Sunday, February 17, 13 - European Commission · 2016-06-03 · Electrons drawn tightly together by...

Sunday, February 17, 13

Actinide Electron Physics: A new frontier of magnetism and superconductivity

Piers Coleman (Rutgers, USA, Royal Holloway, UK)

AAAS, E.Commission

Boston Feb 17, 2013




AAAS, E.Commission

Boston Feb 17, 2013

PuCoGa5 : 20 K Superconductor




AAAS, E.Commission

Boston Feb 17, 2013


UBe13


•Fruit-fly of 21st C.•From the nucleus to the periodic table.•Magnetism and Superconductivity•A new convergence•Mysteries of magnetic pairing.•Hidden Order




Fruit-Fly : 20th C.


Fruit-Fly : 20th C.


Fruit-Fly : 20th C.


Fruit-Fly : 20th C.

Fruit-Fly of the 21st CSunday, February 17, 13

Fruit-Fly : 20th C.


Actinide Electron Physics


Fruit-Fly : 20th C.nm μm




�Fruit-Fly : 20th C.

nm μmQUANTUMEMERGENCE






Fruit-Fly of the 21st C


Fruit-Fly : 20th C.

�




Fruit-Fly of the 21st C


Fruit-Fly : 20th C.

�



H =

✓ 1

2

◆



H =

✓ 1

2

◆

Å μm



H =

✓ 1

2

◆

Atom

Å μm



H =

✓ 1

2

◆

Atom Cooper pair

Å μm



H =

✓ 1

2

◆

Atom Cooper pair

Å μmLife


Mycoplasma mycoides

250nm


H =

✓ 1

2

◆

Atom Cooper pair

Å μmLife

�


http://mgl.scripps.edu/people/goodsell/illustration/mycoplasma

Mycoplasma mycoides

250nm


H =

✓ 1

2

◆

Atom Cooper pair

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�

While we understand most of the physicsat the scale of a nanometer, the emergentproperties that develop between the nanometer and the micron are only just beginning to be understood.

Actinide electron physics is playing an important role in exploring this new frontier.





+

++

+

++

Electron physics of the actinides


+

++

+

++

Electron physics of the actinidesConventional Nucleus Z = no protons small


+

++

+

++


+Ze

electron cloud

-

-

--

-

-


+

++

+

++


Electrons far apart:weakly interacting

+Ze

-

-

electron

electron

-

-

--

-

-


+

++

+

++

+

+

+ +

++

+

++

+

+

+


Actinide Nucleus Z~ 90 HUGE!


+Ze

-

-

electron

electron

-

-

--

-

-


+

++

+

++ +

+

+ +

++

+

++

+

+

+




+Ze +Ze

-

-

electron

electron

-

-

--

-

- -

-

--

-

-

-

-

--

-

-

Electrons drawn tightly together by nucleus:


+

++

+

++ +

+

+ +

++

+

++

+

+

+




-

-+Ze +Ze

-

-

electron

electron

-

-

--

-

- -

-

--

-

-

-

-

--

-

-

Electrons drawn tightly together by nucleus:strongly interacting


+

++

+

++ +

+

+ +

++

+

++

+

+

+




-

-+Ze +Ze

-

-

electron

electron

-

-

--

-

- -

-

--

-

-

-

-

--

-

-


prone to correlated behavior



-

-




+Ze



-

-




One effect is that the electrons tend to localize, forming large magnetic moments

Nd/Fe Magnet Z=60



-

-





But when the quantum mechanical jostlingsof the electrons are too large, magnetismmelts and new kinds of order, with newkinds of properties, may develop.



-

-






Composite pairedsuperconductor PuCoGa5



-

-






Composite pairedsuperconductor PuCoGa5


Rare Earth

Transition metal

Actinide

4f

3d

5fIncreasing localization

FIGURE 1. Depicting localized 4 f , 5 f and 3d atomic wavefunctions.

represented by a single, neutral spin operator

�S =h

2�σ

where �σ denotes the Pauli matrices of the localized electron. Localized moments de-

velop within highly localized atomic wavefunctions. The most severely localized wave-

functions in nature occur inside the partially filled 4 f shell of rare earth compounds

(Fig. 1) such as cerium (Ce) or Ytterbium (Yb). Local moment formation also occurs

in the localized 5 f levels of actinide atoms as uranium and the slightly more delocal-

ized 3d levels of first row transition metals(Fig. 1). Localized moments are the origin

of magnetism in insulators, and in metals their interaction with the mobile charge car-

riers profoundly changes the nature of the metallic state via a mechanism known as the“Kondo effect”.

In the past decade, the physics of local moment formation has also reappeared in

connection with quantum dots, where it gives rise to the Coulomb blockade phenomenon

and the non-equilibrium Kondo effect.

Smith and Kmetko (1983)

Increasing localization


Rare Earth

Transition metal

Actinide

4f

3d




�S =h

2�σ















Rare Earth

Transition metal

Actinide

4f

3d




�S =h

2�σ















Rare Earth

Transition metal

Actinide

4f

3d




�S =h

2�σ












Smith and Kmetko (1983)Increasing localization



Rare Earth

Transition metal

Actinide

4f

3d




�S =h

2�σ















Rare Earth

Transition metal

Actinide

4f

3d




�S =h

2�σ














Low temperatureSuperconductors




New quantum ground-states at the brink of localization.






Magnetism.






Magnetism.







Cuprates Tc=11-135K





High Temperature Superconductivity


Cuprates Tc=11-135K





High Temperature Superconductivity


Iron based High Tc Tc= 6 - 53 ++ ? K







Actinide and Rare Earth Materials. Superconductors, Magnets, Hidden Order.Tc=0.2 -18.5 K

Local

NpAl2Pd5







Local

NpAl2Pd5





High quality crystals, highly tunable,



Local

NpAl2Pd5





High quality crystals, highly tunable, Gateway to quantum materials of the future


Superconductivity and MagnetismLong thought to be separate.


Magnetism


Magnetism


M = order parameter

Magnetism


1911: the discovery

Heike Kammerlingh Onnes(1853-1926)


1911: the discovery

H. K. Onnes, Commun. Phys. Lab.12,120, (1911)



1911: the discovery

“Mercury has passed into a new state, which on account of its extraordinary electrical properties may be called thesuperconductive state.”

H. K. Onnes, Commun. Phys. Lab.12,120, (1911)



Walther Meißner1882 - 1974

Robert Ochsenfeld1901 - 1993

Meissner effect (1934)

Superconductors expelmagnetic fields.


Walther Meißner1882 - 1974

Robert Ochsenfeld1901 - 1993

Meissner effect (1934)

Superconductors expelmagnetic fields.


Superconductivity and MagnetismA new convergence.


1906 Weiss FM

Fron

tier o

f Res

earc

h

Rese

arch

Are

a

1911 OnnesHgDiscovery of SC

1933Meissner Effect

1979CeCu2Si2(d-wave sc)

1986Cuprate SCHigh Tc(d-wave)

AFM Spin Flucs

AttractiveForce Origin

RepulsiveForce Origin

Unconventional SC(Pairing induced by repulsive interaction)

1976CeAl3: Kondo Lattice

2008Iron SCHigh Tc(s+-)

2001Ce/Pu 115

SC

2007NpPd2Al3SC

Year

Magnetism and Superconductivity

Magnetism

1911/21Bohr van Leeuwen

After K. Miyake

Superconductivity


1906 Weiss FM

Fron

tier o

f Res

earc

h

Rese

arch

Are

a


1933Meissner Effect



AFM Spin Flucs






2001Ce/Pu 115

SC

2007NpPd2Al3SC

Year


Magnetism


After K. Miyake

Superconductivity

Superconductivity and Magnetism are Quantum Phenomena.

Neils Bohr 1911


PLAN

CK

EIN

STEI

N Q

UAN

TUM

1906 Weiss FM

Fron

tier o

f Res

earc

h

Rese

arch

Are

a


1933Meissner Effect



AFM Spin Flucs






2001Ce/Pu 115

SC

2007NpPd2Al3SC

Year


Magnetism


After K. Miyake

Superconductivity

Superconductivity and Magnetism are Quantum Phenomena.

Neils Bohr 1911

Some of the many who failed to solve the riddle of superconductivity


PLAN

CK

EIN

STEI

N Q

UAN

TUM

1906 Weiss FM

Fron

tier o

f Res

earc

h

Rese

arch

Are

a


1933Meissner Effect



AFM Spin Flucs






2001Ce/Pu 115

SC

2007NpPd2Al3SC

Year


Magnetism


1957 BCS Theory

BCS 1957

After K. Miyake

| i =Y

k

(uk + vkc†�k#c

†k")|0i

Superconductivity


PLAN

CK

EIN

STEI

N Q

UAN

TUM

1906 Weiss FM

Fron

tier o

f Res

earc

h

Rese

arch

Are

a


1933Meissner Effect



AFM Spin Flucs






2001Ce/Pu 115

SC

2007NpPd2Al3SC

Year


Magnetism


1957 BCS Theory

BCS 1957

After K. Miyake

| i =Y

k

(uk + vkc†�k#c

†k")|0i

Superconductivitycondensation of electron pairs ...


PLAN

CK

EIN

STEI

N Q

UAN

TUM

1906 Weiss FM

Fron

tier o

f Res

earc

h

Rese

arch

Are

a


1933Meissner Effect



AFM Spin Flucs






2001Ce/Pu 115

SC

2007NpPd2Al3SC

Year


Magnetism


1957 BCS Theory

BCS 1957

After K. Miyake

| i =Y

k

(uk + vkc†�k#c

†k")|0i


VOLUME 1, NUMBKR PHYSICAL REVIEW LETTERS AuGUST I, 1958

Z

I- IP0

Z

8z003

Z

6

OI-ujK0UJ

Z2

SPIN ~ EFF

/

/

7j&

5/2

2ZQV)

3/2

an increase in effective moment should go hand

in hand with a decrease of the superconducting

transition temperature does not work at all. In-

stead, the depression of the superconductivity

seems to be correlated only with the spin of the

solute atoms. (The anomalous behavior of ce-rium is probably caused by the rather easy shift

of part of the 4f electron into the 5d band; this

occurs in the pure metal at low temperature orunder pressure. )

The change in the superconducting transition

temperature of lanthanum caused by varying the

dissolved amounts of gadolinium was investigat-

ed in more detail and the results are shown in

Fig. 3. The superconducting transition tempera-

0:.La Ce Pr Nd

aPPrn Srn Eu Gd Tb Dy Ho Er Trn Yb Lu

FIG. l. Effective magnetic moments and spins of the

rare earth elements (see reference 2).

The effective magnetic moments of the rareearth elements follow Van Vleck's well-known

curve, ' Fig. 1. These moments, which originate

in the low-lying 4f shell, are usually assumed

to remain undisturbed in almost all chemi. cal

compounds which include these elements. It was

therefore our hope that by dissolving small

amounts of the magnetic rare earth elements in

lanthanum, the superconducting transition would

be affected by the dipole field from the moment

of the rare earth atoms. In Fig. 2 we show the

superconducting transitions of lanthanum samples

in which 1 at.% of various rare earth elements

has been dissolved. It is immediately apparent

from. these data that the simple assumption that

LU

O

0La Ce Pr Nd Pm Sm Eu Gd Tb Dg Ho Er Trn Yb Lu

FIG. 2. Superconducting transition temperatures of

1 at 90 rare earth solid solutions in lanthanum.

—SUPERCONDUCTI NG

TRA N SIT I ON

FERROMAGNETIC

CURIEPOINT

~00 4 5 6

PER CENT Gd

FIG. 3. Ferromagnetic and superconducting tran-

sition temperatures of solid solutions of gadolinium

in lanthanum.

ture seems to be a strictly linear function of the

amounts of dissolved gadolinium. 2.5 at.% or

more of gadolinium in lanthanum causes this

solid solution to become ferromagnetic above

1'K. The Curie points within this range are an

approximately linear function of the percentage

of gadolinium. This suggests the presence of a

coupling which aligns the moments spontaneous-

ly in these materials and which is different from

overlap exchange forces usually considered

since the coupling extends over several lattice

spacings and is proportional in magnitude to the

amount of gadolinium added. By dissolving ga-dolinium in yttrium, a nonsuperconducting metal,

only moderate paramagnetism was observed and

solid solutions with even as much as 10 at.$gadolinium did not show any ferromagnetism. On

the other hand, solid solutions of gadolinium in

thorium, another superconductor, were again

ferromagnetic.These data suggest that an exchange over con-

duction electrons' leading to ferromagnetism iseasy to bring about in an element which by itself

93

VOLUME 1, NUMBER 3 PHYSICAL REVIEW LETTERS AUGUST 1, 1958

from the data. The results are given in Table I.

