NCTS One-Day Workshop

Emergence of a spin-singlet resonance statewith Kondo temperature well beyond 1000K
in the proton-embedded electron gas:
New route to room-temperature superconductivity
Yasutami Takada
The Institute for Solid State Physics, The University of Tokyo
5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581
takada@issp.u-tokyo.ac.jp
Rm. 916, Condensed Matter and New Physics Building, NTU, Taipei
10:20-11:20, Friday, 16 December 2016
cf. YT, Ryo Maezono, & Kanako Yoshizawa, Phys. Rev. B 92, 155140 (2015)
Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)

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Features of This Work
(1) I have been asking myself for more than 40 years,“How can a theorist contribute to discovering a room-temperaturesuperconductor?”
 Establish a reliable framework for the first-principles calculation of the transition
temperature Tc, with which I can confirm the mechanism for superconductivity fora particular group of superconducting materials and estimate the optimum Tc forthe group. [ ex. Graphite-Intercalation Compounds, YT, JPSJ78, 013703 (2009);
YT, Reference Module in Materials Science and Materials Engineering (Elsevier,2016) doi: 10.1016/B978-0-12-803581-8.00774-8]

 New route: Exploit an empirical relation between Tc and somephysical parameter (for example, Kondo temperature TK). Look forthe system with a favorable value for the physical parameter by
first-principles calculations.
(2) Accurate quantitative determination of TK only through informationon the ground-state electron density distribution n(r)Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)2

Outline
1. Introduction
Heavy fermion superconductors:
Empirical relation between TK and Tc
Concept of metallic screening:
Paradigm shift from the Thomas-Fermi theory
2. Proton Embedded in Metals as Modeled by the Electron Gas
Transition from a positive H+ ion to a negative closed-shell H- ion

3. Emergence of a Spin-Singlet Resonance State
Appearance of Kondo resonance between H+ and H- stateswith TK ~2000K at rs=4.
4. Summary and Discussion
Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)

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Heavy Fermion Superconductors◎ Uemura Plot [cf. Y. J. Uemura, Physica B 374-375, 1 (2006)]:
Heavy-fermion superconductors
 Tc (and also TF) is low, but Tc / TF is large, about 0.04.

Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)4

Tc vs TK◎ Some of the heavy fermion superconductors are considered as
Kondo-lattice systems andstrong correlation exists between
Tc and the Kondo temperature TK.
 Spin-fluctuation mechanism ofsuperconductivity in the Kondo
lattice; Tc / TK ~ 0.1.
Idea: “If we can find a very high-TK system, we can obtain a very high-Tc
superconductor by suitably synthesizing its Kondo lattice.” Making all energy scales up!
 Recent Pu (Z=94)-based superconductors seem to realize this idea.

Let us pursue this idea by looking for a new Kondo system with TK>1000K,
leading to room-temperature superconductivity.
 Theoretical evaluation of TK for an atom embedded in metals
Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)

5Charged Point Impurity in a Metal◎ Put an impurity of charge +Z in a metal with its density
n or the rs parameter, defined by rs=(3/4n)1/3+Zrs=(3/4n)1/3+Zrs⦿ Conduction electrons respond to screen the charge, as described by
the linear-response theory (Debye-Hückel or Thomas-Fermi)
 Charge resonance of metallic electrons and their Friedel oscillation
⦿ Basic Question:
If Z is small enough, this is always a valid concept, but
if Z becomes large, the effects of nonlinear response become
important. Then, what about the effect of spin resonance?
This spin effect is enhanced for an odd positive integer Z.
Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)

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Proton-Embedded Electron Gas◎ Case of a proton (Z=1) embedded in the electron gas

Compare the energy scales of the electron gas and the H atom:(i) rs<2:(ii) rs>10“Charge resonance state”“H- ion confinement state”
screening bare Z=1 charge screening composite Z= 1 charge What is the state in between?
Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)

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Kondo Spin-Singlet Resonance State◎ Case of A<F<1s:○ Naively, we may imagine a single-electron localized statewith either up or down spin.  Spin polarized!
○ Because of Kondo physics, this cannot be a true ground state.
Rather, a spin-singlet resonance state is expected in which
the conduction electrons at the Fermi level hybridize withthe electron in the 1s level with changing spin orientation.

