escolaestruturaeletronica minicurso 2016 aula2luisdias/files/...ebee 2016 lds , m tiago, s ulloa, f...
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![Page 1: EscolaEstruturaEletronica Minicurso 2016 Aula2luisdias/Files/...EBEE 2016 LDS , M Tiago, S Ulloa, F Reboredo E. Dagotto PRB 80 155443 (2009).-Experimental range Spatially extended](https://reader033.vdocument.in/reader033/viewer/2022041818/5e5bf11cf861cc1599643de1/html5/thumbnails/1.jpg)
Minicourse contents:
Lecture 1:
• Intro: “More is Different”.• The Kondo effect: a true “More is Different” phenomenon.• Wilson’s numerical renormalization group method.
Lecture 1 (pdf) available at http://www.fmt.if.usp.br/~luisdias
See link on Twitter: @ProfLuisDias
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
Lecture 2:
• Applications I: Magnetic molecules on surfaces.• Applications II: Vacancies in graphene.
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Application 1:Magnetic molecules on surfaces
Combining NRG with Ab-initio methods
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
Combining NRG with Ab-initio methods
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Kondo: Magnetic molecules on surfacesV. Iancu, A. Deshpande, Saw W. HlaNano Lett. 6 820 (2006)STM measurements
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
CoCo d3z2-1 level (at ~-0.7 eV)
Zero-bias dip: Kondo effect. TK~130-170K
Co
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Cobalt Porphyrin molecule on a surface.
Co atomic configuration: 3d7
dx2-y2
d
+ D4h crystalline field:
?
� Molecule: TBrPP-Co (CoBr4N4C44H24)
� Which microscopic model describes it?
� Naively one might expect S=3/2.
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
Is the simple “Hund’s rule filling” picture correct?
d3z2-r2
dyz dxz
dxy
Co ?
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First-principles calculations (GW): hints for a microscopic model.
The answer: not quite!
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
“Co-like” levelsGW: Molecular levelsLDS, M Tiago, S Ulloa, F Reboredo E. DagottoPRB 80 155443 (2009).
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First-principles calculations (GW): hints for a microscopic model.
The answer: not quite!
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
LDS, M Tiago, S Ulloa, F Reboredo E. DagottoPRB 80 155443 (2009).
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d3z2-r2
LUMO
First-principles calculations (GW): hints for a microscopic model.
dx2-y2 dx2-y2
Model: Anderson-like Hamiltonian
U=6.3 eVEM
2 eV
S=1/2 model
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
d3z2-r2
HOMO dyz dxzdxy
dyz dxzdxy
“Co-like” levelsGW: Molecular levels
Ed
1.8 eV
Not easy to calculate with GW!
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NRG calculations (Kondo temperature).d3z2-r2
d3z2-r2
U=6.3 eV
EM
Ed1.8 eV
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016 LDS, M Tiago, S Ulloa, F Reboredo E. DagottoPRB 80 155443 (2009).
-Experimental range
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Spatially extended Kondo effect
100
120
140
160
180
200
220
Kon
do T
empe
ratu
re (K
)
Path 1 Path 2
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
� Co-porphyrin structures: low-temperature STM dI/dV.� Kondo resonance found away from central Co atom!� Spatially-dependent Kondo temperature.
-1 0 1 2 3 4 5 6 7
100
0
Kon
do T
Distance from centre (A)
UGE Perera, et al. – PRL 105 106601 (2010).
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1
2
Spatially extended Kondo effect� DFT calculations: molecular spin density nDFT(x) .� NRG calculations (Kondo model): � Good agreement with experimental data.
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016UGE Perera, et al. – PRL 105 106601 (2010).
nDFT(x)
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Application 2:Kondo effect in graphene.
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
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Graphene basics: triangular lattices.Direct lattice
Reciprocal lattice (k)
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
Reciprocal lattice (k)
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Electronic structure of graphene.
Tight-binding model
…1st and 2nd neighbours
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
…1st and 2nd neighbours
Wallace, P.R., Phys. Rev. 71 622 (1947)
Phil R. Wallace
Energy dispersion for graphene (derived in the 40’s !)
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Dirac cones: massless fermions
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
Analogy: linear dispersion of massless “relativistic” particles (“E=pc”)
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Graphene: density of states
ρρρρ(E) ∝ ∝ ∝ ∝ |E| (linear)Close to the charge neutrality point.
Wallace, P.R., Phys. Rev. 71 622 (1947)
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
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Graphene: How we think it is…
“Ideal” graphene: planar, 2D, ordered, nice…
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
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… and how it really is.
