How to model holes doped into a cuprate layer

Transcription

1 How to model holes doped into a cuprate layer Mona Berciu University of British Columbia With: George Sawatzky and Bayo Lau Hadi Ebrahimnejad, Mirko Moller, and Clemens Adolphs Stewart Blusson Institute

2 Why holes in a CuO2 layer? à cuprates = family of high-temperature superconductors (high-tc record) à layered compounds, common feature are CuO2 layers believed to host superconductivity upon doping Cu O

3 Temperature AFM Pseudogap Unusual metal SC Fermi metal hole density

4 Temperature AFM Pseudogap Unusual metal SC Fermi metal hole density parent compound = Mott insulator with AFM order

5 Why holes in a CuO2 layer? à cuprates = family of high-temperature superconductors (high-tc record) à layered compounds, common feature are CuO2 layers believed to host superconductivity upon doping à origin of pairing still not resolved, although it is widely believed to have something to do with the AFM order

6 First step: understand what happens when one hole is added

7 First step: understand what happens when one hole is added Temperature AFM Pseudogap Unusual metal SC Fermi metal hole density Today

8 How to model this problem? à parent compound: each Cu has one hole in 3dx2-y2 orbital, while O 2p are full unpaired spin,afm order

9 How to model this problem? à parent compound: each Cu has one hole in 3dx2-y2 orbital, while O 2p are full à if we dope, we remove some spins (very simplistic view, will be revisited)

10 How to model this problem? à parent compound: each Cu has one hole in 3dx2-y2 orbital, while O 2p are full à if we dope, we remove some spins (very simplistic view, will be revisited) U Use Hubbard or t-j model H = t X c i, c j + h.c. + U X ˆn i"ˆn i# hi,ji i 2 t! P 4 t X 3 c i, c j + h.c. 5 P + J X ~S i ~S j hi,ji hi,ji

11 Hole in an AFM background: the t-j model results à could use ED or some flavour of MC, but not so good for gaining intuition à instead, use a variational method not exact but we can understand the important phenomenology because we can identify the key aspects + we can study infinite layers + generalize to few holes M. Berciu, PRL 107, (2011);

12 Hole in an AFM background: the t-j model results à could use ED or some flavour of MC, but not so good for gaining intuition à instead, use a variational method not exact but we can understand the important phenomenology because we can identify the key aspects + we can study infinite layers + generalize to few holes M. Berciu, PRL 107, (2011); à Challenge: the 2D AFM is a very complicated state because of the quantum spin fluctuations ~S i ~S j = S z i S z j S + i S j + S i S+ j "# $ #"

13 Hole in an AFM background: the t-j model results à could use ED or some flavour of MC, but not so good for gaining intuition à instead, use a variational method not exact but we can understand the important phenomenology because we can identify the key aspects + we can study infinite layers + generalize to few holes M. Berciu, PRL 107, (2011); à Challenge: the 2D AFM is a very complicated state because of the quantum spin fluctuations à Two-step solution: 1. ignore background spin fluctuations, ie use a semi-classical Neel state 2. allow for local spin fluctuations only in the vicinity of the doped hole

14 Hole in the t-jz model (spin fluctuations not allowed) à Common wisdom : the hole is confined by the string of wrongly oriented spins (= magnons) generated through its motion

15 Hole in the t-jz model (spin fluctuations not allowed) à Common wisdom : the hole is confined by the string of wrongly oriented spins (= magnons) generated through its motion

16 Hole in the t-jz model (spin fluctuations not allowed) à Common wisdom : the hole is confined by the string of wrongly oriented spins (= magnons) generated through its motion

17 Hole in the t-jz model (spin fluctuations not allowed) à Common wisdom : the hole is confined by the string of wrongly oriented spins (= magnons) generated through its motion

18 Hole in the t-jz model (spin fluctuations not allowed) à Common wisdom : the hole is confined by the string of wrongly oriented spins (, magnons) generated through its motion à Common wisdom is wrong: the hole can move using Trugman loops

