Scholarly article on triuranium tetraphosphide 12037-84-2 from Physical review. B, Condensed matter p. 423

PHYSICAL REVIEW B

VOLUME 55, NUMBER 1

1 JANUARY 1997-I

Near-surface effects at the antiferromagnetic phase transition in uranium phosphide

A. Stunault

ˆ

European Synchrotron Radiation Facility, Boıte Postale 220, F38043 Grenoble, France

S. Langridge

ˆ

European Synchrotron Radiation Facility, Boıte Postale 220, F38043 Grenoble, France

and Commission of the European Communities, Joint Research Centre, Institute for Transuranium Elements,

D-76125 Karlsruhe, Germany

C. Vettier

ˆ

European Synchrotron Radiation Facility, Boıte Postale 220, F38043 Grenoble, France

D. Gibbs

Department of Physics, Brookhaven National Laboratory, Upton, New York 11973

N. Bernhoeft

ˆ

European Synchrotron Radiation Facility, Boıte Postale 220, F38043 Grenoble, France

͑Received 6 March 1996; revised manuscript received 21 March 1996͒

A careful investigation of the resonant x-ray magnetic-scattering response in uranium monophosphide re-

veals the crucial role played by magnetization density correlations in the discontinuous paramagnetic to

antiferromagnetic phase transition in this material. The magnetization density correlations, which are measured

on the short time scale typical of the resonant x-ray-scattering technique, appear highly anisotropic in their

inferred spatial distribution. Such correlated regions exist in the near-surface regime probed by the low-energy

x-ray beam ͑3.278 keV͒ and exhibit strongly temperature-dependent behavior at and above the bulk phase

transition. For specular reﬂections this leads to well deﬁned two length scale diffraction proﬁles in the vicinity

of the phase transition and, when taken with the off-specular data, to a phenomenological model which

suggests that the extended length scale of near-surface antiferromagnetism may arise from a localized surface

distortion. This type of model may be more generally useful both in the two length scale problem and in other

systems where defects or ﬂuctuations may, in a similar manner, drive pseudo-long-range order.

͓S0163-1829͑97͒08401-4͔

I. INTRODUCTION

parallel to the propagation vector. That is, the antiferromag-

netic state consists of alternate up and down sheets of ferro-

magnetically aligned moments, wherein the moment direc-

tion is perpendicular to the sheet and parallel to one of the

original cubic axes. Although the phase transition must be

accompanied by a tetragonal distortion, in practice it is so

weak that we continue to use cubic indexing in discussion.

On account of the symmetry such layers are expected to

occur in three equally populated domains along the three,

orthogonal, cubic axes in the antiferromagnetic state of an

inﬁnite crystal. Any one domain is thus an approximate re-

alization of the antiferromagnetic metamagnetic state or of

‘‘super-antiferromagnetism’’^12,13on which much theoretical

attention has been placed, especially with regard to the na-

ture of the phase transition, possible tricritical points under

the application of an external ﬁeld, and the high-temperature

series expansions of the bulk susceptibility.^14,15

A. Motivation for the study

Unusual and strongly temperature-dependent diffraction

proﬁles observed in the vicinity of structural and magnetic

phase transitions continue to arouse interest.^1–5Pioneering

studies by neutron-diffraction,⁶x-ray diffraction,^7–10and

theoretical methods¹¹revealed the possibility of non-

Lorentzian line shapes in the immediate vicinity of the phase

transition and acted as a stimulus to a host of recent studies

on magnetic systems. These experimental studies have be-

come possible through the combined discovery of resonant

magnetic x-ray diffraction and the availability of

synchrotron-radiation light sources. In turn the data have ac-

tivated much theoretical interest. In the following we report

on the contribution that resonant magnetic x-ray-diffraction

studies of the discontinuous paramagnetic to antiferromag-

netic phase transition in the uranium monopnictides may

make to this problem.

