3) Coulomb interaction between excited electron-hole pair would lead to an instability of the ground state depletion of states near E F Soft Coulomb gap in the DOS near E F : N(E) ~ (E E F ) 2 Universität (TH)

23 (TH) Hopping conductivity in Si:P M. Hornung et al., phys. stat. sol. (b) 218, 75 (2000)

24 (TH) AC conductivity in the VRH regime Independent electrons: Mott VRH (constant DOS) Long-range Coulomb interactions: Efros-Shklovskii VRH Conductivity (Ωcm) -1 1 M. Hering et al., PRB 75, (2007) N = cm -3 n= cm -3 n= cm -3 N = cm -3 0,1 N 0 Density of states at E F x Localization length I o Prefactor of the overlap integral ( 45 mev) 0,02 Frequency (GHz) σ ~ ω

25 (TH) AC conductivity in the VRH regime Independent electrons: Mott VRH (constant DOS) M. Hering et al., Physica B , 1469 (2005) Quantitative disagreement with theory In the region of Mott-ES crossover Long-range Coulomb interactions: Efros-Shklovskii VRH N 0 Density of states at E F Conductivity (Ωcm) -1 3 ve l, N1- x I o Localization length Prefactor of the overlap integral ( 45 mev) Frequency (GHz)

26 (TH) Magnetic moments in Si:P Magnetic susceptibility: Curie-Weiss-like behaviorfrom insulating well into the metallic region, Maximum of χ at N ~ 0.5 N c, i.e., well below N c H. G. Schlager, HvL, EPL 40, 661 (1997) Similar previous experiments: Y.Ootsuka et al., K. Andres et al.

27 (TH) Antiferromagnetically coupled spin pairs in Si:P Broad distribution of exchange couplings J between pairs Freezing of pairs renormalizes remaining couplings Only spin pairs with contribute to and effective density N(T) Bhatt and Lee, PRL 1983

28 (TH) Magnetic susceptibility in the insulating regime ~ 1/T Strong deviations from simple Curie law because of pair interactions J(r i r j ) H. G. Schlager, PhD thesis (1996)

29 (TH) Specific heat of Si:P M. Lakner, HvL, PRL 63, 648 (1989), PRB 50, (1994) C = γt + βt 3 + ΔC Plot of C/T vs. T 2 yields γ(n), β = const. and ΔC

30 (TH) Specific heat ΔC due to local magnetic moments measured ( ) and calculated ( ) after Bhatt-Lee: C ~ dχ/dt H. G. Schlager, HvL, EPL 40, 661 (1997)

31 Magnetic-field dependence of the specific heat of Si:P well in the metallic region Universität (TH) M. Lakner et al., PRB 50, (1994)

32 (TH) Localized magnetic moments in Si:P from specific heat M. Lakner, HvL, PRL 63, 648 (1989) Main results N C exchange splitting 0 for N 0 N S : from entropy in zero field N Sch : from fits to Schottky anomaly interactions broaden Schottky anomaly Theory for metallic phase: Mildenberger, Wölfle Langenfeld,

33 (TH) Localized magnetic moments in the metallic state Some moments survive because of disorder Simple picture: a few P atoms are sufficiently distant from other P atoms that the overlap of donor wave function with conduction electron sea formed by other donor electrons is small Criterion for local-moment behavior? Anderson model of magnetic impurity in a metal N c H. G. Schlager, PhD thesis (1996)

34 (TH) Magnetic moments in metallic Si:P induced by disorder and U Langenfeld, Wölfle, Ann. Phys. (Leipzig) 1995 See also Dobrosavljevic, Kotliar, PRL 1997

35 (TH) Thermoelectric power free electrons, τ ~ E -1/2 (l = const.) Si:P: effective bandmass valley degeneracy phonon drag negligible below 1k M. Lakner, HvL, PRL 70, 3475 (1993)

36 (TH) Thermoelectric power of metallic Si:P at very low temperatures Maximum in S(T): evidence for Kondo effect M. Lakner, HvL, PRL 70, 3475 (1993) Magnetic-field dependence

37 (TH) Long-range and short-range e-e interactions in Si:P long-range loss of screening due to diffusive electron motion arising from disorder metal: Altshuler-Aronov anomalies in DOS and σ (T) insulator: soft Coulomb gap, Efros-Shklovskii VRH short-range on-site Hubbard U metal: formation of magnetic moments, Kondo effect insulator: Hubbard splitting of 1s(A 1 ) impurity band

38 (TH) Uncompensated and compensated semiconductors Uncompensated and compensated semiconductors Electrical resistivity X. Liu et al., PRL 77, 3395 (1996) Si:P uncompensated: half-filled impurity band Si:(P,B) compensated: away from halffilling

39 Uncompensated and compensated semiconductors Universität (TH) Thermoelectric power X. Liu et al., PRL 77, 3395 (1996) Si:P uncompensated: half-filled impurity band Si:(P,B) compensated: away from halffilling