19 Hole in the t-jz model (spin fluctuations not allowed) à Common wisdom : the hole is confined by the string of wrongly oriented spins (, magnons) generated through its motion à Common wisdom is wrong: the hole can move using Trugman loops

20 Hole in the t-jz model (spin fluctuations not allowed) à Common wisdom : the hole is confined by the string of wrongly oriented spins (, magnons) generated through its motion à Common wisdom is wrong: the hole can move using Trugman loops

21 Hole in the t-jz model (spin fluctuations not allowed) à Common wisdom : the hole is confined by the string of wrongly oriented spins (, magnons) generated through its motion à Common wisdom is wrong: the hole can move using Trugman loops

22 Hole in the t-jz model (spin fluctuations not allowed) à Common wisdom : the hole is confined by the string of wrongly oriented spins (, magnons) generated through its motion à Common wisdom is wrong: the hole can move using Trugman loops

23 Hole in the t-jz model (spin fluctuations not allowed) à Common wisdom : the hole is confined by the string of wrongly oriented spins (, magnons) generated through its motion à Common wisdom is wrong: the hole can move using Trugman loops

24 Hole in the t-jz model (spin fluctuations not allowed) à Common wisdom : the hole is confined by the string of wrongly oriented spins (, magnons) generated through its motion à Common wisdom is wrong: the hole can move using Trugman loops

25 Hole in the t-jz model (spin fluctuations not allowed) à Common wisdom : the hole is confined by the string of wrongly oriented spins (, magnons) generated through its motion à Common wisdom is wrong: the hole can move using Trugman loops

26 Hole in the t-jz model (spin fluctuations not allowed) à Common wisdom : the hole is confined by the string of wrongly oriented spins (, magnons) generated through its motion à Common wisdom is wrong: the hole can move using Trugman loops à Effective 2nd nn hopping (and 3rd nn) are dynamically generated through these loops, so the hole can move anywhere on its sublattice à What is the resulting dispersion?

27 Hole in the t-jz model à very heavy qp M. Berciu and H. Fehske, PRB 84, (2011) J. Riera and E. Dagotto, PRB 47, 15346(R) (1993)

28 Hole in the t-jz model à very heavy qp + wrong dispersion à In LR-AFM, qp dispersion MUST be E qp (k) =4t 2 cos k x cos k y +2t 3 [cos(2k x ) + cos(2k y )] à For Trugman loops t2 >> t3 ; then the curve has a saddle point at (p/2,p/2)

29 Can we measure the dispersion of one hole in parent compound? à YES, using ARPES B.O. Wells et al, PRL 74, 964 (1995)

30 Hole in the t-jz model à very heavy qp + wrong dispersion à In LR-AFM, qp dispersion MUST be E qp (k) =4t 2 cos k x cos k y +2t 3 [cos(2k x ) + cos(2k y )] à For Trugman loops t2 >> t3 ; then the curve has a saddle point at (p/2,p/2)

31 Hole in the t-j model (with local spin fluctuations) à Common wisdom: spin fluctuations can remove pairs of wrongly oriented spins so mass should be much lower

32 Hole in the t-j model (with local spin fluctuations) à Common wisdom: spin fluctuations can remove pairs of wrongly oriented spins so mass should be much lower

33 Hole in the t-j model (with local spin fluctuations) à Common wisdom: spin fluctuations can remove pairs of wrongly oriented spins so mass should be much lower

34 Hole in the t-j model (with local spin fluctuations) à Common wisdom: spin fluctuations can remove pairs of wrongly oriented spins so mass should be much lower

35 Hole in the t-j model (with local spin fluctuations) à Common wisdom: spin fluctuations can remove pairs of wrongly oriented spins so mass should be much lower