… versus “real” graphene: wavy, corrugated, membrane -like.
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
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Graphene: “long-range disorder”.
Disorder potential
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
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Short-range disorder (vacancies)
Localized state at the vacancy site
Tight-binding calculations: single vacancy leads to midgap state
Typical delocalized state
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
Vacancy tight-binding model:
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Vacancies = localized spins?
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
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Resistivity ρ(T) in irradiated graphene.
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
Jian-Hao Chen et al., Nature Phys. 7 535 (2011)
Vacancies (short-range disorder) intentionally caused by irradiation.Resistivity increases at low temperatures !! We have seen that before…
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ρρ ρρ(T
)/ρρ ρρ(
0)
U εεεε0/ΓΓΓΓ=-4εεεε0/ΓΓΓΓ=-3εεεε ΓΓΓΓ
EF
T
Kondo effect: NRG calculationsΓ=πV2ρ(EF)
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
T.A. Costi, et al., J Phys Cond Mat.6 2519 (1994).
NRG calculations: scaling with TKρ(T)
T/TK
εεεεo
εεεε0/ΓΓΓΓ=-2
ΓΓΓΓ
TK
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Kondo-like ρ(T) features in irradiated graphene
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
Jian-Hao Chen et al., Nature Phys. 7 535 (2011)
Resistivity vs Temperature measurements in disordered graphene.Vacancies (short-range disorder) intentionally caused by irradiation.Left: Kondo-like scaling in T/TK. Right: TK vs gate voltage.
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Vacancy magnetism in graphene: STM
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
Yu Zhang et al., Phys. Rev. Lett. 117, 166801 (2016)
STS spectra (right) showing Hubbard peaks with U~30 meV.
Our calculations: U~640 meV V. Miranda, LDS, C.H. Lewenkopf PRB 94 075114 (2016) – Editor’s Suggestion
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Kondo effect in graphene: a few questions.
Where does the localized (magnetic) state come from?
How does it couple to the continuous band?
R: Vacancies (=mid-gap states) V. M. Pereira et al., PRB 77 115109 (2008)
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
Does this system retain features of an Anderson model coupled to Dirac fermions?
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Mid-gap state in the presence of vacancies.
Localized state at the vacancy site
Tight-binding calculations: single vacancy leads to midgap state
Typical delocalized state
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
Agreement with previous results
Vacancy tight-binding model: V. M. Pereira et al., PRB 77 115109 (2008)
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Mid-gap state is not coupled!!Localized state at the vacancy site Typical delocali zed state
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
Localized and delocalized states are decoupled:
No Kondo possible .
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Where does the localized (magnetic) state comes from?
How does it couple to the continuous band?
R: Vacancies (=mid-gap states) V. M. Pereira et al., PRB 77 115109 (2008)
Kondo effect in graphene: a few questions.
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
Does this system retain features of an Anderson model coupled to Dirac fermions?
R1: Rippling, Jahn-Teller-like distortion
R2: Long-range disorder (this work)
M. A. Cazallila et al., arXiv 1207.3135 (2012)
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How to couple? Long-range disorder.
How to couple the localized state to the graphene band? Disorder (weak)
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
Our “basis”: localized and extended states
Projectors:
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The trick: projecting into different sectors!How to couple the localized state to the graphene band? Disorder (weak)
Extended LocalizedCoupling
Effective band density of states.
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
Coupling to the localized state
Renormalized state energy
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What about the Hubbard U?Tight-binding (single orbital) contributions to the charging energy of the localized state |0> give:
with
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
“Diagonal” “Off-diagonal”
Our estimates:
In the NRG calculations, we use:
consistent with recent DFT+cRPA calculations
M. Schüler et al., PRL 111 036601 (2013)
V. Miranda, LDS, Caio Lewenkopf PRB 94 075114 (2016)
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What about the Hubbard U?Tight-binding (single orbital) contributions to the charging energy of the localized state |0> give:
“Diagonal”
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
“Off-diagonal”
V. Miranda, LDS, Caio Lewenkopf PRB 94 075114 (2016)
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Where does the localized (magnetic) state comes from?
How does it couple to the continuous band?
R: Vacancies (=mid-gap states) V. M. Pereira et al., PRB 77 115109 (2008)
Kondo effect in graphene: a few questions.
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
Does this system retain features of an Anderson model coupled to Dirac fermions?
R1: Rippling, Jahn-Teller-like distortion
R2: Long-range disorder (this work)
M. A. Cazallila et al., arXiv 1207.3135 (2012)
NRG calculations: TB-derived Anderson model
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Kondo effect with “massless Dirac Fermions”?