B. Previous work on uranium phosphide „UP…

The monopnictides of interest have a simple NaCl crystal

structure and transform in an apparently discontinuous man-

ner from the disordered paramagnetic state to a simple ͑com-

mensurate͒ type-I antiferromagnetic state with a propagation

Of the systems which may be taken as physical realiza-

tions of the super-antiferromagnetic phase, we have studied a

series of single crystals of high crystalline perfection of com-

position UAs and UP. Preliminary results for the line shapes

vector along equivalent 001 -type directions and moments

͗

͘

0163-1829/97/55͑1͒/423͑16͒/$10.00

55

423

424

A. STUNAULT et al.

55

of resonantly magnetically diffracted x-ray beams from UAs

have been reported;¹⁶in this communication we focus on the

compound UP. While the monopnictide and arsenide readily

form crystals to sizes of the order of millimeters the mono-

phosphide is much more reluctant to grow and is conse-

quently less studied. Nevertheless, previous work on UP in-

cludes neutron diffraction, NMR, magnetization, resistivity,

heat-capacity studies, and volumetric measurements.¹⁷Both

through bulk properties such as the magnetic susceptibility

and through microscopic measurement of the staggered mag-

netization ͑order parameter͒ by magnetic neutron scattering,

a discontinuous transition from paramagnetic to antiferro-

magnetic order has been established at a temperature close to

T_nϳ120 K. In common with data collected on the neighbor-

ing arsenide and pnictide these probes give evidence for

anomalous dynamical behavior of the magnetization density

in both the antiferromagnetically ordered and paramagnetic

phases of UP. Such effects are already apparent in the elec-

trical resistivity, which increases slightly with decreasing

temperature in the paramagnetic state,^18,19and the NMR in-

vestigations which yield evidence for a large energy scale of

magnetic ﬂuctuations which may be characteristic of

strongly coupled 5f and conduction-electron states.^20,21

These ﬁndings are substantiated in the ordered state by an

investigate a discontinuous phase transition one would hope

to distinguish the following regimes: a near-surface magnetic

layer above a paramagnetic bulk ͑the short length scale ther-

modynamic correlations should give a relatively weak and

temperature-independent signal in this interval͒, a region of

coexistence of surface and bulk ͑ordered͒ magnetism where

one may hope to observe a two-component diffraction proﬁle

and ﬁnally, at low temperatures, a diffraction proﬁle domi-

nated by the ordered bulk. In view of this we have examined

in some detail the contribution that resonant magnetic x-ray-

diffraction data of the discontinuous paramagnetic to antifer-

romagnetic phase transition in the uranium monopnictides

may make to our understanding of the antiferromagnetic

phase transition and the observation of two length scale dif-

fraction proﬁles.