36 Hole in the t-j model à much lighter qp, but shape of dispersion is still wrong! ED: PW Leung, RJ Gooding, PRB 52 R15711 (1995)

37 Hole in the t-j model à much lighter qp, but shape of dispersion is still wrong! ED: PW Leung, RJ Gooding, PRB 52 R15711 (1995) à In LR-AFM, qp dispersion MUST be E qp (k) =4t 2 cos k x cos k y +2t 3 [cos(2k x ) + cos(2k y )] à For spin fluctuations t2=2t3; then the curve is flat along (0,p)-(p,0)

38 How to fix the wrong shape? à add longer range hopping

39 Hole in the t-t -t -J model à interplay of spin fluctuations + longer range hopping correct dispersion emerges ED: PW Leung et al, PRB 56, 6320 (1997)

40 Hole in the t-t -t -J model à interplay of spin fluctuations + longer range hopping correct dispersion emerges ED: PW Leung et al, PRB 56, 6320 (1997) but why?!

41 Hole in the t-t -t -Jz model (without spin fluctuations) à Longer range hopping should open additional options à Hole can propagate freely, so t2 = t, t3= t?

42 Hole in the t-t -t -Jz model (without spin fluctuations) à Longer range hopping should open additional options à Hole can propagate freely, so t2 = t, t3= t?

43 Hole in the t-t -t -Jz model (without spin fluctuations) à Longer range hopping should open additional options

44 Hole in the t-t -t -Jz model (without spin fluctuations) à Longer range hopping should open additional options

45 Hole in the t-t -t -Jz model (without spin fluctuations) à Longer range hopping should open additional options

46 Hole in the t-t -t -Jz model (without spin fluctuations) à Longer range hopping should open additional options

47 Hole in the t-t -t -Jz model (without spin fluctuations) à Longer range hopping should open additional options

48 Hole in the t-t -t -Jz model (without spin fluctuations) à Longer range hopping should open additional options

49 Hole in the t-t -t -Jz model (without spin fluctuations) à Longer range hopping should open additional options à Such processes also contribute, so t2 = t ( 1+.)

50 Hole in the t-t -t -Jz model à The longer range hopping integrals get renormalized by similar amounts! à t2/t3 = t /t (= -1.5) à In LR-AFM, qp dispersion MUST be E qp (k) =4t 2 cos k x cos k y +2t 3 [cos(2k x ) + cos(2k y )]

51 Hole in the t-t -t -J model à interplay of spin fluctuations + longer range hopping correct dispersion emerges à isotropic dispersion around GS is an accident! On (0,0)-(p,p) bandwidth set by J On (p,0)-(0,p) bandwidth set by t One needs to chose t =J to get isotropy H. Ebrahimnejad, G. Sawatzky and M. Berciu, JPhys:CondMat 28, (2016)

52 Hole in the t-t -t -J model à dispersion agrees with that measured experimentally

53 Hole in the t-t -t -J model à dispersion agrees with that measured experimentally à so problem solved, move on!

54 Hole in the t-t -t -J model à dispersion agrees with that measured experimentally à so problem solved, move on! à or maybe not, maybe we get the right curve for the wrong reasons? (this is not a robust result!)

55 Hole in the t-t -t -J model à dispersion agrees with that measured experimentally à so problem solved, move on! à or maybe not, maybe we get the right curve for the wrong reasons? (this is not a robust result!) à How to check?! Go back to the model

56 What really happens when we dope the cuprate layer? à not Mott insulators, but instead a charge-transfer insulator! à doped holes go into the O band, not onto Cu orbitals à doped system described (minimally) by three-band Emery model VJ Emery, PRL 58, 2794 (1987) H = T pd +T pp + Δ n i +U dd n i n i + U pp n i n i +... i O i Cu i O O 2p y t pd = 1.3eV, t pp = 0.65 ev, Δ pd =3.6 ev, U pp = 4 ev, U dd = 8 ev Cu 3d x y 2 2 O 2p x

57 What really happens when we dope the cuprate layer? à not Mott insulators, but instead a charge-transfer insulator! à doped holes go into the O band, not onto Cu orbitals à doped system described (minimally) by three-band Emery model VJ Emery, PRL 58, 2794 (1987) à why should we expect to be able to continue to ignore the O and use a one-band model instead?