Table I. Electron spin resonance results for NH,

and ND~

Radical A (Mc/sec) B (Mc/sec)

NH2

ND2

67.03(20)

10.27 (20)

28.90(20)33.28(20)

2.00481 (8)2.00466 (8)

l I I

5240 52SO 3260OERSTEDS

NH~ ~I

t II II I

I I I

5280 5290 5500

THEORETICALPA TTERN

~ND2

FIG. 2. Electron spin resonance spectrum of ND

in an argon matrix at 4. 2 K. Also present arespectra of D and NH2 and weak traces of NHD.

The ratio of the hydrogenic coupling constants,/A = 6.526, is in excellent agreement with

the predicted ratio, gi(H)/gr(D) = 6.514. Thedifference between the values for the nitrogen

coupling constants is unexpected, indicating that

the electronic wave functions for the two radi-cals are somewhat different. It is clear that the

inclusion of higher-order terms in the solution

of the spin Hamiltonian would not bring the Bvalues into closer agreement. Apparently, the

hyperfine interaction with the nitrogen atom israther sensitive to some small perturbation in

the electronic state. Thus far, we have been un-

able to account for the discrepancy by consider-ing zero-point vibration and the differences in

the rotational states of these molecules.

coupling constants were evaluated, as discussed

later, and the complete spectrum calculated.

The predicted positions of the spectral lines areindicated in the bottom of the figure. Eleven of

the lines are clearly recognized in the record-ing. The others are too close to other lines to be

resolved. In addition, one sees the center deu-

terium atom line, slight traces of NHD, and

several lines from NH, arising from some re-manent NH, in the system. The lines of NH, and

ND, were recorded individually on expanded

sweeps to determine their field positions with

high precision.If one solves the spin Hamiltonian for the mag-

netic energy, 8', to the first order approxima-

tion, one obtain, s

W=M g p, ,H+AM Zm. + BM M (N)Jz

z J I

+p, H g gm. +g (N)M (N)Ig

z I

where m. is the nuclear magnetic quantum num-

ber of hydrogen (+ 1/2) in the case of NH, or ofz

deuterium (1,0, -1) in the case of ND„and the

other symbols have their usual significance.

The hyperfine coupling constants A and B and

the electronic g -factor, g, can be calculated

+ This work supported by Bureau of Ordnance, De-partment of the Navy.

Jen, Foner, Cochran, and Bowers, Phys. Hev.104, 846 (1956).

2 Foner, Jen, Cochran, and Bowers, J. Chem. Phys.28, 851 (1958).

'Jen, Foner, Cochran, and Bowers, (to be publish-

ed) .

SPIN EXCHANGE IN SUPERCONDUCTORS

B. T. Matthias, H. Suhl, and E. Corenzwit

Bell Telephone Laboratories,Murray Hill, New Jersey(Received July 15, 1958)

The only known superconductor among the rareearth elements is lanthanum. The elements fol-lowing lanthanum in the periodic system areeither strongly paramagnetic or ferromagnetic,with magnetic moments which are due to their

4f electrons. In lutetium, 14 electrons have

filled this 4f shell entirely and the element doesnot show pronounced paramagnetism. Lutetium,

however, is not superconducting above 1.02'K

because its metallic radius has become much

smaller and at the same time it is much heavierthan lanthanum.


PLAN

CK

EIN

STEI

N Q

UAN

TUM

1906 Weiss FM

Fron

tier o

f Res

earc

h

Rese

arch

Are

a


1933Meissner Effect



AFM Spin Flucs






2001Ce/Pu 115

SC

2007NpPd2Al3SC

Year


Magnetism


1957 BCS Theory

BCS 1957

After K. Miyake

| i =Y

k

(uk + vkc†�k#c

†k")|0i



Z

I- IP0

Z

8z003

Z

6

OI-ujK0UJ

Z2

SPIN ~ EFF

/

/

7j&

5/2

2ZQV)

3/2














0:.La Ce Pr Nd


















LU

O




—SUPERCONDUCTI NG

TRA N SIT I ON

FERROMAGNETIC

CURIEPOINT

~00 4 5 6

PER CENT Gd



in lanthanum.




















93




and ND~


NH2

ND2

67.03(20)

10.27 (20)

28.90(20)33.28(20)

2.00481 (8)2.00466 (8)

l I I


NH~ ~I

t II II I

I I I

5280 5290 5500

THEORETICALPA TTERN

~ND2
























tion, one obtain, s


z J I


z I











ed) .










“Magnetism is BAD for SC !”1% mag impurities usually kill Tc


Pair condensation.


Pair condensation.


PLAN

CK

EIN

STEI

N Q

UAN

TUM

1906 Weiss FM

Fron

tier o

f Res

earc

h

Rese

arch

Are

a


1933Meissner Effect



AFM Spin Flucs






2001Ce/Pu 115

SC

2007NpPd2Al3SC

Year

A surprise from the actinides

Magnetism


1957 BCS Theory

1937 London

After K. Miyake

Superconductivity


Z

I- IP0

Z

8z003

Z

6

OI-ujK0UJ

Z2

SPIN ~ EFF

/

/

7j&

5/2

2ZQV)

3/2














0:.La Ce Pr Nd


















LU

O




—SUPERCONDUCTI NG

TRA N SIT I ON

FERROMAGNETIC

CURIEPOINT

~00 4 5 6

PER CENT Gd



in lanthanum.




















93




and ND~


NH2

ND2

67.03(20)

10.27 (20)

28.90(20)33.28(20)

2.00481 (8)2.00466 (8)

l I I


NH~ ~I

t II II I

I I I

5280 5290 5500

THEORETICALPA TTERN

~ND2
























tion, one obtain, s


z J I


z I











ed) .










Magnetism is BAD for SC !1% mag impurities usually kill Tc

UBe13


PLAN

CK

EIN

STEI

N Q

UAN

TUM

1906 Weiss FM

Fron

tier o

f Res

earc

h

Rese

arch

Are

a


1933Meissner Effect



AFM Spin Flucs






2001Ce/Pu 115

SC

2007NpPd2Al3SC

Year


Magnetism


1957 BCS Theory

1937 London

After K. Miyake

Superconductivity


Z

I- IP0

Z

8z003

Z

6

OI-ujK0UJ

Z2

SPIN ~ EFF

/

/

7j&

5/2

2ZQV)

3/2














0:.La Ce Pr Nd


















LU

O




—SUPERCONDUCTI NG

TRA N SIT I ON

FERROMAGNETIC

CURIEPOINT

~00 4 5 6

PER CENT Gd



in lanthanum.




















93




and ND~


NH2

ND2

67.03(20)

10.27 (20)

28.90(20)33.28(20)

2.00481 (8)2.00466 (8)

l I I


NH~ ~I

t II II I

I I I

5280 5290 5500

THEORETICALPA TTERN

~ND2
























tion, one obtain, s


z J I


z I











ed) .











UBe13


PLAN

CK

EIN

STEI

N Q

UAN

TUM

1906 Weiss FM

Fron

tier o

f Res

earc

h

Rese

arch

Are

a


1933Meissner Effect



AFM Spin Flucs






2001Ce/Pu 115

SC

2007NpPd2Al3SC

Year


Magnetism


1957 BCS Theory

1937 London

After K. Miyake

Superconductivity


Z

I- IP0

Z

8z003

Z

6

OI-ujK0UJ

Z2

SPIN ~ EFF

/

/

7j&

5/2

2ZQV)

3/2














0:.La Ce Pr Nd


















LU

O




—SUPERCONDUCTI NG

TRA N SIT I ON

FERROMAGNETIC

CURIEPOINT

~00 4 5 6

PER CENT Gd



in lanthanum.




















93




and ND~


NH2

ND2

67.03(20)

10.27 (20)

28.90(20)33.28(20)

2.00481 (8)2.00466 (8)

l I I


NH~ ~I

t II II I

I I I

5280 5290 5500

THEORETICALPA TTERN

~ND2
























tion, one obtain, s


z J I


z I











ed) .











UBe13Magnetic moments


PLAN

CK

EIN

STEI

N Q

UAN

TUM

1906 Weiss FM

Rese

arch

Are

a

1931Resistance Minimum

Year

Superconductivity

Magnetism

1963Kondo Theory

1911 HgDiscovery of SC

1933Meissner Effect

1957 BCS Theory

1933 Landau NeelAFM1925 Pauli

SPIN


1972He-3

(Superfluid)

FM Spin FlucsAfter K. Miyake

We tried to detect any possible magnetic orderingbelow 1K. Instead we found a sharp superconductingtransition at 0.97K, which was reduced by about0.3K only in a field of 60kOe. (1974)

UBe13 VOr. UMZ 50, NUMSZR 20 PHYSICAL REVIEW LETTERS 16 Mwv 1983

240—

[p,Q, cm] ~

200 '—

l60—

l20 -2oo-

80—

40—

00 40

~ ~

~~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

I I I I I I I I

80 l 20 l 60 200 240 280T [K]

p~, „=(2l + 1)c(2Iz/e'kFz), (2)

where N(Ez) is the density of electronic states

per spin direction at the Fermi energy EF, p, &

is the Bohr magneton, and k& is Boltzmann's

constant. If we insert the above-quoted value for

y = 1.1 J/mole K' into Eq. (1), we obtain X =1.51

x10 ' emu/mole, in extremely good agreement

with our experimental. vat. ue of X at about 1 K,again confirming the claim above, that we aredealing with an electronic system that can bedescribed as a Fermi liquid.

The maximum resistivity is thought to be due

to incoherent scattering of conduction electronsat the U ions and may, according to Friedel, "be described by

FIG. 2. Temperature dependence of the electricalresistivity of single-crystalline UBe&3. Inset: The low-

temperature part on an extended temperature scale.

superconducting transition at 0.86 K. As may be

seen from the inset in Fig. 2, the resistive tran-

siti.on to the superconducting state is much more

narrow in temperature than the transitions shown

in Fig. 1. These features are probably due to

residual inhomogeneities in the not yet optimized

samples.

For the room-temperature lattice constant of

the UBe» single crystal. s used in the present in-

vestigation we obtained 10.2607 A, resulting in a

nearest U-U distance of 5.130A in this compound.

According to Hil. l. 's earlier arguments" it may

therefore be expected that the 5f electrons of

the U' ions are fairly well localized and with

any conventional view, certainly no occurrence

of superconductivity in such a system is antici-

pated. On the contrary, the common enhanced

increase of c~/T and X (not shown explicitly here),as well as of p, with decreasing temperature be-

low 10 K rather indicate precursor effects to a

possibl. e magnetic phase transition.

This pronounced temperature dependence of all.

these properties just above T, makes a clear-cut

interpretation of the experimental. data somewhat

difficult. Nevertheless it is interesting to quote

some val. ues for physically important parameterswhich we calculate from our experimental data.

