Electro-nuclear transition into a spatially modulated magnetic state in YbRh 2Si2 J. Knapp L. V. Levitin J. Ny eki A. F. Ho B. Cowan and J. Saunders Department of Physics Royal Holloway University of London TW20 0EX Egham UK.

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Electro-nuclear transition into a spatially modulated magnetic state in YbRh2Si2

J. Knapp, L. V. Levitin, J. Ny´eki, A. F. Ho, B. Cowan, and J. Saunders

Department of Physics, Royal Holloway University of London, TW20 0EX, Egham, UK. ∗

M. Brando and C. Geibel

Max Planck Institute for Chemical Physics of Solids,

N¨othnitzer Straße 40, 01187 Dresden, Germany.

K. Kliemt and C. Krellner

Physikalisches Institut, Max-von-Laue-Straße 1, 60438 Frankfurt am Main, Germany.

(Dated: 9 January 2023)

The nature of the antiferromagnetic order in the heavy fermion metal YbRh2Si2, its quan-

tum criticality, and superconductivity, which appears at low mK temperatures, remain open

questions. We report measurements of the heat capacity over the wide temperature range

180 µK - 80 mK, using current sensing noise thermometry. In zero magnetic ﬁeld we observe

a remarkably sharp heat capacity anomaly at 1.5 mK, which we identify as an electro-nuclear

transition into a state with spatially modulated electronic magnetic order of maximum amplitude

0.1 µB. We also report results of measurements in magnetic ﬁelds in the range 0 to 70 mT,

applied perpendicular to the c-axis, which show eventual suppression of this order. These re-

sults demonstrate a coexistence of a large moment antiferromagnet with putative superconductivity.

The interplay of magnetism and superconductivity is a

central question in the study of strongly correlated elec-

tronic systems. In heavy fermion (HF) metals a par-

ticular advantage is the ability to tune the system to

a quantum critical point (QCP), by pressure or some

other tuning parameter, at which superconductivity can

emerge. In YbRh2Si2magnetic ﬁeld provides a conve-

nient tuning parameter, at ambient pressure, and with-

out recourse to doping. However superconductivity in

YbRh2Si2only appears at low mK temperatures, imply-

ing extremely low thermodynamic critical ﬁelds. The on-

set of strong magnetic screening and a heat capacity peak

observed in the vicinity of 2 mK [1] have been interpreted

in terms of a simultaneous superconducting and electro-

nuclear magnetic phase transition. The experiment we

report in this Letter focuses on a detailed and precise

investigation of this transition, on establishing the mag-

netic ground state, and its evolution with magnetic ﬁeld.

YbRh2Si2has tetragonal symmetry and a theoretically

predicted highly anisotropic, three dimensional Fermi

surface [2–6]. Antiferromagnetic (AFM) electronic or-

der appears in zero applied ﬁeld at TN= 70 mK and

features ultra-small ordered moments, µe≈0.002 µB[7],

which develop out of partially Kondo-screened Yb local

moments 1.4µB[8]. The nature of this order is not estab-

lished, with an interesting possibility of the ordered mo-

ments aligned with the magnetically-hard c-axis [9]. Neu-

tron scattering, above TN, shows incommensurate AFM

ﬂuctuations which emerge from ferromagnetic (FM) ﬂuc-

tuations [10]. Static magnetic susceptibility [11], NMR

[12] and ESR [13–15] also provide evidence of FM ﬂuc-

tuations.

The observed suppression of TNby magnetic ﬁeld at

ambient pressure on high quality samples ﬁrst led to

the proposal of a QCP, induced by an in-plane ﬁeld of

Bc= 60 mT, or ten times larger ﬁeld along the c-axis [8],

reﬂecting the highly anisotropic electronic magnetism.

The nature of the putative QCP remains a matter of de-

bate, including theories of local quantum criticality [16–

19], see also [20–24] and theories invoking strong coupling

of fermions and spin ﬂuctuations into critical quasiparti-

cles [25–27]. Negative chemical pressure, achieved by Ge

doping, shifts the QCP to smaller ﬁelds [11, 28], cobalt

doping induces ferromagnetism [9, 29].