58 The Zhang-Rice singlet à F. C. Zhang and T. M. Rice, Phys. Rev. B 37, 3759(R) (1988) ( ) (S P ) iσ = 1 1 l p 2 lσ (5) l i (not orthogonal to each other!) φ iσ = 1 N k e ikr i P kσ (7) P kσ = 1 N β k e ikr i P iσ β k = i 1 (S ) (8) ( ) cos(k x ) + cos(k y ) Zhang-Rice singlet: ψ i = 1 ( 2 φ i d i φ i d i ) (10) k 0 (9)

59 The Zhang-Rice singlet (ZRS) à F. C. Zhang and T. M. Rice, Phys. Rev. B 37, 3759(R) (1988) à doped hole goes into x 2 -y 2 linear combination of O orbitals and locks into a singlet with the hole on the central Cu (spins on Cu sites!) à Zhang and Rice argued that ZRS dynamics is described by the t-j model. à Can the ZRS = a composite object that combines both charge and spin degrees of freedom, properly describe the low-energy properties of the three-band Emery model?

60 The Zhang-Rice singlet (ZRS) à F. C. Zhang and T. M. Rice, Phys. Rev. B 37, 3759(R) (1988) à doped hole goes into x 2 -y 2 linear combination of O orbitals and locks into a singlet with the hole on the central Cu (spins on Cu sites!) à Zhang and Rice argued that ZRS dynamics is described by the t-j model. à Can the ZRS = a composite object that combines both charge and spin degrees of freedom, properly describe the low-energy properties of the three-band Emery model? à Direct comparison beyond our abilities to perform. à What we can do is compare to an (intermediary) model where we have spins on the Cu sites, but doped hole is on O sublattice

61 (Simplified) three-band model à strongly-correlated limit of the Emery model, with spins at Cu sites and doped holes on the O àin between Emery model and one-band models à For a single doped hole the Hamiltonian is (for more holes, there are additional terms): B. Lau, M. Berciu and G. A. Sawatzky, PRL 106, (2011)

62 (Simplified) three-band model à strongly-correlated limit of the Emery model, with spins at Cu sites and doped holes on the O àin between Emery model and one-band models à For a single doped hole the Hamiltonian is (for more holes, there are additional terms): B. Lau, M. Berciu and G. A. Sawatzky, PRL 106, (2011) J dd ~ mev = unit of energy from now on

63 (Simplified) three-band model à strongly-correlated limit of the Emery model, with spins at Cu sites and doped holes on the O àin between Emery model and one-band models à For a single doped hole the Hamiltonian is (for more holes, there are additional terms): B. Lau, M. Berciu and G. A. Sawatzky, PRL 106, (2011)

64 (Simplified) three-band model à strongly-correlated limit of the Emery model, with spins at Cu sites and doped holes on the O àin between Emery model and one-band models à For a single doped hole the Hamiltonian is (for more holes, there are additional terms): B. Lau, M. Berciu and G. A. Sawatzky, PRL 106, (2011) t pp = 4.1, t pp =2.4

65 (Simplified) three-band model à strongly-correlated limit of the Emery model, with spins at Cu sites and doped holes on the O àin between Emery model and one-band models à For a single doped hole the Hamiltonian is (for more holes, there are additional terms): B. Lau, M. Berciu and G. A. Sawatzky, PRL 106, (2011)

66 (Simplified) three-band model à strongly-correlated limit of the Emery model, with spins at Cu sites and doped holes on the O àin between Emery model and one-band models à For a single doped hole the Hamiltonian is (for more holes, there are additional terms): B. Lau, M. Berciu and G. A. Sawatzky, PRL 106, (2011) t sw = t 0 sw = t2 pd =3.0