If, as indicated above, the specific heat up to

about 1 K is interpreted as being of electronic

origin we can calculate the corresponding mag-

netic susceptibility of that electronic system us-

ing

X =2p B &(EF)= 3|zan y/" tza

where l =3 for f electrons, c=~z is the concen-

tration of scattering centers, Z is the number

of conduction electrons per atom, and kz = (3zz'Z/

0)' ' with 0 as the mean volume per atom. Fromit we can calcul. ate Z and subsequently k F through

Z = [2(2 l+ 1)tzc/e2p ] ~4[@/3&2]z~4 (3)

From the experimental. value of p „we obtainZ =0.81 per atom and k F=1.36&&10 em . Wi,thi, n

the Fermi-1. iquid model. we then deduce an effec-tive mass of the fermions of m*= 192m, . Thestill. rather high el.ectrieal resistivity at T, in-

dicates that superconducting parameters of the

present material should be cal.cul.ated in the dirtylimit. According to Hake, '~

(BH,2/&T)r is then

given by

(&H 2/&T)r = —4.48&104py, (4)

wherey is given in cgs units and p in 0 cm. In-

serting our experimental values for y and (BH„/&T)r we obtain p =42 p,Q cm, the expected value

of the residual resistivity for T-0. Once ongoing

additional experiments give more information on

other supercondueting parameters of UBe» we

shall discuss them by comparing them with the

presently available normal- state properties.In conclusion we feel. that the experimental. data

presented and described above show convincingly

that, as was anticipated, CeCu, Si, is not a singu-

larity of nature. " It seems again quite cl.ear thatthe presence of f electrons is essential for the

occurrence of superconductivity in UBe», sinceno traces of superconductivity were found in

LaBe», LuBe», and ThBe» down to 0.45 K.'Since UBe» shows al. l. the interesting featuresnot onl.y in polycrystalline but also in its single-crystal. line form at zero pressure, "this mater-ial. is very well suited to investigation of the mi-

1597



PLAN

CK

EIN

STEI

N Q

UAN

TUM

1906 Weiss FM

Rese

arch

Are

a


Year

Superconductivity

Magnetism

1963Kondo Theory


1933Meissner Effect

1957 BCS Theory


SPIN


1972He-3

(Superfluid)




240—

[p,Q, cm] ~

200 '—

l60—

l20 -2oo-

80—

40—

00 40

~ ~

~~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

I I I I I I I I

80 l 20 l 60 200 240 280T [K]

p~, „=(2l + 1)c(2Iz/e'kFz), (2)


















samples.























ing






Z = [2(2 l+ 1)tzc/e2p ] ~4[@/3&2]z~4 (3)





(BH,2/&T)r is then

given by

(&H 2/&T)r = —4.48&104py, (4)













1597



PLAN

CK

EIN

STEI

N Q

UAN

TUM

1906 Weiss FM

Rese

arch

Are

a


Year

Superconductivity

Magnetism

1963Kondo Theory


1933Meissner Effect

1957 BCS Theory


SPIN


1972He-3

(Superfluid)




240—

[p,Q, cm] ~

200 '—

l60—

l20 -2oo-

80—

40—

00 40

~ ~

~~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

I I I I I I I I

80 l 20 l 60 200 240 280T [K]

p~, „=(2l + 1)c(2Iz/e'kFz), (2)


















samples.























ing






Z = [2(2 l+ 1)tzc/e2p ] ~4[@/3&2]z~4 (3)





(BH,2/&T)r is then

given by

(&H 2/&T)r = —4.48&104py, (4)













1597



PLAN

CK

EIN

STEI

N Q

UAN

TUM

1906 Weiss FM

Rese

arch

Are

a


Year

Superconductivity

Magnetism

1963Kondo Theory


1933Meissner Effect

1957 BCS Theory


SPIN


1972He-3

(Superfluid)




240—

[p,Q, cm] ~

200 '—

l60—

l20 -2oo-

80—

40—

00 40

~ ~

~~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

I I I I I I I I

80 l 20 l 60 200 240 280T [K]

p~, „=(2l + 1)c(2Iz/e'kFz), (2)


















samples.























ing






Z = [2(2 l+ 1)tzc/e2p ] ~4[@/3&2]z~4 (3)





(BH,2/&T)r is then

given by

(&H 2/&T)r = —4.48&104py, (4)













1597

This suggests that thesuperconductivity is not an intrinsic property ofUBe13.



PLAN

CK

EIN

STEI

N Q

UAN

TUM

1906 Weiss FM

Rese

arch

Are

a


Year

Superconductivity

Magnetism

1963Kondo Theory


1933Meissner Effect

1957 BCS Theory


SPIN


1972He-3

(Superfluid)




240—

[p,Q, cm] ~

200 '—

l60—

l20 -2oo-

80—

40—

00 40

~ ~

~~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

I I I I I I I I

80 l 20 l 60 200 240 280T [K]

p~, „=(2l + 1)c(2Iz/e'kFz), (2)


















samples.























ing






Z = [2(2 l+ 1)tzc/e2p ] ~4[@/3&2]z~4 (3)





(BH,2/&T)r is then

given by

(&H 2/&T)r = —4.48&104py, (4)













1597

This suggests that thesuperconductivity is not an intrinsic property ofUBe13. WRONG!



PLAN

CK

EIN

STEI

N Q

UAN

TUM

1906 Weiss FM

Rese

arch

Are

a


Year

Superconductivity

Magnetism

1963Kondo Theory


1933Meissner Effect

1957 BCS Theory


SPIN


1972He-3

(Superfluid)




240—

[p,Q, cm] ~

200 '—

l60—

l20 -2oo-

80—

40—

00 40

~ ~

~~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

I I I I I I I I

80 l 20 l 60 200 240 280T [K]

p~, „=(2l + 1)c(2Iz/e'kFz), (2)


















samples.























ing






Z = [2(2 l+ 1)tzc/e2p ] ~4[@/3&2]z~4 (3)





(BH,2/&T)r is then

given by

(&H 2/&T)r = —4.48&104py, (4)













1597

Ott, Fisk & Smith, (1983)

This suggests that thesuperconductivity is not an intrinsic property ofUBe13.

Ott, Fisk & Smith, (1983)

Ott 1976

Steglich1979

Fisk1983

“Heavy Fermion Superconductor



Fron

tier o

f Res

earc

h



AFM Spin Flucs






2001Ce/Pu 115

SC

2007NpPd2Al3SC

PLAN

CK

EIN

STEI

N Q

UAN

TUM

1906 Weiss FM

Rese

arch

Are

a


1933Meissner Effect

Year

Magnetism


1957 BCS Theory

1937 London

After K. Miyake

Superconductivity

UBe13

A new era of discovery

Ott 1976

Steglich1979

Fisk1983



PLAN

CK

EIN

STEI

N Q

UAN

TUM

1906 Weiss FM

Fron

tier o

f Res

earc

h

Rese

arch

Are

a








2001Ce/Pu 115

SC

2007NpPd2Al5SC

Year


Magnetism

1963Kondo Theory


1933Meissner Effect

1957 BCS Theory


SPIN


1972He-3

(Superfluid)

FM Spin Flucs

A remarkable convergence of two fields.

After K. Miyake1983UBe13

1979CeCu2Si2

(d-wave sc)

AFM Spin Flucs

Superconductivity

Ott 1976

Steglich1979

Fisk1983

A new era of discovery


Hg Pb NbNbNi Nb3Sn

conventional s.c. Nb3Ge

Limit of Tc

The search for High Tc.


Hg Pb NbNbNi Nb3Sn

heavy fermions


Limit of Tc



Hg Pb NbNbNi Nb3Sn

heavy fermions

copper-oxides


Limit of Tc



Hg Pb NbNbNi Nb3Sn

heavy fermions

copper-oxides


The search for High Tc.?


Hg Pb NbNbNi Nb3Sn

heavy fermions

copper-oxides


organics



Hg Pb NbNbNi Nb3Sn

MgB2

heavy fermions

copper-oxides


organics



Hg Pb NbNbNi Nb3Sn

MgB2

iron based

heavy fermions

copper-oxides


organics



Applications of superconductivity


magnetic resonance imaging

huge current densities

urban setting for power distribution

Large currents without loss…. production of large magnetic fields

Applications of superconductivity


SC Levitation




Acoustic vs Magnetic pairing.



Conventional superconductors:pairing driven by virtual lattice vibrations

“phonons”





Anomalous superconductors:Pairing driven by magnetic fluctuations?


But what is the nature of the magnetic glue?

33


Nature © Macmillan Publishers Ltd 1998

8

in good thermal contact with both the pressure cell and the sampleleads. To try to ensure adequate pressure and temperature homo-geneity, a slow cooling rate from room temperature, typically0.2 K min−1, was employed. Measurements were carried out fromroom temperature to the millikelvin temperature range within apumped 4He cryostat, an adiabatic demagnetization refrigeratorand a top-loading dilution refrigerator.

Our key experimental results are summarized in Figs 2, 3. In thematerials studied, the antiferromagnetic ordering temperature TN,at which there is a discontinuity in the gradient of the resistivity r(not shown), was found to decrease slowly and monotonically withincreasing pressure, p. Over a wide region of the CePd2Si2 phasediagram, TN is close to being linear in p, and extrapolation from thisregime to absolute zero allows us to define an effective criticalpressure pc of 28 kbar. In CeIn3, the variation of TN with p is morerapid, and we estimate pc to be ,26 kbar. The behaviour of TN as itfalls below 1 K has not been resolved in these studies.

In the case of CePd2Si2, the resistivity r does not exhibit thestandard T 2 form expected of a Fermi liquid. Careful analysis showsthat near pc it in fact varies as T 1.260.1 over nearly two decades intemperature down to the millikelvin range (Fig. 2, inset). Below500 mK and in a narrow region near pc, we observe an abrupt dropin r to below the detection limit, consistent with the occurrence of asuperconducting transition, as discovered during our initial obser-vations in September 199437,38. At a given pressure, this transitionmay be characterized by a temperature Tc, at which r falls to 50% ofits normal state value. The width of this transition grows markedlyas the pressure is varied away from pc. We stress that experimentally,r is found to actually vanish only close to pc. By energizing a Nb–Ticoil placed in our pressure cell, it was established that the uppercritical field Bc2 varies as dBc2ðpcÞ=dT, 2 6 T=K near Tc. This is ahigh rate of change for such a small value of Tc—much higher thanthe expected figure for a conventional superconductor. However,it is the same order of magnitude as the value found in theheavy fermion superconductor CeCu2Si2 (ref. 27). We note that ina traditional analysis, the slope of Bc2(T ) at Tc implies a super-conducting coherence length of 150 A, a value which is below thevalue of lmfp that we estimate for our best samples. No super-conductivity has been observed in specimens with residual resistiv-ities above several mQ cm, namely those with an estimated lmfp that is

substantially below y (for a similar example, see ref. 39).In the case of cubic CeIn3, we find that very close to pc the normal

state resistivity assumes a non-Fermi liquid form, but this time40,41

varies as T 1.660.2. Thus, near their respective critical pressures, theresistivity exponent in the cubic material is significantly higher thanit is in tetragonal CePd2Si2. In a very narrow region near pc, we againsee a sharp drop in r to below the detection limit, but at somewhatlower temperatures than the transitions observed in CePd2Si2. Thisis consistent with the occurrence of superconductivity in yetanother cerium compound on the edge of long-range magneticorder40,41.

We stress that in each material studied, both the form of thetemperature dependence of the normal state resistivity, and thenature and existence of the superconducting transition are sensitiveto sample quality. In particular, the superconducting transitionsappear only in samples with residual resistivities in the low mQ cmrange, as expected in the case of anisotropic pair states withcoherence lengths of the order of a few hundred angstroms.

Magnetic interactionsThe observed temperature–pressure phase diagrams for bothCePd2Si2 and CeIn3 are at least qualitatively consistent with whatis expected in terms of the magnetic interactions model (Fig. 1). Wenow consider a more quantitative comparison. In the following it isassumed that the magnetic transition is continuous and that n isclose to nc. The incoherent scattering of quasiparticles via magneticinteractions is then expected to lead to a resistivity of the form

r ¼ r0 þ ATx ð1Þ

where r0 and A are constants and the exponent x is smaller than two,that is, smaller that it is in a conventional Fermi liquid at low T

articles

NATURE | VOL 394 | 2 JULY 1998 41

0

5

10

0 10 20 30 40

Tem

pera

ture

(K)

Pressure (kbar)

TN

3Tc

anti-ferromagnetic

state superconductingstate

0

20

40

0 20 40T1.2 (K1.2)

28 kbar

Tc

ρ (µΩ

cm

)

Figure 2 Temperature–pressure phase diagram of high-purity single-crystal

CePd

2

Si

2

. Superconductivity appears below Tc

in a narrow window where the

Neel temperature TN

tends to absolute zero. Inset: the normal state a-axis

resistivity above the superconducting transition varies as T 1.260.1

over nearly two

decades in temperature

27,30

. The upper critical field Bc2

at the maximum value of Tc

varies near Tc

at a rate of approximately −6T/K. For clarity, the values of Tc

have

been scaled by a factor of three, and the origin of the inset has been set at 5K

below absolute zero.