Most recently, the report of superconductivity in

YbRh2Si2[1, 30] led to the proposal that an important

role is played by the coupling of electronic and nuclear

magnetism. The strong hyperﬁne interaction and pres-

ence of active Yb nuclei distinguishes YbRh2Si2from Ce-

based HF systems, for which the nuclear moments are

zero. Thus YbRh2Si2provides a model system to inves-

tigate the inﬂuence of nuclear spins in a Kondo lattice ex-

hibiting quantum criticality [31]. The ground state dou-

blet of the Yb ion in the crystalline electric ﬁeld (CEF),

also distinguishes this system from systems with strong

hyperﬁne interactions based on non Kramers ions such as

Pr and Ho [32]. The work reported here presents a ﬁrst

step to precisely thermodynamically characterize the in-

terplay of electronic and nuclear magnetism in YbRh2Si2.

Our experimental set-up exploits advances in cur-

rent sensing noise thermometry [34]. This includes im-

provements in the speed of measurement achieved by a

relatively-high sensor resistance (a 0.2 Ω PtW wire), cou-

pled with the ability to limit the heat leak into the noise

thermometer to below 1 fW by appropriate shielding and

ﬁltering of the leads [35]. The single crystal of YbRh2Si2

from batch 63129 with RRR = 50 [36] is thermalised via

an aluminium wire, operating as a superconducting heat

switch. A superconducting solenoid both provides the

sample ﬁeld and operates the heat switch. The heat ca-

arXiv:2210.03673v2 [cond-mat.str-el] 19 Feb 2023

10−1100101102

Temperature (mK)

10−1

100

101

CM(J/(mol K))

t

0.0 mT

data

FIG. 1. Molar heat capacity in zero ﬁeld exhibits two sharp

anomalies at TAand TNthat we identify with magnetic tran-

sitions. The data between 1.85 and 30 mK are ﬁtted to the nu-

clear and heavy-electron heat capacity. The ﬁt curve is plot-

ted outside of the ﬁtting interval as a dashed line. The small

ordered electronic moments found above TA[7] would signif-

icantly aﬀect the nuclear heat capacity only below 0.2 mK.

0.0 0.5 1.0 1.5 2.0

Temperature (mK)

CM(J/(mol K))

(a)

21.1 mT

14.7 mT

10.5 mT

8.4 mT

0.0 mT

1.2 1.3 1.4 1.5

14.7 mT

0 2 4 6

T(mK)

CM(J/(mol K))

(b)

35.9 mT

10.5 mT

0 50 100

Bext (mT)

0.00

0.05

0.10

0.15

µ4f(µB)

(c)

χat 20 mK

Schottky ts

FIG. 2. Molar heat capacity in ﬁeld applied in the ab-plane.

(a) Suppression of TAanomaly with ﬁeld. (b) Examples of

ﬁtting the Schottky model. Below 35.9 mT, where TAis ob-

served, only the data above this anomaly are ﬁtted. (c) The

static electronic moment of Yb determined from ac suscepti-

bility χ[33] and from the Schottky model.

pacity is determined by the adiabatic heat pulse method

below 10 mT, the critical ﬁeld of aluminium, and by the

relaxation method above it.

The molar heat capacity in zero applied ﬁeld, Fig. 1,

shows the well-known N´eel anomaly at TN= 70.5 mK,

and another sharp anomaly at TA= 1.5 mK. The heat ca-

pacity measured around 1 mK exceeds the heavy-electron

term γSTby 3 orders of magnitude. We demonstrate

that this large heat capacity originates from Yb nuclear

degrees of freedom, however the TAanomaly reﬂects a co-

operative transition involving both nuclei and electrons.

On the other hand, above a few mK the nuclear heat

capacity decreases as T−2, while the electronic heat ca-

pacity increases linearly with temperature. As a result,

above 20 mK the electronic part dominates (see SI, Fig.