εd
εd+U
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
Theory:D. Withoff and E. Fradkin, PRL 64 1835 (1990).C. Gonzalez-Buxton, K. Ingersent, PRB 57, 14254 (1998)P.S. Cornaglia et al. PRL 102 046801 (2009).M. Vojta, et al., Europhys. Lett. 90, 27006 (2010).Review: L. Fritz and M. Vojta, arXiv:1208.3113 (2012) =Quantum phase transition!Q: Do we see this in our model?
εd
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Long-range disorder model: Hybridization.
For each realization:
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
And also:
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Long-range disorder model: Hybridization.
For each realization:
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
And also:
PRB 90 201101(R) (2014)
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Disorder: model parameters distributions
For each realization:
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
And also:
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Effective Anderson model: disorder-mediated coupling.
Γdis(ω) δε+U
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
Anderson model with disorder-mediated coupling.
δε: impurity state energy; Γdis(ω): hybridization
µ(Vg): Fermi energy (gate-dependent)
µ(0): Fermi energy at charge neutrality. (∆µ=µ(Vg)-µ(0))
δε
δε+U
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Anderson model + massless Dirac fermionsAnderson impurity coupled to
a “Dirac band” with linear dispersion.
δε: impurity state energy
∆µ= µ(Vg)- µ(0)
µ(Vg): Fermi energy (gate-dependent)
“metallic” model “Pseudogap” model
δε+U∆µ
δε+U
: density of states
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
(gate-dependent)
µ(0): Fermi energy at charge neutrality
Realization of the “pseudogap Anderson model” for Vg=0.
δε∆µ
∆µ
δεΓ(ω) Γ(ω)
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Pseudogap model: quantum phase transition.
δε>δεc
δεc=-0.00253724
δε<δεc
δε
δε+Uµδεc
δε+U
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
T*(δε) (not TK)
δε
δε+U
µδεc
Also: C. Gonzalez-Buxton, K. Ingersent, PRB 57, 14254 (1998)
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Metallic x Pseudogap: crossover x QPTMetallic (“usual”) Pseudogap
Occupation
∆µ ∆µ
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
Magneticmoment(local)
Kondo
T*=TK
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Pseudogap x Disorder: NRG calculations
∆µ=0 Kondo
δε
δε+Uµδεc δε
δε+U
µδεc
LM
EO
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016PRB 90 201101(R) (2014)
QPT
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Kondo temperature distributions.
EO
EO
δε+U
∆µ
∆µ=0
TK
N(T
K),
N(T
*)
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
Kondo
Kondo
δε
Kondo
Kondo
1000 realizations
N(T
PRB 90 201101(R) (2014)
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P(TK) distributions: Griffiths phase.
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
E. Miranda and V. Dobrosavljevic, PRL 86 264 (2001)
Hallmark of a “Griffiths phase”
PRB 90 201101(R) (2014)
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In a few words…“More is Different!”
“ The behavior of large and complex aggregates of elementary particles, it turns out, is not to be understood in terms of simple extrapolation of the properties of a few particles.
Instead, at each level of complexity entirely new
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
Instead, at each level of complexity entirely new properties appear and the understanding of the new behaviors requires research which I think is as fundamental in its nature as any other.“
Phillip W. Anderson, “More is different”, Science 177 393 (1972)
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Current Group Members
Dimy NanclaresGraduate student
David Ruiz-TijerinaLuis Dias da SilvaProfessor
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
Graduate studentPost-doc (now at Manchester)
Marcos MedeirosGraduate student
Raphael LevyGraduate student
Professor
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Acknowledgements
Elbio Dagotto (Oak Ridge)Fernando Reboredo (Oak Ridge)Murilo Tiago (Oak Ridge)Caio Lewenkopf (UFF)Vladimir Miranda (UFF)
CollaboratorsCollaborators onon thethe worksworks presentedpresented
Sergio Ulloa (Ohio U,)Saw Hla (Ohio U. /Argonne)Gayani Perera (Ohio U.)Nichola Marzari (MIT)Heather Kulik (MIT)
Luis Gregório Diashttp://www.fmt.if.usp.br/~luisdias
EBEE 2016
SupportSupport:: FAPESP FAPESP (2016/18495(2016/18495--4); 4); CAPES (PNPD CAPES (PNPD programprogram), ), CNPqCNPq ((307107/2013307107/2013--2 and 2 and 449148/2014449148/2014--9); 9); USPUSP--PRP NAP QPRP NAP Q--Nano.Nano.