0

5

10

0 10 20 30

Tem

pera

ture

(K)

Pressure (kbar)

TN

10 TSuperconductivity c

0

0.4

0.8

1.2

0 0.4 0.8 1.2T (K)

24.0 kbar

Tc

ρ (µΩ

cm

)

0

1

2

0.6 1 1.4log

10 (T(K))

27 kbar

d (ln

Δρ)

/ d

(ln T

)


CeIn

3

. A sharp drop in the resistivity consistent with the onset of super-

conductivity below Tc

is observed in a narrow window near pc

, the pressure at

which the Neel temperature TN

tends to absolute zero. Upper inset: this transition

is complete even below pc

itself. Lower inset: just abovepc

, where there is noNeel

transition, a plot of the temperature dependence of d(ln Dr)/d(ln T) is best able to

demonstrate that the normal state resistivity varies as T1.660.2

below several

degrees K (ref. 29) (Dr is the difference between the normal state resistivity and its

residual value—which is calculated by extrapolating the normal-state resistivity to

absolute zero). For clarity, the values of Tc

have been scaled by a factor of ten. The

resistivity exponents of CeIn

3

and CePd

2

Si

2

may be understood by taking into

account the underlying symmetries of the antiferromagnetic states and using

the magnetic interactions model. Superconductivity near nc

in pure samples is

expected to be a natural consequence of the same model.


8

NATURE | VOL 394 | 2 JULY 1998 39

articles

Magnetically mediatedsuperconductivity inheavy fermion compoundsN. D. Mathur*, F. M. Grosche*, S. R. Julian, I. R. Walker, D. M. Freye, R. K. W. Haselwimmer & G. G. Lonzarich

Cavendish Laboratory and the Interdisciplinary Research Centre for Superconductivity, University of Cambridge, Cambridge CB3 0HE, UK

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

In a conventional superconductor, the binding of electrons into the paired states that collectively carry thesupercurrent is mediated by phonons—vibrations of the crystal lattice. Here we argue that, in the case of the heavyfermion superconductors CePd2Si2 and CeIn3, the charge carriers are bound together in pairs by magnetic spin–spininteractions. The existence of magnetically mediated superconductivity in these compounds could help shed light onthe question of whether magnetic interactions are relevant for describing the superconducting and normal-stateproperties of other strongly correlated electron systems, perhaps including the high-temperature copper oxidesuperconductors.

The ancient observation of action at a distance between magneticmaterials such as lodestone inspired William Gilbert to postulate ananalogous force between the Earth and the Sun1. Later work showedthat a parallel force holds atoms together. This charge–chargeinteraction is also ultimately responsible, in a subtle way, for thebinding of Cooper pairs in superconductors. In traditionalsuperconductors2, this binding is most simply described in termsof the emission and absorption of waves of lattice density. Here weconsider instead whether this binding could in some materials bemost simply described in terms of the emission and absorption ofwaves of the electron spin density: that is, whether a type of boundstate that was only identified this century can arise because of aneffective spin–spin interaction (see, for example, refs 3–7) that isreminiscent of one of the oldest known forces.

The low-temperature properties of interacting electrons in con-ductors are normally described in terms of excited states known as‘quasiparticles’8,9. These entities are liable to be very different frombare electrons. Although they may obey the same ‘fermion’ quan-tum statistics, they can behave as though they are very ‘heavy’. Weare interested in materials that support such quasiparticles, namelythe ‘heavy fermion’ compounds. Exotic interactions can take placebetween quasiparticle charge carriers via their host medium, leadingto new and subtle states of matter. One may imagine that, at eachinstant in its motion, a quasiparticle emits a wave that perturbs itsenvironment in the manner of a stone entering a pond. Anothercharge carrier elsewhere can receive the signals, and thus chargecarriers are able to communicate with each other. It is possible totune the nature of these communications by altering the density ofthe crystal. In this way it is possible to alter dramatically thetransport behaviour and other measured properties of a material.And in sufficiently calm conditions at low temperatures, boundstates may form leading, in particular, to superconductivity.

We consider whether it is possible to tune the nature of chargecarrier interactions such that magnetic coupling demonstrablydominates the more conventional non-magnetic channels of com-munication. In particular, we examine whether in such a scenario itwould be possible for a magnetically mediated form of super-conductivity to exist at sufficiently low temperatures. We presentnew evidence to suggest that this kind of superconductivity mayindeed exist in some heavy fermion compounds, but that it isnormally only viable extremely close to the critical lattice density nc

at which long-range magnetic order is suppressed, and in specimensof extremely high purity. We discuss a magnetic-interactions modelthat provides a natural description of the temperature–densityphase diagrams we have observed in two particularly simple heavyfermion compounds, namely cubic CeIn3 and tetragonal CePd2Si2.In each of these two systems, we find that both the variation of themagnetic and superconducting transition temperatures with latticedensity, and the anomalous forms of the normal state resistivities,are qualitatively consistent with the magnetic-interactions model,which uses independently estimated parameters. This agreement,together with other features described here, provides compellingevidence for the probable existence of magnetically mediated super-conductivity. Moreover, the magnetic interactions model, whichmay account for superconductivity in the materials studied,suggests how the properties of a material might be modified inorder to increase the superconducting transition temperature.Interestingly, the changes required would lead us to materialswith some of the features of the copper oxide superconductors.

The edge of magnetic orderWe consider how to tune the charge carrier interactions so that themagnetic channel may become dominant. During the course of itsmotion, a quasiparticle nucleates various waves in the mediumwhich affect other quasiparticles. The amplitudes of such wavesdepend on the strength of the coupling of a quasiparticle with themedium, and on the ease with which such waves can be excited inthe medium. Magnetic waves of interest may be readily excitedwhen a magnetic medium is near a critical lattice density, nc, thatrenders it close to entering a state of continuous long-rangemagnetic order. Such waves in the non-magnetically ordered stateare usually damped and may be described in terms of a relaxationfrequency spectrum, °

q

. And this quantity, which describesthe characteristic rate of decay of a spontaneous fluctuation ofthe magnetization of wavevector q, characterizes the dynamics ofquasiparticle interactions (for a review, see ref. 10; see also refs 11–14).

In the simplest model, the retarded interaction produced by onequasiparticle on another depends on the product of three factors4,namely the relative orientation and magnitudes of the magneticmoments involved, 2 m1⋅m2, the square of a coupling parameter,and the space and time-dependent magnetic susceptibility, whichitself depends on °

q

. When it is dominant, this interaction can leadto two striking consequences. First, near nc, magnetic waves tend topropagate over a long range. If the coupling parameter for suchwaves remains finite, the magnetic interactions are therefore also

* Present address: Department of Materials Science, University of Cambridge, Cambridge CB2 3QZ, UK(N.D.M.); MPI Chemische Physik fester Stoffe, Bayreuther Str. 40, 01187 Dresden, Germany (F.M.G.).


8

NATURE | VOL 394 | 2 JULY 1998 39

articles



. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .







q



q



Lonzarich, 1998

CeIn3Tc



8











articles

NATURE | VOL 394 | 2 JULY 1998 41

0

5

10

0 10 20 30 40

Tem

pera

ture

(K)

Pressure (kbar)

TN

3Tc

anti-ferromagnetic


0

20

40

0 20 40T1.2 (K1.2)

28 kbar

Tc

ρ (µΩ

cm

)


CePd

2

Si

2



Neel temperature TN



over nearly two


27,30



varies near Tc


have



0

5

10

0 10 20 30

Tem

pera

ture

(K)

Pressure (kbar)

TN


0

0.4

0.8

1.2

0 0.4 0.8 1.2T (K)

24.0 kbar

Tc

ρ (µΩ

cm

)

0

1

2

0.6 1 1.4log

10 (T(K))

27 kbar

d (ln

Δρ)

/ d

(ln T

)


CeIn

3




, the pressure at








below several






3

and CePd

2

Si

2




in pure samples is



8

NATURE | VOL 394 | 2 JULY 1998 39

articles



. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .







q



q




8

NATURE | VOL 394 | 2 JULY 1998 39

articles



. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .







q



q



Lonzarich, 1998

CeIn3 Tc Tc

0.2K CeIn3’98

115 Compounds

variation of the Cooper state can be properly adjusted to match theoscillations of the spin interaction. Generally, the instability is foundin the spin-singlet state, which must necessarily have even angularmomentum. As the interaction is repulsive at the origin, a non-zeroangular momentum state, typically a d-wave state, is favoured(Fig. 1c).

We note that the analogues of the ferromagnetic and antiferro-magnetic instabilities can exist in the density (described by xn(r,t)) aswell43. The pattern of oscillations in xn(r,t) is determined by thewavevector at which the density response is maximum, and thistoo can lead to unconventional pairing states. In contrast to themagnetic interaction where the sign of the interaction is differentfor spin-triplet and spin-singlet Cooper pairs, the density interactionis oblivious to the spin state of the Cooper pairs. Hence, the magneticinteraction offers more freedom to match the attractive regions of theoscillations of the interactions with the quasiparticle states near theFermi surface available to construct a Cooper-pair wavefunction.

Some surprisesOn the border of ferromagnetism the induced interaction is purelyattractive at short distances, whereas on the border of antiferromag-netism it is likely to have repulsive regions when the two interactingquasiparticles are close to each other. It might therefore be expectedthat the search for exotic pairing states on the former border wouldhave proved the more fruitful. This seemed to have been borne out bythe discovery of the superfluidity of liquid 3He in which the uniformmagnetic susceptibility is strongly enhanced44. Given the abundanceof metals that exhibit strong ferromagnetic correlations, it is morethan a little surprising that it took a quarter of a century to discover asuperconducting analogue of liquid 3He, namely, the layered perovs-kite Sr2RuO4 that has a Tc value two orders of magnitude below thatof the high Tc copper oxides26.

Even more perplexing on the other hand is the fact that manyexamples of superconductivity on the border of antiferromagnetismhave been found in the intervening period4–7,28–32. These findingswould suggest at first sight that the magnetic interaction isgiving us little or no insight on where to look for exotic forms ofsuperconductivity.

However, the great lesson of the past decade is that subtleties in themagnetic interaction model only come to the surface after a careful

examination of its properties. Although the idea of a magnetic inter-action goes back nearly half a century, the computer algorithms andhardware necessary for an exploration of the detailed predictions ofthe model have only become available more recently.

The results of these theoretical investigations have led to an intui-tive understanding of the following: (1) why superconductivity canbe particularly robust on the border of antiferromagnetism in aquasi-two-dimensional tetragonal system with high characteristicspin fluctuation frequencies30,45–48; (2) how the charge–charge andspin–spin interactions can in some cases work coherently to stabilizeanisotropic Cooper-pair states; and (3) why pairing on the border offerromagnetism is hampered by quite a number of effects, and maydepend on subtle details of the electronic structure, that is, features(absent in liquid 3He) of the energy band of the periodic crystalpotential49–51. Illustrations of these ideas are given below.

The first reason for the robustness of pairing in the presence ofantiferromagnetic correlations in a tetragonal structure is that theamplitude of the oscillations in the interaction is strong because ofthe low dimensionality. The energy density of the interaction wavescreated by the polarizer falls off more gradually in two dimensions (as1/distance) than in three dimensions (as 1/distance2). The secondreason is that the repulsive regions of the interaction in real spaceare along the diagonals of the lattice given that one quasiparticle is atthe origin (see Fig. 3). In this case, the crystal symmetry allows one tochoose a d-wave Cooper state with nodes along the diagonals, therebyneutralizing most of the repulsive regions while retaining the attrac-tive regions. One can easily imagine that it will not always be possibleto choose a Cooper-pair state in such an optimal way, and that theinitial impression that the oscillations of the interaction are detri-mental to superconductivity may only be wrong in special cases. Inparticular, as the tetragonal structure becomes more and more iso-tropic under otherwise similar conditions, the model predicts adecrease in the robustness of the pairing. The range in temperatureand pressure over which superconductivity is observed was increasedby about one order of magnitude in going from cubic CeIn3 (refs 30,52) to its tetragonal analogues CeMIn5, where M stands for Rh, Ir orCo (refs 31, 32, 53–56; Fig. 4), as anticipated by the magnetic inter-action model.