7), and thus at TNthe nuclear spin degrees of freedom

play no role [37], contributing less than 1% to the heat

capacity there. While the electronic moments form a reg-

ular lattice, only a minority of Yb sites carry a nuclear

moment. Thus the low temperature heat capacity arises

from the nuclei of 171Yb (I= 1/2) and 173 Yb (I= 5/2)

isotopes with 0.1431 and 0.1613 natural abundances re-

spectively, distributed randomly across Yb sites. The nu-

clear spins are subject to an eﬀective hyperﬁne magnetic

ﬁeld Bhf =−Ahf µe, produced by the ordered static part

of Yb electronic moments µe. Here Ahf = 102 T/µB

is the hyperﬁne constant [38–40]. The “fast relaxation

regime”, realised in YbRh2Si2[37, 41–44], enables us to

ignore the hyperﬁne ﬁeld due to the ﬂuctuating part of

the electronic moments and treat µeas a mean ﬁeld. Ad-

ditionally the 173Yb nuclei experience quadrupolar split-

ting in the crystalline electric ﬁeld gradient, that points

along the c-axis.

We neglect interactions between nuclei and consider a

single-spin Hamiltonian

H=gµNAhfb

I·µe+e2qQ

4I(2I−1)(3b

z−I(I+ 1)),(1)

where g,µN,Qare the nuclear g-factor, magneton and

quadrupole moment and eq represents the electric ﬁeld

gradient. In general the nuclear spin b

Iis not aligned

with the c axis and Eq. (1) is diagonalised numerically, to

calculate the partition function Zof the nuclear system,

and hence thermodynamic quantities. These are summed

over a random distribution of 171Yb and 173Yb nuclei

according to their natural abundance.

In the simplest case, the Schottky model, we assume

uniform µeon all Yb sites. Fitting the data above TA

unambiguously proves that the size of the static elec-

tronic moment in zero magnetic ﬁeld is small, in agree-

ment with measurements of muon spin resonance [7].

We put an upper bound µe0.01 µB(in any di-

rection) and directly determine the parameters of the

quadrupole splitting. We ﬁnd a positive electric ﬁeld

gradient eq = (2.06 ±0.01) ·1021 Vm-2, less than half of

the previously used estimates [37, 42, 45], and obtain the

Sommerfeld coeﬃcient γS= (1.65 ±0.01) J/(mol K2), in

good agreement with previous work [46].

Fig. 2(a) shows the evolution of the TAanomaly with

magnetic ﬁeld Bext applied in the ab-plane in the range

0.0-21.1 mT. The anomaly shifts to lower temperatures

10−1100

Temperature (mK)

102

103

104

CM/T(J/(mol K2))

Schottky t

SMO t

x(a)

−0.1

0.0

0.1

µe(µB)

FIG. 3. Zero ﬁeld CM/T with the ﬁt according to the SMO

model below 0.5 mK; the “best ﬁt” according to spatially ho-

mogeneous Schottky model clearly disagrees with the data.

The inset: electronic moments in SMO, black arrows repre-

sent randomly distributed Yb sites with active nuclei.

with increasing applied ﬁeld, broadens, and possibly de-

velops a structure (a split into a double peak is observed

at 14.7 mT). Measurements in ﬁelds in the range 35.9-

69.7 mT do not display any anomaly and their overall

shape resembles a typical Schottky peak.

In all magnetic ﬁelds the data above TA(or down

to the lowest temperatures at Bext >35.9 mT, where

the anomaly was not observed) are well described by

the Schottky model, assuming paramagnetic polarisation

µekBext, see Fig. 2(b). Fixing the quadrupolar param-

eters for 173Yb at the zero ﬁeld value, we ﬁnd approx-

imately linear growth of µewith ﬁeld up to ≈0.1µB

at Bc= 60 mT, with weaker increase at higher ﬁelds,

Fig.2(c), consistent with the magnetic susceptibility mea-

surements [33, 47, 48]. More subtle eﬀects, such as the

temperature dependence of µe, may improve the agree-

ment between the data and the model.

We now discuss the transition at TAand the data down

to the lowest temperatures. The entropy release below

10 mK matches well the full nuclear entropy of 171Yb and

173Yb, SYb = 3.22 J/(mol K) in all magnetic ﬁelds, see

Fig. S1 in SI. Under the conditions of our experiments the

nuclear spins 29Si and 103 Rh remain disordered and do

not contribute to the heat capacity due to weak hyperﬁne

constants for these elements.

