Collective electronic transport close to metal-insulator or superconductor-insulator transitions

Transcription

1 Colleve eleron ranspor lose o meal-nsulaor or superonduor-nsulaor ransons Markus Müller Unversy of Geneva Lev Ioffe (Rugers Unversy) dsussons wh Mkhal Fegelman (Landau Insue) Newon Insue, Cambrdge, 17 h Deember, 008

2 Oulne Revew of sngle-parle and many-body loalzaon. Expermens suggesng purely eleron onduon n nsulaors (.e. many-body deloalzaon ). Theory of eleron-asssed ranspor Maor ngreden: srongly orrelaed, quanum glassy sae of elerons lose o he meal-nsulaor ranson. Remnans of many-body loalzaon lose o he superonduor-o-nsulaor ranson?

3 Revew of loalzaon and nsulaors

4 Revew of loalzaon and nsulaors Sll lle undersandng beyond he smple model!!

5 Anderson loalzaon (3D) Connuous sperum Pon sperum Mobly edge

6 Anderson loalzaon (3D) On he Behe lae: (Abou-Chara, Thouless, Anderson (1973)) P ( ε ) 1 δ δ δ < # # k d δ δ > # # k d loalzed exended

7 Loalzaon wh neraon? L. Fleshman and P. W. Anderson, PRB, 1, 366 (1980). H Q: Does loalzaon perss n he presene of neraons? In oher words: Does onduvy vansh exaly whou phonons? ψ ( x) ψ ( x) ψ ( x) V ( x) ψ ( x) ψ ( x) ψ ( x ) V ( x x' ) ψ ( x' ) ψ ( x) ' n

8 Loalzaon wh neraon? L. Fleshman and P. W. Anderson, PRB, 1, 366 (1980). H Q: Does loalzaon perss n he presene of neraons? In oher words: Does onduvy vansh exaly whou phonons? ψ ( x) ψ ( x) ψ ( x) V ( x) ψ ( x) ψ ( x) ψ ( x ) V ( x x' ) ψ ( x' ) ψ ( x) ' n Creae exra harge bump a orgn: Ψ / -x a e dx ψ π a ( x) ΨGS

9 Loalzaon wh neraon? L. Fleshman and P. W. Anderson, PRB, 1, 366 (1980). H Q: Does loalzaon perss n he presene of neraons? In oher words: Does onduvy vansh exaly whou phonons? ψ ( x) ψ ( x) ψ ( x) V ( x) ψ ( x) ψ ( x) ψ ( x ) V ( x x' ) ψ ( x' ) ψ ( x) ' n δ ( x, ) ˆ ρ ( x) ( x) ( ) ˆ ρ q Ψ Ψ GS Tme evoluon? Dynam loalzaon?

10 Loalzaon wh neraon? L. Fleshman and P. W. Anderson, PRB, 1, 366 (1980). H Q: Does loalzaon perss n he presene of neraons? In oher words: Does onduvy vansh exaly whou phonons? ψ ( x) ψ ( x) ψ ( x) V ( x) ψ ( x) ψ ( x) ψ ( x ) V ( x x' ) ψ ( x' ) ψ ( x) ' n δ ( x, ) ˆ ρ ( x) ( x) ( ) ˆ ρ q Ψ Ψ GS Tme evoluon? Dynam loalzaon? x δq δ ( x, ) x dx q( x, ) dx ~ D α D, α < 1 < ons. Deloalzaon Anomalous dffuson Manybody loalzaon

11 Loalzaon wh neraon? L. Fleshman and P. W. Anderson, PRB, 1, 366 (1980). Q: Does loalzaon perss n he presene of neraons? In oher words: Does onduvy vansh exaly whou phonons? A: Fleshman and Anderson: 1 s order perurbaon heory: Yes: for shor range neraons. No: for long range neraons: Eleron-asssed hoppng s possble.