○ Here the spin-screening length K is long, i.e., K >>kF-1.
cf. The Thomas-Fermi screening length -1
in the charge resonance state is short, i.e., -1 ~ kF-1.
Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)

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Purpose of Research○ Usually, Kondo physics is discussed in the impurity Andersonmodel,
but in our many-electron system with a proton at the origin ofcoordinates,(i) no predetermined localized d state exists;(ii) the long-range Coulomb interaction works even among
conduction electrons.
 Must confirm the existence of the Kondo spin-singlet statein the actual proton embedded electron gas.
○ Because the long-range Coulomb interaction is the origin forthe charge resonance state, competition between charge andspin resonances is also an important issue to investigate.
Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)

9Strategy for Confirmation
Our basic strategy:
We focus on the ground-state electron density n(r)which will be calculated by both DMC and LSDA.
cf. ○ DMC can treat systems with a finite number of electrons.
○ In principle, it is proved that DFT can provide the exactn(r) even for strongly-correlated systems.

○ Check n(r) in LSDA at finite N in comparison with DMC.
○ Obtain n(r) in either LSDA at N∞ or extrapolation in DMC.
○ Check the obtained n(r) at N∞ reproducing the behavior

characteristic to the Kondo spin-singlet resonance state, or“the anomalous (shortened-period) Friedel oscillation”.
Note that TK is quantitatively determined only throughthis anomalous Friedel oscillation or n(r)!
Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)

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LSDA
LSDA Calculation

M. Shimomoto & K. Yoshizawa (ISSP)

n0: the uniform density of
the host electron gas

Immersion Energy

Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)

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VMC
VMC＋Diffusion Monte Carlo Ryo Maezono (JAIST)(1) Variational Monte Carlo (VMC)
N-electron systems confined in a sphere of R (in which theembedded atom is situated at the center of the sphere) withthe fixed boundary condition at the surface.

R = (N-1)1/3rs

Slater-Jastrow trial function for VMC

Jastrow factor

Kohn-Sham basetwo-electron electron-proton
correlation
correlation

two-electron-proton
correlation

Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)

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DMC
VMC＋Diffusion Monte Carlo Ryo Maezono (JAIST)(2) Diffusion Monte Carlo (DMC)
Improve on the variational wave function  by DMC into .

Fixed-node DMC to avoid the fermion sign problem N-independent quantities are supposed to be independent of
the position of fixed nodes, because the node-positions varywith R or N for a given value of rs.

R = (N-1)1/3rs

Casino code developed by Needs’ group at Cambridge as well as the codesdeveloped by C. Umrigar et al. at Cornell.
Calculated mostly at N=58 or 60 (which has been the case for this kind ofsystems, including Ceperley’s group at Illinois), but for checking size effects,
N is increased up to132 and 170 in the present work.
Expectation value:
Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)

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Comparison between LSDA & DMC○ No stable spin-polarizedstate is obtained in either
LSDA and DMC.
Thus LSDA gives thesame results as thesimple LDA.
○ Rather strong N
dependence is seen.
cf. This should be, aslong as R is smaller than
K.
○ At each N, DMC andLSDA agree surprisingly
well in the whole rangeof r, though VMC
deviates much fromboth.
Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)

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Cusp Theorem and n(0)○ For r ~ 0, the cusp theorem around the proton (Z=1) predicts:n(r) ~ n(0) exp(-2Zr) , giving the cusp line in the semi-log plot.

○ From the cusp line, we can determine n(0) accurately. For rs=4,n(0)=0.388±0.004 in atomic units in DMC, while it is 0.369 inLSDA. Note that it is 0.329 for an isolated H- ion.
○ In the vicinity of r=0, N=60 is already large enough to obtainthe convergent results for n(r) in the limit of N∞ .
Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)15

Two Sequential Transitions○ Plot as a function of rs○ Plot of the transitions in (rs,N) space

Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)

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“Kondo Resonance State” in Finite Systems
In DMC, we find that the results converge well at N=58 if rs is
less than 2.6, while the convergence is obtained at N=60 forlarger rs; we have checked the sequence of the occupied levelsin the finite jellium sphere to find the following:
rs=1 (N=58) rs=4 (N=60) rs=4 (N=170)
4d4d5d3s11n9l6h6h6g------ Fermi level ----------------------3s4s
At high densities, the state without ns
5g
3p4p
3p5g8korbital is stabilized, but for low
4f
4f
5f
2s
3d7idensities, the state with ns orbital near
3d2s
6h
2p
2p
4d
the Fermi level is stabilized, indicating
1s
1s5g
3s
that the ns state turns into the Kondo
3p
4f
resonance one in the bulk system.
3d2s
 “The Kondo resonance state” is2pthought to appear at rs=2.6 at N=60!
1sProton-Embedded Electron Gas as a Novel High-T_K System (Takada)

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Friedel Oscillation○ For metals without the Kondo resonance state:↑Consider the s-wave part. ↓ Affleck, Borda & Saleur, PRB77, 180404(R) (2008)○ For metals with the Kondo resonance state:

Put F(r/K) here.