Continuous warm-up measurements in zero magnetic

ﬁeld suggest, despite the sharpness of the heat capac-

ity anomaly, that the phase transition is continuous, see

Fig. S2 in SI. The majority of the Yb nuclear entropy is

released below TA, leaving only 0.06SYb for the transition

region. This points to gradual ordering of Yb nuclear

spins in the hyperﬁne ﬁeld produced by the electronic

moments and supports the picture of a nuclear-assisted

electronic transition, developed later.

In zero magnetic ﬁeld, the relatively slow decrease of

the heat capacity with decreasing temperatures cannot

be accounted for by the Schottky model with uniform µe,

10−1100101

Temperature (mK)

10−1

100

101

CM(J/(mol K))

CM0.00

0.02

0.04

0.06

0.08

0.10

µA(µB)

µA

FIG. 4. Model of the zero-ﬁeld heat capacity using a simple

ansatz for the temperature dependence of the order parame-

ter, Eq. (3).

see Fig. 3. We therefore postulate a spatially modulated

electronic order (SMO) state, with a sinusoidal distribu-

tion of the hyperﬁne ﬁeld on the randomly distributed

171Yb and 173 Yb nuclei (see inset in Fig. 3) produced by

the electronic moments

µe(r) = µAˆx sin(r·q),(2)

where ˆx is a unit vector, that we assume to lie in the

easy ab plane. The heat capacity derived from the SMO

model is insensitive to q, as long as it is small or incom-

mensurate, leaving a single free parameter, the maximum

value of the modulated electronic moment µA. The ﬁt to

the data below 0.5 mK yields µA= (0.093 ±0.001) µB.

It is noteworthy that this is comparable with the size of

the moment induced by the critical ﬁeld of the TNorder,

see Fig. 2(c). Nuclear magnetic resonance is an estab-

lished tool to conﬁrm the existence of a SMO, or a spin

density wave (SDW), in the absence of direct evidence

from neutron scattering. Here we demonstrate that the

heat capacity of nuclei responding to electronic order is

another powerful probe of SMO, albeit it does not allow

us to determine qin Eq. (2).

To account for the shape of the heat capacity anomaly,

we make a simple ansatz for the temperature dependence

of the order parameter

µA(T) = µA(T= 0) |1−T/TA|βc.(3)

For βc≈0.07 the calculated heat capacity ﬁts the zero-

ﬁeld data well across the whole temperature region, as

shown in Fig. 4. The smallness of the critical exponent

βcand the resulting sharpness of the heat capacity peak

demonstrate signiﬁcant critical point ﬂuctuations.

We now move to a simple mean-ﬁeld model that cap-

tures the TAtransition. We argue that the mechanism

behind this transition in the electronic magnetism is the

hyperﬁne coupling of the static Yb electronic moments

µeto the active Yb nuclei. The key is to recognize the

fragility of the AFM order that emerges at TN. The

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Electro-nucleartransitionintoaspatiallymodulatedmagneticstateinYbRh2Si2J.Knapp,L.V.Levitin,J.Nyeki,A.F.Ho,B.Cowan,andJ.SaundersDepartmentofPhysics,RoyalHollowayUniversityofLondon,TW200EX,Egham,UK.M.BrandoandC.GeibelMaxPlanckInstituteforChemicalPhysicsofSolids,NothnitzerStrae40,01187Dresden,Germa...

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Electro-nuclear transition into a spatially modulated magnetic state in YbRh 2Si2 J. Knapp L. V. Levitin J. Ny eki A. F. Ho B. Cowan and J. Saunders Department of Physics Royal Holloway University of London TW20 0EX Egham UK..pdf

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Electro-nuclear transition into a spatially modulated magnetic state in YbRh 2Si2 J. Knapp L. V. Levitin J. Ny eki A. F. Ho B. Cowan and J. Saunders Department of Physics Royal Holloway University of London TW20 0EX Egham UK.

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