12 Loalzaon wh neraon? L. Fleshman and P. W. Anderson, PRB, 1, 366 (1980). Q: Does loalzaon perss n he presene of neraons? In oher words: Does onduvy vansh exaly whou phonons? A: Fleshman and Anderson: 1 s order perurbaon heory: Yes: for shor range neraons. No: for long range neraons: Eleron-asssed hoppng s possble. Reason: Energy onservaon mpossble f here s no onnuous bah! Sngle hop: Energy msmah beause of loal pon sperum. No harge ranspor a hs level

13 Loalzaon wh neraon? L. Fleshman and P. W. Anderson, PRB, 1, 366 (1980). Q: Does loalzaon perss n he presene of neraons? In oher words: Does onduvy vansh exaly whou phonons? A: Fleshman and Anderson: 1 s order perurbaon heory: Yes: for shor range neraons. No: for long range neraons: Eleron-asssed hoppng s possble. Reason: Energy onservaon mpossble f here s no onnuous bah! Mulparle rearrangemens: Transon energes reman dsree for weak neraons and low T

14 Loalzaon wh neraon? Invesgaon o all orders n perurbaon heory: I. V. Gorny, A. D. Mrln, and D. G. Polyakov, PRL 95, (005). D. M. Basko, I. L. Alener, and B. L. Alshuler, Ann. Phys. 31, 116 (006). Assumpon: Very weak neraons: V n << level spang δ ξ. Conluson: An energy rss (.e., a meal-nsulaor ranson whou phonons) ours a hgh emperaure due o loalzaon n Fokspae. Argumen: Same as Anderson loalzaon: 1) Ses many body saes Ψ Ψ 0 1 a a α γ a Ψ β a GS α Ψ e. ) Perurbaon heory n hoppng Perurbaon heory n neraons GS

15 Loalzaon wh neraon? Invesgaon o all orders n perurbaon heory: I. V. Gorny, A. D. Mrln, and D. G. Polyakov, PRL 95, (005). D. M. Basko, I. L. Alener, and B. L. Alshuler, Ann. Phys. 31, 116 (006). Assumpon: Very weak neraons: V n << level spang δ ξ. Conluson: An energy rss (.e., a meal-nsulaor ranson whou phonons) ours a hgh emperaure due o loalzaon n Fokspae. Argumen: Same as Anderson loalzaon: 1) Ses many body saes T MIT Ψ Ψ 0 1 ) Perurbaon heory n hoppng Perurbaon heory n neraons ( δ ) V a a ξ n α γ a Ψ β a GS α >> δ Ψ ξ GS T e. Mo hoppng

16 Loalzaon wh neraon? Invesgaon o all orders n perurbaon heory: I. V. Gorny, A. D. Mrln, and D. G. Polyakov, PRL 95, (005). D. M. Basko, I. L. Alener, and B. L. Alshuler, Ann. Phys. 31, 116 (006). Assumpon: Very weak neraons: V n << level spang δ ξ. Conluson: An energy rss (.e., a meal-nsulaor ranson whou phonons) ours a hgh emperaure due o loalzaon n Fokspae. Could here be nsanons?? Argumen: Same as Anderson loalzaon: 1) Ses many body saes T MIT Ψ Ψ 0 1 ) Perurbaon heory n hoppng Perurbaon heory n neraons ( δ ) V a a ξ n α γ a Ψ β a GS α >> δ Ψ ξ GS T e. Mo hoppng

17 Implaons of manybody loalzaon A rue quanum glass: non-ergod sysems, despe of neraons! Defea of ardnal assumpon of hermodynams: ha nfnesmal neraons wll evenually lead o equlbraon Perfe, olleve nsulaors a fne T Quanum ompung/nformaon: Preserved quanum oherene due o lmed enanglemen of loal degrees of freedom

18 Wha abou expermen? No meal-nsulaor ranson observed a fne T Raher: Evdene for e-asssed hoppng (many-body deloalzaon) Why hs dfferene from heoreal predons!?