Another case where subtle features of the model considered herecould explain puzzling superconducting properties is the first of theheavy-fermion superconductors, CeCu2Si2 (ref. 4), and the related

Repulsion Attraction

Figure 3 | Magnetic interaction potential in a lattice. Graphicalrepresentation of the static magnetic interaction potential in real space seenby a quasiparticle moving on a square crystal lattice given that the otherquasiparticle is at the origin (denoted by a cross). The spins of the interactingquasiparticles are taken to be antiparallel, such that the total spin of theCooper pair is zero. The dashed lines show the regions where the d-waveCooper-pair state has vanishing amplitude. This is the state that bestmatches the oscillations of the potential, in that a quasiparticle has minimalprobability of being on lattice sites when the potential induced by thequasiparticle at the origin is repulsive. The size of the circle in each lattice siteis a representation of the absolute magnitude of the potential (on alogarithmic scale). This picture is appropriate for a system on the border ofantiferromagnetism in which the period of the real space oscillations of thepotential is precisely commensurate with the lattice.

10 Celn3

CeRhIn5TN

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ture

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Figure 4 | Effect of electronic anisotropy. The schematictemperature–pressure phase diagram of two related heavy fermioncompounds, CeIn3 (refs 30, 52) and CeRhIn5 (refs 53–55). These twomaterials differ in particular in the degree of anisotropy of the low energyexcitation spectrum. As one would expect, the thermal fluctuations in thelocal magnetization lead to a smaller value of the magnetic transitiontemperature (Neel temperature, TN) in the anisotropic material. By contrast,perhaps unexpectedly, Tc is greatly suppressed in the isotropic compoundCeIn3 (red lines) compared with CeRhIn5 (blue lines). Both of these featuresare in qualitative agreement with the magnetic interaction model.

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Lonzarich, 1998

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115 Compounds












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115 Compounds

100 fold increase in Tc from mother compound.












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PuCoGa5 : 20 K Superconductor..............................................................

Plutonium-based superconductivitywith a transition temperature above18KJ. L. Sarrao*, L. A. Morales*, J. D. Thompson*, B. L. Scott*,G. R. Stewart*†, F. Wastin‡, J. Rebizant‡, P. Boulet‡, E. Colineau‡& G. H. Lander*‡

* Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA†Department of Physics, University of Florida, Gainesville, Florida 32611, USA‡ European Commission, JRC, Institute for Transuranium Elements, Postfach2340, 76125 Karlsruhe, Germany.............................................................................................................................................................................

Plutonium is a metal of both technological relevance and funda-mental scientific interest. Nevertheless, the electronic structureof plutonium, which directly influences its metallurgical proper-ties1, is poorly understood. For example, plutonium’s 5f electronsare poised on the border between localized and itinerant,and their theoretical treatment pushes the limits of currentelectronic structure calculations2. Here we extend the range ofcomplexity exhibited by plutonium with the discovery of super-conductivity in PuCoGa5. We argue that the observed supercon-ductivity results directly from plutonium’s anomalous electronicproperties and as such serves as a bridge between two classes ofspin-fluctuation-mediated superconductors: the known heavy-fermion superconductors and the high-Tc copper oxides. Wesuggest that the mechanism of superconductivity is unconven-tional; seen in that context, the fact that the transition tempera-ture, Tc < 18.5 K, is an order of magnitude greater than themaximum seen in the U- and Ce-based heavy-fermion systemsmay be natural. The large critical current displayed by PuCoGa5,which comes from radiation-induced self damage that createspinning centres, would be of technological importance forapplied superconductivity if the hazardous material plutoniumwere not a constituent.

Large, well-faceted single crystals of PuCoGa5 were grown bycombining Pu and Co with excess Ga in an alumina crucible. Thecrucible was encapsulated in an evacuated quartz ampoule, whichwas then heated to 1,100 8C and cooled over forty hours to 600 8C.At this point the excess flux was removed with a centrifuge. Single-crystal X-ray diffraction measurements reveal that the resultingmaterial is tetragonal with room-temperature lattice constantsa ! 4.232 A and c ! 6.786 A. This structure consists of alternatingPuGa3 and ‘CoGa2’ layers and is identical to the HoCoGa5 structurereported for uranium-based gallides with quite similar latticeconstants3. This is also the structure type in which a family ofunconventional Ce-based superconductors CeMIn5 (whereM is Co,Ir, Rh) crystallizes4. Magnetic susceptibility, specific heat, andelectrical resistivity measurements on PuCoGa5 were made asfunctions of temperature and magnetic field. Some of the resultsobtained at Los Alamos have been reproduced at Karlsruhe (usingdifferent starting 239Pu) by extracting small single-crystal plateletsfrom arc-melted materials. The excellent reproducibility of theresults from the two laboratories demonstrates the robust natureof the observed ground-state properties.

Low-temperature magnetic susceptibility and specific heatmeasurements on PuCoGa5 are shown in Fig. 1. Zero-field cooledmagnetization measured in a field of 10Oe reveals a sharp diamag-netic transition at 18.5 K. At low temperature this signal corre-sponds to almost 100% of perfect diamagnetism. Heat capacitymeasurements confirm a bulk phase transition. The inferred Som-merfeld coefficient g ! 77mJmol21 K22 for PuCoGa5 is compar-able to the value observed for d-Pu (g ! 50mJmol21 K22) (refs 5,6) and is indicative of modest quasiparticle mass enhancement.

Taken together, these data provide unambiguous evidence for bulksuperconductivity in PuCoGa5, the first such observation in aplutonium compound. The approximately 0.25-K difference in Tc

between the magnetic and heat capacity measurements provides asimple indication that the compound in question contains Pu: theTc of PuCoGa5 decreases at a rate of about 0.2 K per month,presumably as a result of radiation-induced self damage. As a result,non-simultaneous measurements yield slightly different Tc values.This effect is also evident inmagnetizationmeasurements.Measure-ments made on the same sample, but separated in time byapproximately two months, reveal a decrease in Tc of 0.4 K.Such a high Tc is unusual for an intermetallic compound (only a

small number of intermetallics, led by MgB2 with a Tc of 39 K(ref. 7), have Tc values exceeding 18K). The upper critical field inPuCoGa5 is correspondingly large. Figure 2 shows field-dependentresistivity data and the resulting upper critical field Hc2(T) phasediagram inferred from these data. In particular, we find an initialslope dHc2=dT of 259 kOeK21. In the WHH approximation8,the orbital upper critical field Hc2"0# !20:69TcdHc2=dT !740 kOe. This estimate is quite large and exceeds the Pauli limit(HP ! 18.6 (kOeK21)Tc ! 340 kOe) (ref. 9). From this value ofthe orbital Hc2, we infer the Ginzburg–Landau coherence lengthyGL ! [F0/2pHc2(0)]

0.5 ! 2.1 nm. Further, assuming the Bardeen–Cooper–Schreiffer (BCS) coherence length yBCS < yGL and thatPu is trivalent (see below), we estimate g < 58mJmol21 K22

in the free-electron approximation, consistent with the data ofFig. 1.The lower inset of Fig. 2 presents a complete magnetization loop

M(H) for PuCoGa5 at 5 K. We estimate the lower critical field

Figure 1 Crystal structure and evidence for superconductivity in PuCoGa5. PuCoGa5crystallizes in the P4/mmm space group as follows: Pu, 1a, (0,0,0); Co, 1b, (0,0,0.5); Ga1,

1c, (0.5,0.5,0); Ga2, 4i, (0,0.5,0.312). Zero-field-cooled and field-cooled magnetic

susceptibility of PuCoGa5 were measured in 10 Oe for a fresh (filled symbols) and an aged

(open symbols) single crystal. Magnetization measurements were performed in a

Quantum Design superconducting quantum interference device (SQUID) magnetometer

with the sample sealed in an alumina holder designed to minimize background signal and

prevent spread of radioactive contamination. The inset shows heat capacity, plotted as

heat capacity, C, divided by temperature versus temperature for PuCoGa5. Heat capacity

measurements were made in a Quantum Design Physical Property Measurement System

on a 27-mg single crystal. Self-heating limited the lowest measurement temperature to

6.6 K, although the calorimeter reached a base temperature of 1.5 K. The jump in C/T

at T c corresponds to DC/T c ! 110 ^ 4mJmol21 K22. Assuming the BCS value for

DC/gT c yields g ! 77mJmol21 K22. Fitting the heat capacity data above T c yields an

estimate of the Debye temperature of 240 K.

letters to nature

NATURE |VOL 420 | 21 NOVEMBER 2002 | www.nature.com/nature 297© 2002 Nature Publishing Group

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115 Compounds

H c1 < 350Oe from the field at which M deviates from its initiallinear H dependence. The approximate linearity of M(H) to muchhigher fields is indicative of strong flux pinning and consistent withthe absence of a strong Meissner effect in the field-cooled data ofFig. 1. The geometric mean ofHc1 andHc2 gives the thermodynamiccritical fieldHc < 16 kOe. These values ofHc2,Hc1, andHc indicatethat PuCoGa5 is a strongly type II superconductor with Ginzburg–Landau parameter k < 32, and London penetration depth about124 nm. From the condensation energy 21=2N!0"D2!0" #H2

c=8pm0 and the BCS result D(0) # 1.76 kTc, we estimate g#73mJmol21 K22; again consistent with the data of Fig. 1, whereN(0) is the density of electronic states,D is the superconducting gap,m0 is permeability and k is Boltzmann’s constant. Magnetizationdata also have been obtained for a larger, more well-shaped singlecrystal; however, the data are limited to T . 0.9 Tc owing tosaturation of the SQUID detection system. A simple estimate ofthe critical current can be made from these higher temperature datausing the Bean critical state model10. We find low-field values ofJc . 104 A cm22 for T . 0:9Tc: Such values are competitive withthe best available applied superconductors. Further, in contrast tothe degradation of Tc, Jc increases with time by a factor of nearly 2over the same time period that Tc decreases by 0.4 K, evidentlyowing to the increasing number of radiation-induced pinningcentres.The superconducting properties of PuCoGa5 are surprisingly well

suited to the self-damage mechanism of Pu. The radioactive decayof 239Pu results in the formation of a high-energy alpha particle anda U nucleus. The principal damage is done by the U nucleus, whichis displaced by approximately 12 nm and creates about 2,300 Frenkelpairs of vacancies and displaced interstitials distributed over a rangeof 7.5 nm (ref. 11). This damage is randomly distributed through-out the bulk of the material because it arises from spontaneousdecay of Pu. Defects with spatial dimensions of the order of thesuperconducting coherence length create effective flux pinning andhigh critical currents (ref. 12).A detailed understanding of what gives rise to such high-tem-

perature superconductivity in PuCoGa5 must await comprehensivestudies of both its superconducting and normal states; however,available data are suggestive. Figure 3 shows magnetic susceptibilityand electrical resistivity data over a broad temperature range forPuCoGa5. The temperature dependence of the electrical resistivity isreminiscent of that of UMGa5 (refs 3, 13) and suggests the presence

of spin-disorder scattering at high temperature. The temperaturedependence of the magnetic susceptibility of PuCoGa5 is indicativeof local-moment behaviour close to that expected for Pu3$. A local-moment susceptibility is also found in CeCoIn5 (ref. 4) butisostructural UCoGa5 (ref. 3) displays temperature-independentparamagnetism consistent with itinerant f-electron behaviour.Taken together, these data suggest that the degree of 5f-electronlocalization in PuCoGa5 falls between that of its Ce-based andU-based analogues. This conclusion is consistent with our severalestimates of the low-temperature enhancement of electronic heatcapacity and leads to the reasonable speculation that the supercon-ductivity in PuCoGa5may be unconventional. Although this may bequestioned, the alternative, 18-K phonon-mediated superconduc-tivity in the presence of local-moment susceptibility, is equallychallenging. Moreover, the isostructural UCoGa5, which shows nosign of a local moment, is not a superconductor.

In a scenario of unconventional superconductivity, the nearlyorder-of-magnitude-higher Tc in PuCoGa5 relative to that ofCeCoIn5 would be attributable to increased hybridization consist-ent with predictions for models of magnetically mediated super-conductivity14. Further, the 5f electrons of the actinides areintermediate between the more localized 4f electrons of the rareearths and the itinerant d electrons of the transition metals. Thecombination of layered crystal structure with greater delocalizationmay suggest that the Tc < 200mK observed in CeIn3 (ref. 14), thethree-dimensional, low-Tc analogue of CeCoIn5 (ref. 4), is notunrelated to the Tc < 20Kof PuCoGa5. As a result, the transuranicsmay represent a promising field for superconductivity, intermediatebetween the known heavy-fermion superconductors and the high-Tc copper oxides. A

Received 2 September; accepted 14 October 2002; doi:10.1038/nature01212.