40 (TH) Hubbard splitting of the impurity band in Si:P 8 l ihn6 H1ud6 N- Nv NdLNd

41 (TH) Total DOS at the metal-insulator transition of an interacting electron system HvL, unpublished Dynamical mean field theory (on-site U) Efros-Shklovskii Coulomb gap (long-range el-el interaction) Note: Specific heat measured at 0.1 K includes contribution of magnetic moments

42 (TH) Interplay of disorder and on-site interaction Byczuk et al., PRL 94, (2005) Solution of Anderson-Hubbard model with DMFT: Interaction favors delocalization

43 (TH) Total DOS at the metal-insulator transition of an interacting electron system Total DOS at metal-insulator transition of an interacting electron system HvL, unpublished Dynamical mean-field theory (on-site U) Efros-Shklovskii Coulomb gap (long-range el-el interaction) Note: Specific heat measured at 0.1 K includes contribution of magnetic moments

44 Critical behavior of the electrical conductivity in Si:P: The exponent puzzle Universität (TH) σ min Si:P uncompensated Si: (P,B) compensated - - σ min Thomas et al. Hirsch et al.

45 Main issues for the metal-insulator transition in doped semiconductors (d = 3) Universität (TH) No signature of a thermodynamic transition, smooth evolution of magnetic moments and specific heat across MIT. MIT is seen only in transport. What is the order parameter? Spontaneous symmetry breaking? No Ginzburg-Landau mean-field-type model. Theoretical predictions difficult, even for the noninteracting case. Numerics: µ = for noninteracting electrons. Rigorous lower bound ν ³ 2/3 (with Wegner scaling µ = (d 2)ν ). Experimentally, what is the physical distinction between uncompensated and compensated samples (Hubbard U)?

46 (TH) Conductivity of uncompensated Si:P 0== 8 0d=0phi6l, L

47 (TH) Sign change of dσ/dt above N c hu6vl,,, uh σ (T) passes over a maximum, similar to σ (T) in a magnetic field: evidence for the influence of local moments on e-e interactions B = 0 B > 0

48 (TH) Concentration tuning of the metal-insulator transition in Si:P 0==, uh

49 Crossover in the critical behavior of σ in Si:P Universität (TH) critical region limited to concentration region where i. e. Sign change of m coincides with change of critical exponent µ 0==, uh

50 (TH) Dynamical critical scaling of the conductivity 0==, uh ν: localization-length exponent z: dynamical exponent Wegner scaling: µ = ν scaling with ν = 1.3 and z = 2.4

51 (TH) Stress-tuning of the metal-insulator transition in Si:P 1ibid Nn 6 1, 66 0didb. u h, - N1pi 6 u, 6 H udl u1, dn hndb, 1, ib, d6 u, 6 0=Nd 6-e e, 1-1, L0v ind p- 0diumiuh 6 1, 66 ule im 01, Nn l ibl, 1 6 u, 6 N Lo LH L, =, dl6 Nd 6 1, 66 Li1, v ind mue =h, 1, 66 uhndb S t l u6 dn, nn, v Nd uuhud, o d, uh Hl Ne u6, uh unn, d6vl e il, uh

52 Stress-tuning of the metal-insulator transition in Si:P unn, d6vl e il, uh Universität (TH)

53 Dynamic scaling of the conductivity at the MI transition Universität (TH) unn, d6vl e il, uh

54 (TH) Strong increase of current noise at N c c u6 is u1, uh 8iL, dv, nn1 v1i ivuh nh0v 0u ind6

55 (TH) Other systems Ge:Ga (donor system) Transmutation doping: 70 Ge + n 71 Ga (β capture): critical region µ 1 shrinks as compensation decreases Itoh, Haller et al., PRL 1996, PRB 1999, PRB 2000 Si:B (acceptor system) Large difference between concentration tuning µ 0.5 and stress tuning µ 1.6, not understood Bogdanovich, Sarachik et al., PRL 1999

56 (TH) Conclusions and open questions I I Heavily doped semiconductors - a kaleidoscope of solid-state physics Random distribution of donors: a truly amorphous system Both disorder and electron-electron interactions important Disorder: enhanced long-range e-e interaction (loss of screening) Metal: Altshuler-Aronov contribution to δσ (T,B) Insulator: Coulomb gap On-site Coulomb interactions (Hubbard U): Metal: Magnetic moments on the metallic side, Kondo effect Insulator: direct evidence for U from transport measurement Spectroscopic detection of Hubbard U? Explanation of the specific-heat-derived DOS?

57 (TH) Conclusions and open questions II Metal-insulator transition in Si:P at T = 0 - Both disorder and electron-electron interactions important - Origin of crossover from dσ/dt > 0 to dσ/dt < 0 and from µ 1 to E in metallic phase close to N C? - Role of compensation? - Dynamical scaling of the conductivity σ (S,T) - Nature of the order parameter? - Dynamical scaling of σ (ω,t)?