19 Eleron asssed hoppng Doped GaAs/Al x Ga 1-x As heerosruure S. I. Khondaker e al., PRB 59, 4580 (1999)

20 Eleron asssed hoppng Doped GaAs/Al x Ga 1-x As heerosruure Efros-Shklovsk varable range hoppng: h T T ρ D f exp e T0 T 0 1 S. I. Khondaker e al., PRB 59, 4580 (1999) Nearly unversal prefaor! ( 1) O( 1)! f In sark onras wh sandard phonon-asssed hoppng! [ ( ) ( )] 4 f 1 O 10 e ph Mo and Daves (1979), Alener e al. (1994)

21 Open Quesons Theory for eleron-asssed ranspor n nsulaors? Expermenal evdene for e-asssed hoppng Cavea n heores of manybody loalzaon??? Can one have an nsulaor and eleron-eleron neraon-ndued onduvy a fne T? How o explan he nearly unversal eleron prefaor? h e

22 Model sysem Elerons wh dsorder Coulomb neraons n 3d or quas d H H V V kn ds Sngle parle Anderson problem Dagonalze! Assumpon abou dsorder Sngle parle problem d lose o he Anderson ranson ( ) 1 Cb Large loalzaon lengh ξ >> n -1/3, Small level spang νξ δ ξ

23 H Model sysem Elerons wh dsorder Coulomb neraons n 3d or quas d ε n n Jn k nk, H H V V kn ds Sngle parle Anderson problem Dagonalze! Assumpon abou dsorder Sngle parle problem lose o he Anderson ranson Cb Large loalzaon lengh ξ >> n -1/3, Small level spang Hamlonan n sngle parle bass (wavefunons ϕ ):,, k,, k, l u kl δ ξ k l d ( νξ ) 1 P Sngle parle energes 1 νξ δ 3 ( ε )

24 P H Model sysem Elerons wh dsorder Coulomb neraons n 3d or quas d ε n n Jn k nk J, H H V V kn ds Sngle parle Anderson problem Dagonalze! Assumpon abou dsorder Sngle parle problem lose o he Anderson ranson Hamlonan n sngle parle bass (wavefunons ϕ ): Sngle parle energes 1 νξ δ 3 ( ε ),, k,, k, l u kl [ ϕ ( r) ρ] [ ϕ ( r ) ρ] ( ) ( ) ϕ r ϕ r ϕ ( r ) ϕ ( r ) drdr κ r r Cb Large loalzaon lengh ξ >> n -1/3, Small level spang u kl Coulomb neraon (paral sreenng from hgh energy saes) δ ξ k l d ( νξ ) 1 k drdr κ r r l

25 Wavefunons a he mobly edge Egensaes of he non-nerang Anderson problem: Spaally overlappng fraal wavefunons H. Aok, PRB, 33, 7310 (1986). Theory: Mrln e al.; Kravsov e al.;

26 Coulomb neraons are srong a he Meal-nsulaor ranson! Sale of Coulomb neraons: Level spang: J ~ e κξ δ 1 νξ 3 Salng argumens numeral and expermenal ndaons: J ~ δ 3 e νξ κξ ~ ξ ξ α ; wh α > 0. Conluson: Coulomb neraons are srong and non-perurbave n he nsulaor!

27 Quanum eleron glass Theoreal model: Mean feld-lke quanum eleron glass ( ) 3 1 νξ δ ε P κξ ~ e J J ( ) d e J z νξ κξ δ As ξ, k l k l k kl k k u n n J n n H ε

28 Quanum eleron glass Theoreal model: Mean feld-lke quanum eleron glass Expe quanum glass sae: Many loal mnma wh many sof olleve exaons! l k l k kl u n J n n H,,,, ε ( ) δ ε 1 P ( ) d e J z νξ κξ δ Srong neraons GS non-rval Random sgns Frusraon As ξ, ( ) 3 1 νξ δ ε P κξ ~ e J J k l k l k kl k k u n n J n n H ε

29 Quanum eleron glass Theoreal model: Mean feld-lke quanum eleron glass l k l k kl u n J n n H,,,, ε δ δ >> >> J J J T E a Energy range where reshufflng ours: ( ) δ ε 1 P Expe quanum glass sae: Many loal mnma wh many sof olleve exaons! ( ) d e J z νξ κξ δ As ξ, ( ) 3 1 νξ δ ε P κξ ~ e J J k l k l k kl k k u n n J n n H ε Srong neraons GS non-rval Random sgns Frusraon