1. Hecker, S. S. The complex world of plutonium science. MRS Bull. 26, 672–678 (2001).

2. Savrasov, S. Y., Kotliar, G. & Abrahams, E. Correlated electrons in d-plutonium within a dynamical

mean-field picture. Nature 410, 793–795 (2001).

3. Grin, Yu. N., Rogl, P. & Hiebl, K. Structural chemistry and magnetic behavior of ternary uranium

gallides U(Fe,Co,Ni,Ru,Rh,Pd,Os,Ir,Pt)-Ga5. J. Less Common Met. 121, 497–505 (1986).

4. Petrovic, C. et al. Heavy-fermion superconductivity in CeCoIn5 at 2.3 K. J. Phys. Condens. Matt. 13,

L337–L342 (2001).

Figure 2 Upper critical field of PuCoGa5 as a function of temperature. The upper insetshows the field-dependent resistivity data from which the field-temperature phase

diagram was deduced. The lower inset shows a representative magnetization loop for

PuCoGa5, measured at 5 K.Figure 3 Normal-state properties of PuCoGa5. The electrical resistivity r-(circles)

increases approximately as T 1.35 from just above T c to 50 K. The magnetic susceptibility

x ; M/H (squares) as a function of temperature follows x# x0 $ C=!T 2 v" withan effective moment m# !8C "1=2 # 0:68mB and an interaction temperature

v < 2 2 K.

letters to nature

NATURE | VOL 420 | 21 NOVEMBER 2002 | www.nature.com/nature298 © 2002 Nature Publishing Group

PuCoGa5 : 20 K Superconductor..............................................................


























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‘02

‘07

0.2K CeIn3’98

115 Compounds














L337–L342 (2001).






v < 2 2 K.

letters to nature



Superconducting

..............................................................


























letters to nature


..............................................................


























letters to nature


Magnetic



TcTc

2K CeCoIn5 ‘01

18.5K PuCoGa5

NpAl2Pd54.5K

‘02

‘07

0.2K CeIn3’98

115 Compounds














L337–L342 (2001).






v < 2 2 K.

letters to nature



Superconducting

..............................................................


























letters to nature


..............................................................


























letters to nature


Magnetic


12K

92K

Ba2201

YBa2Cu3O7

Oxides

Tc


TcTc

2K CeCoIn5 ‘01

18.5K PuCoGa5

NpAl2Pd54.5K

‘02

‘07

0.2K CeIn3’98

115 Compounds














L337–L342 (2001).






v < 2 2 K.

letters to nature



Superconducting

..............................................................


























letters to nature


..............................................................


























letters to nature


Magnetic


?

12K

92K

Ba2201

YBa2Cu3O7

Oxides

Tc

RT


Cuprates Iron-based superconductors

Heavy fermions

SC SC

Magnetic pairing: ubiquitous


Insights into SC from Entropy.

L. Boltzmann



L. BoltzmannW⎬⎫⎭

Many (W) excitedstates

T>0



Cool L. Boltzmann

Heat lost on cooling = k log W

W⎬⎫⎭


T>0

SINGLEground state T=0

= amount of orderSunday, February 17, 13


Cool L. Boltzmann

Heat lost on cooling = k log W

W⎬⎫⎭


T>0

SINGLEground state T=0

= amount of orderSunday, February 17, 13

L. Boltzmann

Heat lost on cooling = k log W = amount of order



L. Boltzmann⇠ 1

5

R log 2

Area =Amountof orderin SC

Heat lost on cooling = k log W = amount of order



L. Boltzmann⇠ 1

5

R log 2

Area =Amountof orderin SC

This entropy is magnetic “Spins form pairs”



Local Moment

The remarkable case of NpPd5Al2

4.5K Heavy Fermion S.C NpAl2Pd5

Aoki et al 2007Sunday, February 17, 13

Local Moment



Aoki et al 2007

dependence. It is noticed that the resistivity decreaseslinearly below 10 K, as shown in the inset of Fig. 2(a),indicating a non-Fermi liquid character. At Tc ! 5:0 K, theresistivity shows a sharp drop and becomes zero, indicatingthe superconducting transition.

Superconductivity is stable against the magnetic field, asshown in Fig. 2(b), and is found to be highly anisotropicwith respect to the direction of the magnetic field. Thesuperconducting transition is defined as the zero-resistivityin the resistivity measurement under magnetic field, whichcorresponds to the upper critical field Hc2.

Figure 3 shows the temperature dependence of Hc2 forH k "100# and "001#. The value of Hc2 at 0 K, Hc2$0%, and theslope of Hc2 at Tc, &dHc2=dT , are obtained as Hc2$0% !37 kOe and &dHc2=dT ! 64 kOe/K for H k "100#, andHc2$0% ! 143 kOe and &dHc2=dT ! 310 kOe/K forH k "001#. The value of &dHc2=dT is extremely large, butthe upper critical field is strongly suppressed with decreasingtemperature, suggesting the existence of a large Pauliparamagnetic e!ect.

Figure 4 shows the angular dependence of Hc2 at 80 mK.Hc2 is highly anisotropic and large for H k "001#. Herewe assumed that anisotropy of Hc2 is mainly due to thetopology of the Fermi surface. We tried to fit the Hc2 data tothe so-called anisotropic e!ective mass model, as in aheavy-fermion superconductor PuRhGa5.7) The solid line inFig. 4 is the result of fitting, using the following function:

Hc2$!% !Hc2$! ! 90'%!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

sin2 ! (m)cm)a

cos2 !

s ; $1%

where m)c=m)a is the mass anisotropy ratio for [001] and

[100] directions, and ! is the field angle from [001] to [100].The value of m)c=m

)a ! 0:067 or m)a=m

)c ! 14:9 is compared

to the value of m)c=m)a ! 3:9 in PuRhGa5, for example. In

the case of PuRhGa5, the electronic state is considered tobe quasi-two-dimensional, indicating an ellipsoidal Fermisurface elongated along the [001] direction. On the otherhand, the present ellipsoidal Fermi surface in NpPd5Al2 isextremely flat as a pancake, as shown in the inset of Fig. 4.Here we note that Hc2 in the (001) plane possesses four-foldsymmetry, reflecting the tetragonal structure: Hc2 ! 37:0kOe for H k "100# and Hc2 ! 36:6 kOe for H k "110#.

Next we show in Fig. 5 the temperature dependence of thespecific heat C in the form of C=T . The specific heat jump!C at Tc ! 4:9 K is due to the superconducting transition.

[001]

[100][010]

NpPd5Al2

Pd(1)Np

Pd(2)

Al

C

B O

A

Fig. 1. Tetragonal crystal structure of NpPd5Al2.

80

60

40

20

0

(µ!

. cm

)"

3002001000Temperature (K)

NpPd5Al2J // [100]

10

5

0

(µ!

. cm

)"

86420Temperature (K)

1 kOe 711

50

J // [100]H // [010]

2025303335

20

10

0

(µ!

. cm

)"

20151050 K

(a)

(b)

Fig. 2. (a) Temperature dependence of the electrical resistivity and (b) theresistivity under vaious constant magnetic fields in NpPd5Al2.

200

150

100

50

0

Hc2

(kO

e)

6420

Temperature (K)

H // [001]

[100]

NpPd5Al2

Fig. 3. Temperature dependence of the upper critical field Hc2 forH k "100# and [001].

J. Phys. Soc. Jpn., Vol. 76, No. 6 LETTERS D. AOKI et al.

063701-2


Local Moment



Aoki et al 2007

NpPd5Al2 TC = 4.5K

NpPd5Al2 TC = 4.5K


Local Moment



Aoki et al 2007

NpPd5Al2 TC = 4.5K

NpPd5Al2 TC = 4.5K


Local Moment



Aoki et al 2007

NpPd5Al2 TC = 4.5K

NpPd5Al2 TC = 4.5K

Magnetic moments


Local Moment



Aoki et al 2007

NpPd5Al2 TC = 4.5K

NpPd5Al2 TC = 4.5K


Local Moment



Aoki et al 2007

NpPd5Al2 TC = 4.5K

NpPd5Al2 TC = 4.5K

~1/3 R ln(2)


Local Moment



Aoki et al 2007

NpPd5Al2 TC = 4.5K

NpPd5Al2 TC = 4.5K

~1/3 R ln(2)

How does the spin form the condensate?




Aoki et al 2007

NpPd5Al2 TC = 4.5K

NpPd5Al2 TC = 4.5K

~1/3 R ln(2)

How does the spin form the condensate?

“COMPOSITE PAIR”

Abrahams, Balatsky, Schrieffer and Scalapino (1994)

Flint, Dzero, Coleman (2008)


•Fruit-fly of 21st C.•From the nucleus to the periodic table.•Magnetism and Superconductivity•A new convergence•Mysteries of magnetic pairing.•Hidden Order in URu2Si2


URuSi

Hidden Order in URu2Si2


URuSi

�



URuSi

Huge amount of order.

�


=0.14 x 17.5 K =2.45 J/mol/K=0.42 R ln 2


URuSi

Huge amount of order.

�


Yet the hiddenorder has remained unidentifiedfor more than 25 years.

=0.14 x 17.5 K =2.45 J/mol/K=0.42 R ln 2


Cause Célèbre: state of the art spectroscopies

Scanning Tunneling Microscopy



the precise determination of the onset temperature difficult.Regardless, we find the temperature dependence of ΔHO!T" tofollow a mean-field behavior with an onset temperature ofTHO ∼ 16 K (Fig. 4C). Broken symmetry at the surface is likelyto influence the HO state and may account for the slightlyreduced observed onset temperature relative to that of bulk mea-surements. An important aspect of the ΔHO is the fact that it de-velops asymmetrically relative to the Fermi energy and it shiftscontinuously to lower energies upon lowering of the temperature(Fig. 2 C andD). We quantify the changes to ΔHO and its offset byfitting the data to a BCS function form with an offset energy re-lative to EF (Fig. 2 CandD and Fig. 4D; see the caption of Fig. 4).

The low temperature extrapolation, ΔHO!0" # 4.1$ 0.2 meV,yields 2ΔHO!0"∕kBTHO # 5.8$ 0.3, which together with the valueof the specific heat coefficient γc # C∕T for T > THO (8) withinthe BCS formalism results in a specific heat jump at the transitionofΔC # 6.0$ 1.3 JK−1 mol−1, consistent with previous measure-ments (7, 8, 12). The partial gapping of the Fermi surface ob-served in our spectra also corroborates the recently observedgapping of the incommensurate spin excitations by inelastic neu-tron scattering experiments (12). Finally, the spectrum developsadditional, sharper features within ΔHO at the lowest tempera-tures (Fig. 4B). Such lower energy features may be related tothe gapping of the commensurate spin excitations at the antifer-romagnetic wave vector below THO also seen in inelastic neutronscattering at an energy transfer of about 2 meV (11–13).

The spatial variation of the STM spectra provides additionalinformation about the nature of redistribution of the electronicstates that gives rise to ΔHO. In Fig. 5, we show energy-resolvedspectroscopic maps measured above and below THO, all of whichshow modulation on the atomic scale. The measurements aboveTHO show no changes in their atomic contrast within the energyrange where the ΔHO is developed. In fact, the modulations inthese maps (Fig. 5 B–E) are because of the surface atomic struc-

ture but occur with a contrast that is opposite to that of the STMtopographies of the same region (Fig. 5A). However, observationof reverse contrast in STM conductance maps is expected as aconsequence of the constant current condition. Similar measure-ments below THO are also influenced by the constant currentcondition, as shown in Fig. 5 G–J; nonetheless, these maps showclear indication of the suppression of contrast associated withΔHO at low energies (within the gap; see Fig. 5F) and the conse-quent enhancement at high energies (just outside the gap).