The technique of resonant magnetic x-ray diffraction has

been developed extensively over the past ten years and has

proven exceptionally useful in the investigation of large

length scale ﬂuctuations of antiferromagnetic correlations in

the rare-earth and actinide compounds.^31–33The sensitivity

arises from the resonant enhancement of the x-ray scattering

cross section which occurs on tuning the incident x-ray en-

ergy near the L and M absorption edges of the heavy-metal

components coupled with transition selection rules which re-

ﬂect the magnetic state of the heavy-metal ion.^34–36For ura-

nium compounds large enhancements are found to occur at

the M edges. In particular, previous studies at the M_IVedge

of uranium in the related compound UAs have shown the

dominant scattering to be electric dipolar in nature and to

lead to measured magnetic intensities as large as 1% of the

typical charge peaks.^31–33

Our x-ray resonant scattering investigations of the

magnetic-diffraction proﬁles supplement the previous ﬁnd-

ings in UP and strongly suggest the important role that near-

surface-induced magnetism may play in the formation of the

antiferromagnetic state. The idea for these experiments was

prompted by preliminary results obtained on UAs ͑Ref. 16͒

and the consideration of the role played by the discontinuous

phase transition as outlined above. The rest of the paper is

organized as follows: Sec. II gives physical details of the

sample and the diffractometer used in this study in addition

to presenting the key experimental result of the paper, the

observation of an anisotropic two length scale diffraction

proﬁle occurring in the immediate vicinity of the antiferro-

magnetic phase transition. The neutron-scattering data al-

luded to both in this section and elsewhere in this paper has

been instrumental in generating the point of view adopted,

however, it represents a sufﬁciently different and detailed

study that we feel obliged to present it fully in a separate

publication.³⁷Section III offers an interpretation of the reso-

nant two length scale line shape based on as complete as

possible analysis of the available x-ray data. In particular, we

address the anisotropic nature of the scattering proﬁle in the

longitudinal and transverse directions within the scattering

plane and attempt an analysis including the thermal evolu-

tion of the magnitude of the effective sublattice magnetic

scattering moment. In Sec. IV a summary of the interpreta-

tion of the data is given together with an analysis of a simple

Ginzburg-Landau model which illustrates the possibility of

generating near-surface ͑antiferro͒-magnetism above a dis-

continuous phase transition. This type of model may be ex-

anomalously large linear term in the electronic speciﬁc-heat

22

coefﬁcient ␥ϭ32 mJ mol^Ϫ1K^Ϫ2

,

the strongly reduced

low-temperature moment of 1.74 ␮_B/uranium ion in

comparison with a Curie-Weiss-like effective moment of

3.3–3.4␮_B,^23–26and the results of recent low-temperature

NMR studies.²⁷However, as in the arsenide, despite careful

searches no studies to date give direct measure of the tetrag-

onal distortion upon entering the antiferromagnetic phase

from the paramagnetic state.^28,29In summary, from the avail-

able literature one is led to conclude that UP undergoes a

transition of a discontinuous nature from the paramagnetic to

a simple type-I antiferromagnetic state without a severe lat-

tice distortion and that there exist important electronic cor-

relations which may play upon the magnitude of the sublat-

tice magnetization density both below and above T_n.

C. Scope of present measurements

It has been suggested that ‘‘second length scale,’’ that is

to say anomalously sharp, diffraction proﬁles in the vicinity

of a magnetic phase transition reﬂect the presence of near-

surface-induced ͑antiferro͒ magnetism.^1–5The sensitivity of

such a state to sample surface preparation has been

demonstrated²as has the location of the effect to a near-

surface regime with a range variously estimated to extend

from some hundreds of Ångstroms²to some hundred

micrometers^4,30below the nominal sample surface. On ap-

proaching a continuous phase transition the presence of such

a near-surface ͑antiferro͒ magnetic layer taken in conjunction

with the diverging correlation length will tend to polarize the

underlying material with the result that the bulk magnetic

response is merged into that of the surface layers making the

separation of the two effects difﬁcult. A cleaner study of

near-surface magnetism may be possible in materials with

relatively short and temperature-independent thermodynamic

correlation lengths such as may exist at a transition of a

discontinuous nature. Using a near-surface sensitive probe to

55

NEAR-SURFACE EFFECTS AT THE . . .

425

tended to other systems where defects or ﬂuctuations may, in

a similar manner, drive pseudo-long-range order. Finally,

Sec. V offers a summary and comparison with other two

length scale results by way of a conclusion.

II. EXPERIMENTAL TECHNIQUE

AND SUMMARY OF DATA

A. Sample preparation and characterization

Despite the difﬁculty in obtaining single-crystal samples,

UP was chosen for this study since, unlike UAs, it gives no

evidence, neither in its bulk properties nor in magnetic-

diffraction data for an incommensurate precursor magnetic

phase above T_n.³⁸This makes possible an analysis of the

diffraction proﬁles with a minimum of parameters which in

turn motivates a simple qualitative model of the phase tran-

sition ͑Sec. IV͒. The sample is that characterized in the pre-

vious neutron-diffraction studies of Burlet et al.³⁹Details of

sample preparation, given in detail by Horyn, Henry, and

Rossat-Mignot,⁴⁰are brieﬂy as follows. An initial reaction at

800–850 °C to form UP₂was followed by decomposition at

1700 °C to yield the UP precursor. This route was developed

on account of the high chemical reactivity of the constituent

elements and the high melting point of the compound ͑in the

region of 2500 °C͒. The resulting UP precursor was dis-

solved in a gallium ﬂux and slowly cooled in a temperature

gradient at 1100 °C over a period of one month to yield a set

of small single crystals. X-ray microprobe analysis indicated

the resulting crystals to be free of gallium contamination.

Although no detailed studies have been carried out, prelimi-

nary investigations of crystalline quality on a rotating anode

spectrometer and the results presented in this paper, indicate

a crystal mosaic of the order of 0.05° full width at half

maximum ͑FWHM͒.