In accordance with the behaviorof the universal function F(x), (r)= for r≪K, changing into (r)= for r≫Kcf. Without the Kondo resonancestate, F ≡ 1   (r) ≡ 0.
Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)18

Anomalous Friedel Oscillation○  ris a monotonically increasing function of r over 

Write ri as ith zero of n(r)-n0:2kFri+ (ri) = i + /2
Write ri+1 as (i+1)th zero of n(r)-n0: 2kFri+1+ (ri+1) = (i+1)+ /2

 ri+1 - ri = - [ (ri+1) -  (ri)]/}/2kF < /2kF

Shortening of the half period of the Friedel
oscillation by the factor of [(ri+1) - (ri)]/ !
An example of the shortening
factor plotted as a function of
0 for various kFK.

First few shortening
factors are enough touniquely determine
0 and kFK.
Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)

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Oscillation Behavior in DMC○ Size correction in DMC:n(r) = n(r)Z=1,N – (r) n(r)Z=0,N-2 + n0
with = [(N-2)/(N-1)]1/3 and(r) = ln[nZ=0,N-1LSDA(r)/nZ=0,N-2LSDA(r)]/ln

○ For r < 3, oscillation amplitudes
are converged.
○ For r < 13, oscillation nodes areconverged.
Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)

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Shortening of the Oscillation Period in DMC○ The node positions are:2kFr1/= 0.994±0.005
2kFr2/= 1.970±0.008
2kFr3/= 2.949±0.013
2kFr4/= 3.927±0.019○ The corresponding shortening factors [i+1-i]/ are
0.024, 0.021, and 0.022 for i=1, 2, and 3, respectively.
○ These factors are consistent with the choice of kFK~35
and 0 =0.43.

○ The Kondo temperature TK ~ (2/kFK)EF is estimatedas 2100K in DMC with about 10% error at rs=4.
Thus we cannot expect to observe the lnT anomaly
in the usual experiments.
Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)

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Shortening of the Oscillation Period in LSDA
○ 0 =0.07 in LSDA
The first shortening factor
is 0.0133 and all the others
are less than 0.004.
cf. s(kF)=0+/2=0.507
○ TK is estimated as 1900K
in LSDA at rs=4.
○ There appears a very shallow (extending about r ~ 80 or kFr ~ 38)1s bound statein LSDA, but this is not a real physical state, but it appears only to correctly
reproduce n(r) in the Kondo singlet state.
○ The shortening of the oscillation period appears as long as this shallow-state
density nL(r) exists, as seen by comparison with the oscillation induced by thepoint charge of Z 1 at the origin.
rsB(K)

2.07
97

3
2200

43600

63500

Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)

82200

10720
22

Summary
10 Two sequential transitions in the ground state withintervening Kondo spin-singlet resonance state areclarified and identified for a single proton embeddedin metals:

20 This result enriches the century-long paradigm formetallic screening to a point charge of Z in the sense
that, depending on the metallic density and Z, spinresonance takes the place of charge resonance.
cf. YT, R. Maezono & K. Yoshizawa,Phys. Rev. B 92, 155140 (2015)
Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)

23Discussion
10 By referring to heavy-fermion superconductors and by considering
astonishingly high TK in the proton-embedded metal at intermediate
densities, we may expect room-temperature superconductivity in a
metallic hydrogen alloy at ambient pressure, if the electron density
is appropriately given and the quantum criticality condition in theDoniach phase diagram is satisfied.
3 < rs < 8H
rs=(3/4n)1/3
Single-hydrogen Kondo system

Hydrogen Kondo latticecf. H2 in metals: Bonev-Ashcroft, PRB64, 224112 (2001)
HHr○ modified concept of chemical bonding; anomaly for rs>3.2○ related to calculating the RKKY interaction
Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)24

Discussion
20 In the solid metallic hydrogen and surfur hydrides (H2S and H3S)
at high pressures (~150-500GPa), the rs parameter is typically 1.4.
cf. Ashcroft, PRL21, 1748 (1968); PRL92, 187002 (2004);
Drozdov, Eremets and Troyan, Nature 525,73 (2015).
30 Metal hydrides, usually used for secondary battery, may be a goodcandidate, though currently it seems that the rs parameters are toosmall.
cf. MgH2, VH2, ZrH2, TixZr1-xHy, Mg2NiH4, FeTiH1.95, LaNi5H6.7,
ZrMn2H3.6, ZrCoH3, YH3, etc.

Proton-Embedded Electron Gas as a Novel High-T_K System (Takada)25