To isolate the spatial structure associated with ΔHO and toovercome any artifacts associated with the measurement settings,we divide the local conductance measured below THO by thatabove for the same atomic region, as shown in Fig. 5 L–O. Suchmaps for jV j < ΔHO illustrate that the suppression of the spectralweight principally occurs in between the surface U atoms. Thesemaps are essentially the spatial variation of the conductanceratios, shown in Fig. 4A. Therefore, consistent with the BCS-likeredistribution of spectral weight, we find that conductance mapratios at energies just above ΔHO illustrate an enhancement be-tween the surface U atoms. Quantifying these spatial variationsfurther, we also plot the correlation between the conductancemap ratios and the atomic locations above and below THO(Fig. 5K) to show that ΔHO is strongest in between the surfaceU atoms—i.e., at the same sites where tunneling to the Kondoresonance is enhanced (Fig. 3E). Our observation that themodulation in the tunneling amplitude into the Kondo resonancecorrelates with the spatial structure of the HO gap shows that thetwo phenomena involve the same electronic states.

Our finding of an asymmetric mean-field-like energy gapwould naively suggest the formation of a periodic redistributionof charge and/or spin at the onset of the HO because of Fermisurface nesting. However, consistent with previous scattering ex-periments (8, 11–13), we find no evidence for any conventionaldensity wave in our experiments. Recently, it has been suggested

Fig. 4. Temperature dependence of the HO gap. (A and B) The experimental data below THO divided by the 18-K data. The data are fit to the formD!V" # !V − V0 − iγ"∕

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!V − V0 − iγ"2 − Δ2

p, which resembles an asymmetric BCS-like DOS with an offset from EF . V0, γ, and Δ are the gap position (offset from

the Fermi energy), the inverse quasi-particle lifetime, and the gap magnitude, respectively. A quasi-particle lifetime broadening of γ ∼ 1.5 mV was extractedfrom the fits. (C) Temperature dependence of the gap extracted from the fits in A (Black Squares) and from a direct fit to the raw data of Fig. 2C (Blue Circles).Both results are comparable within the error bars. The transition temperature THO # 16.0$ 0.4 K is slightly lower than the bulk transition temperaturepresumably as a consequence of the measurement being performed on the surface. (D) Temperature dependence of the gap position Vo extracted fromthe fits. The line is a guide to the eye.

10386 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1005892107 Aynajian et al.

Pegor Aynajian et al,PNAS (2010)

(Supplementary Fig. 1). In Fig. 2c–e we show the atomically resolvedimages of the parameters of the Fano spectrum. Here e0(r), C(r), andf(r) are determined from fitting g(r, E5 eV) for each pixel within theyellow box in Fig. 2a. Significantly, we find that themaximum in bothe0(r) and f(r) and the minimum in C(r) occur at the U sites (X inFig. 2c–e), as would be expected for a Kondo lattice of U atoms. Theseobservations, in combination with theoretical predictions for such aphenomenology12–14, indicate that the r-space ‘Fano lattice’ elec-tronic structure of Kondo screening in magnetic lattices can nowbe visualized.

Evolution of density of states at Si- and U-termination surfaces

For U-terminated surfaces (see Fig. 3a), the spatially averaged densityof states DOS(E) / ,g(E). spectrum for T.To is less structuredthan that of the Si-terminated surface in Figs 2a and 3c. Typical,g(E). spectra are shown as open squares for each listed temperaturebetween 18.6K and 1.9K in the inset to Fig. 3a, with the top spectrumbeing characteristic of T.To. Upon cooling through To, strongchanges are detected in the DOS(E) in a narrow energy range (insetto Fig. 3a). By subtracting the spectrum forT.To, we determine howthe DOS(E) modifications due to the hidden-order state emergerapidly belowTo (Fig. 3b). They are not particle–hole symmetric, withthe predominant effects occurring between –4meV and13meV. Forthe Si-terminated surfaces upon cooling below To, the overall Fanolineshape of DOS(E) as discussed in Fig. 2c–e is unchanged (Fig. 3c).In the inset to Fig. 3c, we show the evolution of the,g(E). spectrumbetween 19K and 1.7K. In each case, the red line is the fit to the Fanospectrum at each temperature (excluding the data points in the biasrange –7.75mV to 6.75mV) while the measured,g(E). spectra areshown as open squares. Again, by subtracting the fitted Fano spectrumfrom the ,g(E). at each T value we determine the temperaturedependence of the hidden-order DOS(E) modifications (Fig. 3d). Atno E value on either surface do these DOS(E) spectra represent acomplete gap. Finally, no changes are observed in the high-energyDOS(E) as the temperature falls below To (Supplementary Fig. 2),perhaps indicating that the basic Brillouin zone geometry is notaltered by the transition.

Heavy f-electron quasiparticle interference imaging

To determine the evolution of k-space electronic structure throughTo, we use heavy-electron quasiparticle interference43–45 (QPI)imaging. The Si-terminated surface has proved unproductive for thispurpose because the Fourier transform of its g(r, E) images (Sup-plementary Fig. 3) are so complex that the multiple bands cannotyet be disentangled. However, in recent studies of heavy-fermionQPIin Sr3Ru2O7 it was shown that replacing 1% of the Ru atoms by Tiatoms produced intense scattering interference and allowed success-ful k-space determination43. Emulating this approach, we substituted1% Th atoms on the U sites, which results in crystals usually cleavingat the U layer. The average spectrum on this U-terminated surfacedevelops the narrow resonantDOS(E) structure below theTo (reddatabetween vertical arrows in Fig. 3b), within which we observe intenseQPI; see the g(q,E)movies in the Supplementary Information.The 1%Th substitution suppresses To by only,1K (refs 46, 47)and does notalter the basic hidden-order phenomenology (refs 46, 47), so the phe-nomena we report are not caused by our dilute Th doping. Moreover,because the energy scale of DOS(E) alterations is consistent withTh-doped specific heat measurements46 and because these alterationsare already detectable in tunnelling within 1K below the bulk transi-tion (blue line in Fig. 3b), the electronic structure of the U-terminatedsurface appears to be bulk representative of the hidden-order phase.

For QPI studies of the hidden-order transition we thereforemeasureg(r,E5 eV) in a 50nm3 50nmfieldof view (FOV)with 250mVenergyresolution andatomic spatial resolutionon theseU-terminated surfaces(the simultaneous topograph is shown in Supplementary Fig. 4). InFig. 4a–f we show simultaneous images of g(r,E) modulations mea-sured at T5 1.9K for six energies near EF within the energy scale where

d

c

j

i

2 mV

–0.75 mV

a g

–3 mV10 nm

f l

7.25 mV

e k

3.25 mV

b h

–1.25 mV

(0, π/a0)

(π/a0,0)

Figure 4 | Energy dependence of heavy f-electron quasiparticleinterference. a–f, Atomically resolved g(r, E) for six energies measured atthe U-terminated surface. Extremely rapid changes in the interferencepatterns occur within an energy range of only a few millielectronvolts. Datawere acquired at –6mV and 25MV setpoint junction resistance. g–l, Fouriertransforms g(q, E) of the g(r, E) in a–f. The associated g(q, E) movie is shownin the Supplementary Information. The length of half-reciprocal unit-cellvectors are shown as dots at the edge of each image. Starting at energiesbelow EF (g), the predominant QPI wavevectors diminish very rapidly untili; upon crossing a few millielectronvolts above EF, they jump to asignificantly larger value and rotate through 45u. Then they again diminishin radius with increasing energy in j, k and l. This evolution is not consistentwith a fixed Q* conventional density wave state but is consistent with anavoided crossing between a light band and a very heavy band.

NATURE |Vol 465 |3 June 2010 ARTICLES

573Macmillan Publishers Limited. All rights reserved©2010

A. R. Schmidt et al., Nature (2010).




the precise determination of the onset temperature difficult.Regardless, we find the temperature dependence of ΔHO!T" tofollow a mean-field behavior with an onset temperature ofTHO ∼ 16 K (Fig. 4C). Broken symmetry at the surface is likelyto influence the HO state and may account for the slightlyreduced observed onset temperature relative to that of bulk mea-surements. An important aspect of the ΔHO is the fact that it de-velops asymmetrically relative to the Fermi energy and it shiftscontinuously to lower energies upon lowering of the temperature(Fig. 2 C andD). We quantify the changes to ΔHO and its offset byfitting the data to a BCS function form with an offset energy re-lative to EF (Fig. 2 CandD and Fig. 4D; see the caption of Fig. 4).

The low temperature extrapolation, ΔHO!0" # 4.1$ 0.2 meV,yields 2ΔHO!0"∕kBTHO # 5.8$ 0.3, which together with the valueof the specific heat coefficient γc # C∕T for T > THO (8) withinthe BCS formalism results in a specific heat jump at the transitionofΔC # 6.0$ 1.3 JK−1 mol−1, consistent with previous measure-ments (7, 8, 12). The partial gapping of the Fermi surface ob-served in our spectra also corroborates the recently observedgapping of the incommensurate spin excitations by inelastic neu-tron scattering experiments (12). Finally, the spectrum developsadditional, sharper features within ΔHO at the lowest tempera-tures (Fig. 4B). Such lower energy features may be related tothe gapping of the commensurate spin excitations at the antifer-romagnetic wave vector below THO also seen in inelastic neutronscattering at an energy transfer of about 2 meV (11–13).

The spatial variation of the STM spectra provides additionalinformation about the nature of redistribution of the electronicstates that gives rise to ΔHO. In Fig. 5, we show energy-resolvedspectroscopic maps measured above and below THO, all of whichshow modulation on the atomic scale. The measurements aboveTHO show no changes in their atomic contrast within the energyrange where the ΔHO is developed. In fact, the modulations inthese maps (Fig. 5 B–E) are because of the surface atomic struc-

ture but occur with a contrast that is opposite to that of the STMtopographies of the same region (Fig. 5A). However, observationof reverse contrast in STM conductance maps is expected as aconsequence of the constant current condition. Similar measure-ments below THO are also influenced by the constant currentcondition, as shown in Fig. 5 G–J; nonetheless, these maps showclear indication of the suppression of contrast associated withΔHO at low energies (within the gap; see Fig. 5F) and the conse-quent enhancement at high energies (just outside the gap).

To isolate the spatial structure associated with ΔHO and toovercome any artifacts associated with the measurement settings,we divide the local conductance measured below THO by thatabove for the same atomic region, as shown in Fig. 5 L–O. Suchmaps for jV j < ΔHO illustrate that the suppression of the spectralweight principally occurs in between the surface U atoms. Thesemaps are essentially the spatial variation of the conductanceratios, shown in Fig. 4A. Therefore, consistent with the BCS-likeredistribution of spectral weight, we find that conductance mapratios at energies just above ΔHO illustrate an enhancement be-tween the surface U atoms. Quantifying these spatial variationsfurther, we also plot the correlation between the conductancemap ratios and the atomic locations above and below THO(Fig. 5K) to show that ΔHO is strongest in between the surfaceU atoms—i.e., at the same sites where tunneling to the Kondoresonance is enhanced (Fig. 3E). Our observation that themodulation in the tunneling amplitude into the Kondo resonancecorrelates with the spatial structure of the HO gap shows that thetwo phenomena involve the same electronic states.

Our finding of an asymmetric mean-field-like energy gapwould naively suggest the formation of a periodic redistributionof charge and/or spin at the onset of the HO because of Fermisurface nesting. However, consistent with previous scattering ex-periments (8, 11–13), we find no evidence for any conventionaldensity wave in our experiments. Recently, it has been suggested

Fig. 4. Temperature dependence of the HO gap. (A and B) The experimental data below THO divided by the 18-K data. The data are fit to the formD!V" # !V − V0 − iγ"∕

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!V − V0 − iγ"2 − Δ2

p, which resembles an asymmetric BCS-like DOS with an offset from EF . V0, γ, and Δ are the gap position (offset from

the Fermi energy), the inverse quasi-particle lifetime, and the gap magnitude, respectively. A quasi-particle lifetime broadening of γ ∼ 1.5 mV was extractedfrom the fits. (C) Temperature dependence of the gap extracted from the fits in A (Black Squares) and from a direct fit to the raw data of Fig. 2C (Blue Circles).Both results are comparable within the error bars. The transition temperature THO # 16.0$ 0.4 K is slightly lower than the bulk transition temperaturepresumably as a consequence of the measurement being performed on the surface. (D) Temperature dependence of the gap position Vo extracted fromthe fits. The line is a guide to the eye.