30 Quanum eleron glass Theoreal model: Mean feld-lke quanum eleron glass H ε εn n J J n, u kkl,, k, l n k k u l kl k l P ( ε ) 1 k δ l P J 1 νξ δ 3 ( ε ) ~ J e κξ As ξ, z J δ d ( e κξ ) νξ Srong neraons GS non-rval Random sgns Frusraon Energy range where reshufflng ours: E a T J J >> δ J >> δ Number of ave neghbors of gven eleron: N a Expe quanum glass sae: Many loal mnma wh many sof olleve exaons! z >>1 Large onrol parameer!

31 Quanum eleron glass Program: Undersand he olleve modes (plasmons) of he quanum eleron glass whn mean feld heory. Infer he exsene of a gapless phononlke bah whh an resolve he energy onservaon problem n hoppng onduvy.

32 Reduon o a quanum spn glass Idea: Classal frusraed glass quanum fluuaons k l k l k kl k k u n n J n n H ε

33 Reduon o a quanum spn glass Idea: Classal frusraed glass quanum fluuaons Spn represenaon for level oupaon: z σ ± 1 1, 0 n

34 Reduon o a quanum spn glass Idea: Classal frusraed glass quanum fluuaons Spn represenaon for level oupaon: z σ ± 1 n 1,0 Dynamal mean feld desrpon (good for z >> 1) Ineral, non-dsspave dynams vrual exhange proesses of elerons wh he bah of neghborng ses, no deay

35 Reduon o a quanum spn glass Idea: Classal frusraed glass quanum fluuaons Spn represenaon for level oupaon: z σ ± 1 n 1,0 For he purpose of olleve dynams: Desrbe quanum fluuaons by a selfonssen effeve ransverse feld eff wh H eff ( z x ε ) σ effσ 1, σ J z J δ σ >> z eff >> J

36 Reduon o a quanum spn glass Idea: Classal frusraed glass quanum fluuaons Spn represenaon for level oupaon: z σ ± 1 n 1,0 For he purpose of olleve dynams: Desrbe quanum fluuaons by a selfonssen effeve ransverse feld eff wh H eff ( z x ε ) σ effσ 1, σ J Am: Oban olleve deloalzed modes onnuous bah. Show ha he sysem remans an nsulaor (sngle parle exaons reman sharp lose o he Ferm level) Consru he heory of eleron-asssed hoppng. z J δ σ >> z eff >> J

37 Quanum TAP equaons (Thouless, Anderson, Palmer 1977: Classal SK model) H eff z x 1 ( ε σ effσ ) Transverse feld Isng spn glass (quanum Sherrngon Krkpark-model a z ), σ J z σ z Paramagne Glass? (Goldshmd and La, PRL 1990)

38 Quanum TAP equaons (Thouless, Anderson, Palmer 1977: Classal SK model) H eff z x 1 ( ε σ effσ ), σ J Transverse feld Isng spn glass (quanum Sherrngon Krkpark-model a z ) z σ z Paramagne For nfne oordnaon z : Phase ranson no a glass sae: - Broken ergody - Many long-lved measable saes Glass? Goldshmd and La, PRL (1990)

39 Quanum TAP equaons (Thouless, Anderson, Palmer 1977: Classal SK model) H eff z x 1 ( ε σ effσ ), σ J Transverse feld Isng spn glass (quanum Sherrngon Krkpark-model a z ) z σ z Paramagne For nfne oordnaon z : Phase ranson no a glass sae: - Broken ergody - Many long-lved measable saes Glass? Goldshmd and La, PRL (1990) Speral gap loses a he quanum phase ranson and remans zero n he glass phase! Read, Sahdev, Ye, PRL (1993)

40 Quanum TAP equaons (Thouless, Anderson, Palmer 1977: Classal SK model) H eff z x 1 ( ε σ effσ ), σ J Transverse feld Isng spn glass (quanum Sherrngon Krkpark-model a z ) z σ z? Glass Paramagne For nfne oordnaon z : Phase ranson no a glass sae: - Broken ergody - Many long-lved measable saes - Self-organzed raly (margnal sably) of he saes whn he glass phase Goldshmd and La, PRL (1990) Speral gap loses a he quanum phase ranson and remans zero n he glass phase! Read, Sahdev, Ye, PRL (1993)