67 (Simplified) three-band model à strongly-correlated limit of the Emery model, with spins at Cu sites and doped holes on the O àin between Emery model and one-band models à For a single doped hole the Hamiltonian is (for more holes, there are additional terms): B. Lau, M. Berciu and G. A. Sawatzky, PRL 106, (2011)

68 (Simplified) three-band model à strongly-correlated limit of the Emery model, with spins at Cu sites and doped holes on the O àin between Emery model and one-band models à For a single doped hole the Hamiltonian is (for more holes, there are additional terms): B. Lau, M. Berciu and G. A. Sawatzky, PRL 106, (2011) J pd = 2t2 pd U pp + =2.8

69 (Simplified) three-band model à strongly-correlated limit of the Emery model, with spins at Cu sites and doped holes on the O àin between Emery model and one-band models à For a single doped hole the Hamiltonian is (for more holes, there are additional terms): B. Lau, M. Berciu and G. A. Sawatzky, PRL 106, (2011) J dd =1, t pp = 4.1, t pp =2.4, t sw =3.0, J pd =2.8

70 Hole in the (simplified) three-band model ED results à exact diagonalization for 1 hole on clusters of 32 Cu + 64 O sites: Bayo Lau, Mona Berciu and George A. Sawatzky, PRL 106, (2011) Good agreement! Problem solved, no?

71 Hole in the (simplified) three-band model ED results à exact diagonalization for 1 hole on clusters of 32 Cu + 64 O sites: Bayo Lau, Mona Berciu and George A. Sawatzky, PRL 106, (2011) Problem: FM correlations near hole and other facts hard (for us) to explain Bonus: a variational approach might work!

72 Hole in the (simplified) three-band model à Step 1: ignore spin fluctuations (Neel background) à Qp dispersion has the deep isotropic minima around (π/2, π/2) appear even for n m =0! à For nm=3, good quantitative agreement with Bayo s ED

73 Hole in the (simplified) three-band model à Step 1: ignore spin fluctuations (Neel background) à Qp dispersion has the deep isotropic minima around (π/2, π/2) appear even for n m =0! à For nm=3, good quantitative agreement with Bayo s ED à WHAT?!

74 Hole in the (simplified) three-band model à Step 1: ignore spin fluctuations (Neel background) à Qp dispersion has the deep isotropic minima around (π/2, π/2) appear even for n m =0! à For nm=3, good quantitative agreement with Bayo s ED à WHAT?! à Spin fluctuations have no influence on qp dispersion in this model! H. Ebrahimnejad, G. Sawatzky and M. Berciu, Nature Physics 10, 951 (2014)

75 Hole in the (simplified) three-band model à Step 2: add local spin fluctuations à Basically no effect à Spin fluctuations have no influence on qp dispersion in this model! H. Ebrahimnejad, G. Sawatzky and M. Berciu, Nature Physics 10, 951 (2014)

76 What is the physical reason for this qualitative difference? à In the three-band model, the hole can move freely on the O sublattice without changing the spin configuration. The magnons in the qp cloud are created and absorbed by the hole through T swap and J pd processes. This is done on a faster time-scale than 1/J dd of spin fluctuations, so they have no time to act and to influence the dynamics of the hole. à One-band models have very different physics (although same dispersion!) à Longer range t, t hopping needed to mimic the role of Tpp à But these are taken to be comparable with J, not much larger than it à spin fluctuations are fast enough to significantly affect the structure of the qp cloud and therefore the qp dynamics.

77 Conclusions à we find that the (simplified) three-band model describes very different physics than the one-band t-j/hubbard models à we believe that this is because the one-band models are way too simplified (many terms are ignored) à this needs to be tested further à warning: different models can give a similar prediction for a given quantity for very different reasons

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