10386 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1005892107 Aynajian et al.

Pegor Aynajian et al,PNAS (2010)

(Supplementary Fig. 1). In Fig. 2c–e we show the atomically resolvedimages of the parameters of the Fano spectrum. Here e0(r), C(r), andf(r) are determined from fitting g(r, E5 eV) for each pixel within theyellow box in Fig. 2a. Significantly, we find that themaximum in bothe0(r) and f(r) and the minimum in C(r) occur at the U sites (X inFig. 2c–e), as would be expected for a Kondo lattice of U atoms. Theseobservations, in combination with theoretical predictions for such aphenomenology12–14, indicate that the r-space ‘Fano lattice’ elec-tronic structure of Kondo screening in magnetic lattices can nowbe visualized.

Evolution of density of states at Si- and U-termination surfaces

For U-terminated surfaces (see Fig. 3a), the spatially averaged densityof states DOS(E) / ,g(E). spectrum for T.To is less structuredthan that of the Si-terminated surface in Figs 2a and 3c. Typical,g(E). spectra are shown as open squares for each listed temperaturebetween 18.6K and 1.9K in the inset to Fig. 3a, with the top spectrumbeing characteristic of T.To. Upon cooling through To, strongchanges are detected in the DOS(E) in a narrow energy range (insetto Fig. 3a). By subtracting the spectrum forT.To, we determine howthe DOS(E) modifications due to the hidden-order state emergerapidly belowTo (Fig. 3b). They are not particle–hole symmetric, withthe predominant effects occurring between –4meV and13meV. Forthe Si-terminated surfaces upon cooling below To, the overall Fanolineshape of DOS(E) as discussed in Fig. 2c–e is unchanged (Fig. 3c).In the inset to Fig. 3c, we show the evolution of the,g(E). spectrumbetween 19K and 1.7K. In each case, the red line is the fit to the Fanospectrum at each temperature (excluding the data points in the biasrange –7.75mV to 6.75mV) while the measured,g(E). spectra areshown as open squares. Again, by subtracting the fitted Fano spectrumfrom the ,g(E). at each T value we determine the temperaturedependence of the hidden-order DOS(E) modifications (Fig. 3d). Atno E value on either surface do these DOS(E) spectra represent acomplete gap. Finally, no changes are observed in the high-energyDOS(E) as the temperature falls below To (Supplementary Fig. 2),perhaps indicating that the basic Brillouin zone geometry is notaltered by the transition.

Heavy f-electron quasiparticle interference imaging

To determine the evolution of k-space electronic structure throughTo, we use heavy-electron quasiparticle interference43–45 (QPI)imaging. The Si-terminated surface has proved unproductive for thispurpose because the Fourier transform of its g(r, E) images (Sup-plementary Fig. 3) are so complex that the multiple bands cannotyet be disentangled. However, in recent studies of heavy-fermionQPIin Sr3Ru2O7 it was shown that replacing 1% of the Ru atoms by Tiatoms produced intense scattering interference and allowed success-ful k-space determination43. Emulating this approach, we substituted1% Th atoms on the U sites, which results in crystals usually cleavingat the U layer. The average spectrum on this U-terminated surfacedevelops the narrow resonantDOS(E) structure below theTo (reddatabetween vertical arrows in Fig. 3b), within which we observe intenseQPI; see the g(q,E)movies in the Supplementary Information.The 1%Th substitution suppresses To by only,1K (refs 46, 47)and does notalter the basic hidden-order phenomenology (refs 46, 47), so the phe-nomena we report are not caused by our dilute Th doping. Moreover,because the energy scale of DOS(E) alterations is consistent withTh-doped specific heat measurements46 and because these alterationsare already detectable in tunnelling within 1K below the bulk transi-tion (blue line in Fig. 3b), the electronic structure of the U-terminatedsurface appears to be bulk representative of the hidden-order phase.

For QPI studies of the hidden-order transition we thereforemeasureg(r,E5 eV) in a 50nm3 50nmfieldof view (FOV)with 250mVenergyresolution andatomic spatial resolutionon theseU-terminated surfaces(the simultaneous topograph is shown in Supplementary Fig. 4). InFig. 4a–f we show simultaneous images of g(r,E) modulations mea-sured at T5 1.9K for six energies near EF within the energy scale where

d

c

j

i

2 mV

–0.75 mV

a g

–3 mV10 nm

f l

7.25 mV

e k

3.25 mV

b h

–1.25 mV

(0, π/a0)

(π/a0,0)

Figure 4 | Energy dependence of heavy f-electron quasiparticleinterference. a–f, Atomically resolved g(r, E) for six energies measured atthe U-terminated surface. Extremely rapid changes in the interferencepatterns occur within an energy range of only a few millielectronvolts. Datawere acquired at –6mV and 25MV setpoint junction resistance. g–l, Fouriertransforms g(q, E) of the g(r, E) in a–f. The associated g(q, E) movie is shownin the Supplementary Information. The length of half-reciprocal unit-cellvectors are shown as dots at the edge of each image. Starting at energiesbelow EF (g), the predominant QPI wavevectors diminish very rapidly untili; upon crossing a few millielectronvolts above EF, they jump to asignificantly larger value and rotate through 45u. Then they again diminishin radius with increasing energy in j, k and l. This evolution is not consistentwith a fixed Q* conventional density wave state but is consistent with anavoided crossing between a light band and a very heavy band.

NATURE |Vol 465 |3 June 2010 ARTICLES

573Macmillan Publishers Limited. All rights reserved©2010

A. R. Schmidt et al., Nature (2010).


Hybridization of U 5f states developsat T0


electron spinStrange electron spin of URu2Si2


electron spin

magnetic field B

gµBB

Strange electron spin of URu2Si2


electron spin

magnetic field B

gµBB



electron spin

magnetic field B

gµBB

M = g(✓)µB = 2µB

Isotropic moment



electron spin

magnetic field B

gµBB

M = g(✓)µB = 2µB

Isotropic momentS=1/2



electron spin

gµBB

URuSi

URu2Si2



electron spin

gµBB

URuSi

URu2Si2



electron spin

gµBB

URuSi

URu2Si2

No splitting in transverse direction



electron spin

gµBB

URuSi

URu2Si2


M = gµB cos ✓ = Mz

Magnetic moment only along z-axis



electron spin

gµBB

URuSi

URu2Si2




“Ising moment”



electron spin

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ARTICLEdoi:10.1038/nature11820

Hastatic order in the heavy-fermioncompound URu2Si2Premala Chandra1, Piers Coleman1,2 & Rebecca Flint3

The development of collective long-range order by means of phase transitions occurs by the spontaneous breaking offundamental symmetries. Magnetism is a consequence of broken time-reversal symmetry, whereas superfluidity resultsfrom broken gauge invariance. The broken symmetry that develops below 17.5 kelvin in the heavy-fermion compoundURu2Si2 has long eluded such identification. Here we show that the recent observation of Ising quasiparticles in URu2Si2results from a spinor order parameter that breaks double time-reversal symmetry, mixing states of integer andhalf-integer spin. Such ‘hastatic’ order hybridizes uranium-atom conduction electrons with Ising 5f2 states toproduce Ising quasiparticles; it accounts for the large entropy of condensation and the magnetic anomaly observed intorque magnetometry. Hastatic order predicts a tiny transverse moment in the conduction-electron ‘sea’, a colossalIsing anisotropy in the nonlinear susceptibility anomaly and a resonant, energy-dependent nematicity in the tunnellingdensity of states.

The hidden order that develops below THO 5 17.5 K in the heavy-fer-mion compound URu2Si2 is particularly notable, having eluded iden-tification for 25 years1–12. Recent spectroscopic13–17, magnetometric18

and high-field measurements19,20 suggest that the hidden order is con-nected with the formation of an itinerant heavy-electron fluid, as aconsequence of quasiparticle hybridization between localized, spin–orbit-coupled f-shell moments and mobile conduction electrons.Although the development of hybridization at low temperatures isusually associated with a crossover, in URu2Si2 both optical17 andtunnelling14–16 probes suggest that it develops abruptly at the hidden-order transition, leading to proposals9,10 that the hybridization is anorder parameter.

Ising quasiparticlesHigh-temperature bulk susceptibility measurements on URu2Si2

show that the local 5f moments embedded in the conduction-electronsea are Ising in nature1,21, and quantum oscillation experiments deepwithin the hidden-order phase22 reveal that the quasiparticles possessa giant Ising anisotropy20,23,24. The Zeeman splitting DE(h) dependssolely on the c-axis component of the magnetic field: DE 5 g(h)mBB(ref. 24). Here B is the magnetic field, mB is the Bohr magneton and theempirically determined g-factor takes the form g(h) 5 gcos(h), whereh is the angle between the magnetic field and the c axis and g is theIsing g-factor. The g-factor anisotropy exceeds 30, corresponding toan anisotropy of the Pauli susceptibility in excess of 900; this aniso-tropy is also observed in the angle dependence of the Pauli-limitedupper critical field of the superconducting state23,24, showing that theIsing quasiparticles pair to form a heavy-fermion superconductor.This giant anisotropy suggests that the f moment is transferred tothe mobile quasiparticles through hybridization25.

In the tetragonal crystalline environment of URu2Si2, such Isinganisotropy is most natural in an integer-spin 5f 2 configuration ofthe uranium ions4,26. Although a variety of singlet crystal-fieldschemes have been proposed6,27, the observation of paired Ising qua-siparticles in a superconductor with a transition temperature of

Tc < 1.5 K indicates that this 5f 2 configuration is doubly degenerateto within an energy resolution of gmBHc2 < 5 K, where Hc2 is theupper critical field of the superconductor. Moreover, the obser-vation of multiple spin zeroes in the quantum oscillations, result-ing from the interference of Zeeman split orbits in a tilted field,requires that in a transverse field the underlying 5f 2 configura-tion is doubly degenerate to within a cyclotron energy, which isBvc~BeB=m!<1:5 K for the largest extremal orbit20,22 (a)(m*5 12.5me measured in B 5 13.9 T, where me is the electron mass).These tiny bounds suggest that the Ising 5f 2 state is intrinsicallydegenerate. In URu2Si2, tetragonal symmetry protects such a mag-netic non-Kramers C5 doublet28, the candidate origin of the Isingquasiparticles4,29.

The quasiparticle hybridization of half-integer-spin conductionelectrons with an integer-spin doublet in URu2Si2 has profound impli-cations for hidden order; such mixing can not occur without the break-ing of double time-reversal symmetry. Time-reversal, H, is an anti-unitary quantum operator with no associated quantum number30.However double time-reversal, H2, which is equivalent to a 2p rotation,forms a unitary operator with an associated quantum number, the‘Kramers index’, K (ref. 30). For a quantum state of total angularmomentum J, K 5 (21)2J defines the phase factor acquired by itswavefunction after two successive time-reversals: H2 yj i~K yj i~y2p!! "

. An integer-spin state jaæ is unchanged by a 2p rotation, andso ja2pæ 5 1jaæ and K 5 1. However, conduction electrons with half-integer-spin states, jksæ, where k is the vector momentum and s is thespin component, change sign: jks2pæ 5 2jksæ. Hence, K 5 21 forconduction electrons.

Double time-reversal symmetryAlthough conventional magnetism breaks time-reversal symmetry, itis invariant under H2, with the result that the Kramers index is con-served. However, in URu2Si2 the hybridization between integer-spinand half-integer-spin states requires a quasiparticle mixing term ofthe form H~ ksj iVsa k" # ah jzH:c:, where H.c. indicates Hermitian

1Center for Materials Theory, Department of Physics and Astronomy, Rutgers University, 136 Frelinghuysen Road, Piscataway, New Jersey 08854-8019, USA. 2Department of Physics, Royal Holloway,University of London, Egham, Surrey TW20 0EX, UK. 3Department of Physics, Massachusetts Institute for Technology, Massachusetts Avenue, Cambridge, Massachusetts 02139-4307, USA.

3 1 J A N U A R Y 2 0 1 3 | V O L 4 9 3 | N A T U R E | 6 2 1

Macmillan Publishers Limited. All rights reserved©2013

































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While we understand most of the physicsat the scale of a nanometer, the emergentproperties that develop between the nanometer and the micron are only just beginning to be understood.

Actinide electron physics is playing a vital rolein exploring this new frontier.



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