41 Quanum TAP equaons (Thouless, Anderson, Palmer 1977: Classal SK model) H eff z x 1 ( ε σ effσ ), σ J z σ z Consraned free energy as a funon of magnezaons mposed by ex ex exernal auxlary felds h (oal loal feld: h h ε ) a large z G z ({ σ m }) E ( m ) ex 1 1 ( h m ) m Jm J dτχ ( τ ) χ ( τ ) 0 E m χ h de dh ( h ) ( ω 0) dm dh eff E ( m )

42 Quanum TAP equaons (Thouless, Anderson, Palmer 1977: Classal SK model) H eff z x 1 ( ε σ effσ ), σ J z σ z Consraned free energy as a funon of magnezaons mposed by ex ex exernal auxlary felds h (oal loal feld: h h ε ) a large z: G z ({ σ m }) E ( m ) ex 1 1 ( h m ) m Jm J dτχ ( τ ) χ ( τ ) 0 E m χ h de dh ( h ) ( ω 0) dm dh eff E ( m ) Loal mnma ( G m 0) ε Jm m (n sa approxmaon) χ ( m ) ; m m( h ) h J N oupled random equaons for {m } wh exponenally many soluons!

43 Quanum TAP equaons (Thouless, Anderson, Palmer 1977: Classal SK model) H eff ( z x ε ) σ effσ 1, σ J z σ z Loal mnma ( G m 0) ( m ) ; m m( h ) h J ε Jm m χ Envronmen of a loal mnmum (poenal landsape): Hessan: Η G m m J dagonal erms Spe λ [ H ] ρh ( λ) C (a small λ) J eff Gapless sperum (assured by margnal sably) n he whole glass phase!

44 Sof olleve modes Sperum of he Hessan Spe λ C λ Dsrbuon of resorng fores [ ] H ( ) H ρ J eff

45 Sof olleve modes Sperum of he Hessan Dsrbuon of resorng fores Spe λ [ H ] ρh ( λ) C J eff Semlasss: N olleve osllaors wh mass M ~ 1/ eff and frequeny ω λ M Mode densy ω J ( ω) C ρ eff

46 Sof olleve modes Sperum of he Hessan Dsrbuon of resorng fores Spe λ [ H ] ρh ( λ) C J eff Semlasss: N olleve osllaors wh mass M ~ 1/ eff and frequeny ω λ M Mode densy ( ω) C ρ ω J eff Connuous bah wh speral funon (n he regme of deloalzed modes!) 1 Mω ( ω) ρ( ω) ~ χ '' ω J Independen of eff! Generalzaon of known speral funon a he quanum glass ranson. [Mller, Huse (SK model); Read, Ye, Sahdev (roor models)]

47 Loalzaon of olleve modes?

48 Loalzaon of olleve modes? In 3D: Random marx J ouples every loalzed level o z >> 1 lose spaal neghbors. z

49 Loalzaon of olleve modes? Egenvalue and egenveor sperum of a random marx J (3D) Sperum of J z z <

50 Loalzaon of olleve modes? Egenvalue and -veor sperum of TAP Hessan H (3d) TAP: Sperum of Hessan z ~ N ave >>1

51 Loalzaon of olleve modes? Egenvalue and -veor sperum of TAP Hessan H (3d) TAP: Sperum of Hessan z ~ N ave >>1 Deloalzed low-energy plasmons down o E Jz 1 6 << mn ~ J

52 Summary of resuls The quanum eleron glass possesses a onnuous bah of olleve unharged exaons, (whh are beyond perurbaon heory)

53 Summary of resuls The quanum eleron glass possesses a onnuous bah of olleve unharged exaons, (whh are beyond perurbaon heory) Furher, we have heked ha: Sngle parle exaons reman very sharp a he Ferm level: Level broadenng from deay proesses (1/T 1 ) and pure dephasng (1/T ) s smaller han level spang δ.