We further characterized the crystals selected for our ex-

periments by their induced bulk magnetic moment in applied

ﬁelds ranging between 100 and 2000 G over the temperature

interval 4.2–300 K as measured in a superconducting quan-

tum interference device magnetometer of the Quantum de-

sign at the CEA Grenoble. The observed bulk magnetic be-

havior of the UP samples is as follows ͑see Fig. 1͒. On

heating, UP exhibits a sharp discontinuous increase in in-

duced moment at a temperature of 120 K which is followed

by a plateau region extending over some 10 K, smooth fall,

and Curie-Weiss-like behavior. The response of UP at high

temperatures is thus generically of the form of that observed

in antiferromagnetic metamagnets and calculated in, for ex-

ample, the series expansions of metamagnetic Ising-like

models.^12,13This simple high-temperature behavior, in con-

trast to that observed in UAs,³⁸is consistent with the absence

of an incommensurate magnetic precursor phase.

FIG. 1. The induced magnetization in UP under an applied ﬁeld

of 500 G as a function of temperature. The applied ﬁeld direction is

approximately parallel to a fourfold axis. The antiferromagnetic

phase transition at 120 K is clearly deﬁned and from the high-

temperature behavior one extracts a Curie-Weiss moment of

3.4␮ per uranium atom. The estimated error in a given data point

B

is approximately represented by the size of the symbol.

turn was mounted on a four-circle diffractometer scattering

in the vertical plane. Care was taken to minimize back-

grounds and absorption by avoiding air in the optical path-

ways; this is necessary on account of the high attenuation of

the low-energy photon beam by nitrogen at atmospheric

pressure. The incoming polarization of the synchrotron ra-

diation, derived from a bending magnet source, is horizontal

and perpendicular to the vertical scattering plane. In this

scattering geometry the 00l and h00 crystallographic di-

͓

͔

͓

͔

rections lie in the scattering plane, respectively, parallel and

perpendicular to the surface normal. To maximize the inten-

sity in the magnetically sensitive signal we tuned the spec-

trometer to the M_IVabsorption edge of uranium which we

experimentally located at 3.728 keV ͑Fig. 2͒. In view of the

severe penalties in intensity no discrimination of the outgo-

ing polarization was carried out. However, from the resonant

B. X-ray measurements and integrated intensities

The resonant x-ray-diffraction measurements were carried

out on the X22C beamline of the National Synchrotron Light

Source, Brookhaven, which is equipped with a toroidally fo-

cusing mirror and a double-crystal Ge͑111͒ monochromator.

The sample was cleaved along a 001 -type plane to yield a

scattering surface of the order of a few square millimeters

shortly before insertion into a displex refrigerator, which in

͕

͖

FIG. 2. Incident photon energy scan around the uranium M_IV

resonance at the antiferromagnetic reﬂection centered on the ͑003͒

Bragg position.

426

A. STUNAULT et al.

55

cant high-temperature ‘‘tail’’ is seen to develop as has been

reported, for example, in UO₂,²nor is any hysteresis evident.

The integrated intensities for the off-specular ͑102͒ reﬂec-

tions which probe magnetic correlations from those domains

with a propagation vector lying in the surface plane of the

sample are, at a given temperature, signiﬁcantly weaker than

those at the ͑003͒ reﬂection. From our analysis ͑Sec. III͒ this

difference in intensity arises primarily from the experimental

resolution.

C. X-ray magnetic-scattering line shapes

As outlined in the introduction and made evident by the

neutron-diffraction studies,³⁷the antiferromagnetic phase ex-

hibits domains of moment alignment parallel to the three

orthogonal cubic axes. Through the width in reciprocal space

of the measured response ͑down to the resolution limit͒ we

infer the spatial extent of magnetic domains located within

the near-surface volume probed by the x-ray beam. The do-

mains have a propagation wave vector either normal to, or in

the plane of, the surface. The response localized around the

specular antiferromagnetic peak is selectively sensitive to

those domains having an ordering wave vector parallel to the

surface normal, while in the ͑102͒ scattering geometry we

probe the response from domains composed of planes of al-

ternating moments with a propagation vector normal to the

sample surface. From the relative intensities of scans taken at

different temperatures we estimate the effective scattering

moment standardized to the low-temperature value in the

antiferromagnetic phase.