54 Summary of resuls The quanum eleron glass possesses a onnuous bah of olleve unharged exaons, (whh are beyond perurbaon heory) Furher, we have heked ha: Sngle parle exaons reman very sharp a he Ferm level: Level broadenng from deay proesses (1/T 1 ) and pure dephasng (1/T ) s smaller han level spang δ. The sysem remans an nsulaor: ρ A fne emperaure: onduon by hoppng, smulaed by olleve eleron modes. ( T 0)

55 Boom lne: Varable range hoppng Eleron hoppng ou of loalzaon volume A olleve mode (plasmon) an provde he exa energy dfferene n a sngle eleron hop beause of he onnuous sperum of he bah. All eleron levels aqure a fne f small wdh due o her ouplng o plasmons. Hene, here s no manybody loalzaon.

56 Boom lne: Varable range hoppng Varable range hoppng σ ( T ) σ 0 T exp T d ξ 0 1 Srehed exponenal n T: Sngle elerons opmze avaon energy vs ranson probably (lengh of hops) elemenary ressors (Mller-Abrahams) Perolaon problem for he nework of ressors (Ambegaokar e al., Pollak, Shklovsk) As n phonon-asssed hoppng bu wh dfferen prefaor refleng he plasmon bah!

57 Boom lne: Varable range hoppng Varable range hoppng σ ( T ) σ 0 T exp T d ξ 0 1 Srehed exponenal n T: Sngle elerons opmze avaon energy vs ranson probably (lengh of hops) elemenary ressors (Mller-Abrahams) Perolaon problem for he nework of ressors (Ambegaokar e al., Pollak, Shklovsk) Only wo energy sales: T and T J 0 e κξ (quas d) σ α e T 0 α h T 0 0.3

58 Boom lne: Varable range hoppng Varable range hoppng σ ( T ) σ 0 T exp T d ξ 0 1 Srehed exponenal n T: Sngle elerons opmze avaon energy vs ranson probably (lengh of hops) elemenary ressors (Mller-Abrahams) Perolaon problem for he nework of ressors (Ambegaokar e al., Pollak, Shklovsk) Only wo energy sales: T and T J 0 e κξ (quas d) σ α e T 0 α h T Doped GaAs/Al x Ga 1-x As heerosruure S. I. Khondaker e al., PRB 59, 4580 (1999)

59 Many body loalzaon: Two problems: where o fnd bes? Four-fermon saerng nrodues srong quanum fluuaons Long range Coulomb neraons spol loalzaon, even a low densy Possble way ou: nsulaors wh srong superondung orrelaons (fermons bound no preformed pars), wh suppressed/sreened Coulomb neraons

60 Why o expe many body loalzaon a he SIT? Elerons are bound n loalzed pars (Anderson pseudospns) Phase volume for nelas proesses s srongly redued as ompared o he sngle eleron problem MIT Cooper hard ore repulson Coulomb

61 Why o expe many body loalzaon a he SIT? Elerons are bound n loalzed pars (Anderson pseudospns) Phase volume for nelas proesses s srongly redued as ompared o he sngle eleron problem MIT Cooper hard ore repulson Coulomb Pars: doubly ouped loalzed wavefunons (hard ore bosons) z z z H par ε σ σ σ Jσ σ (Anderson, MaLee, FegelmannIoffe) Dsorder ( nsulaor) Kne energy of pars ( superonduvy)

62 Why o expe many body loalzaon a he SIT? Elerons are bound n loalzed pars Phase volume for nelas proesses s srongly redued as ompared o he sngle eleron problem MIT Cooper hard ore repulson Coulomb T, E SC T E (olleve) Ins Coneure (for nsulaor): A T0 all exaons wh E < E are loalzed Expermenal ndaons for suh a senaro! Dsorder

63 Conlusons Model for purely eleron-asssed hoppng n nsulaors. Colleve sof modes provde a bah wh onnuous sperum and ensure energy onservaon durng a hoppng even. No manybody loalzaon expeed lose o he Meal-nsulaor ranson Possbly dfferen, and onepually very neresng suaon lose o dry superonduor-nsulaor ransons