FIG. 3. The integrated intensity in arbitrary units along the

specular rod of scattering at the ͑003͒ reﬂection as a function of

sample temperature.

nature of the observed signal and previous work^31–33we are

conﬁdent that the signal is indeed magnetic in origin. ͑This

conclusion is further corroborated by the overall temperature

dependence of the signal shown in Fig. 3 and discussed be-

low.͒ The combination of a relatively long resonant wave-

length coupled with a moderately high lattice parameter

͑5.588 Å͒ severely limits the accessible area of reciprocal

space for data collection. In particular, in the specular scan

direction only the ﬁrst two antiferromagnetic positions at

͑001͒ and ͑003͒ can be reached, and the off-specular scans

were limited to that at the ͑102͒ Bragg position. We restrict

our analysis of specular scans to the ͑003͒ reﬂection on ac-

count of the superior signal-to-noise ratio attainable com-

pared with ͑001͒.

The measured temperature dependence of the integrated

resonant intensity along the specular rod around the ͑003͒

reﬂection below the antiferromagnetic to paramagnetic phase

transition is shown in Fig. 3 from which we conﬁrm that the

signal reﬂects, in a characteristic manner, the degree of an-

tiferromagnetic order in the sample. Interestingly, no signiﬁ-

Data at the resonant energy from UP taken about the

͑003͒ antiferromagnetic Bragg point is shown at tempera-

tures below and above the bulk phase transition for both the

00l and h00 scan directions in Fig. 4. These data illus-

͓

͔

͓

͔

trate the key components of behavior observed in UP for the

specular reﬂection. For scans along the specular rod one has

a sharp and broad component associated with the low- and

high-temperature response, respectively. As we shall see in

detail, there exists a well deﬁned region of temperatures in

which both the sharp and broad responses coexist and we

have a classic two length scale line shape. In the orthogonal

FIG. 4. Longitudinal scans

along the specular rod at 100 K

well within the antiferromagnetic

phase and, at 121.45 K, above the

´

Neel temperature. The dramatic

decrease in width on cooling re-

ﬂects the increasing number of an-

tiferromagnetically correlated lay-

ers forming parallel to the sample

surface. Scans in the transverse di-

rection through the ͑003͒ lattice

point have more similar widths in-

dicating that the antiferromagnetic

´

layers forming above the Neel

temperature already display a de-

gree of long-range coherence in

the surface plane of the sample.

55

NEAR-SURFACE EFFECTS AT THE . . .

427

FIG. 6. Temperature dependence of the off-specular intensity

for a selection of scans at the ͑102͒ position taken in cooling. The

wave-vector transfer is parallel with the ordering wave vector. The

vertical axes of consecutive temperatures have been displaced by an

order of magnitude to improve visibility. The solid lines are the

results of ﬁts to the model function given in Sec. III. In contrast

with Fig. 5 no broad component is observed.

FIG. 5. Temperature dependence of the specular intensity for a

selection of scans at the ͑003͒ position taken in cooling. The wave-

vector transfer is parallel with the ordering wave vector. The verti-

cal axes of consecutive temperatures have been displaced by an

order of magnitude to improve visibility. The solid lines are the

results of ﬁts to the model function given in Sec. III. An evolution

of rounded maxima at elevated temperatures of a broad component

into a two-component sharp plus broad line shape around T_n͑120.9

K͒ is seen with the sharp component being typically 100 times

narrower than the broad. This two length scale line shape is in turn

dominated by the sharp component at low temperatures.

‘‘broad’’ component either above or below the bulk phase

transition for this reﬂection. The scans taken about ͑102͒ are

illustrated in Fig. 6. The apparent lack of a broad component

is signiﬁcant since it indicates that the broad response seen in

the specular ͑003͒ scans is unlikely to arise from thermody-

namic ﬂuctuations ͑which would be seen for all domains͒ but

rather is a response particular to those planes which form the

free surface. In addition, the data at the ͑102͒ position yields

an estimate of the bulk phase transition temperature and the

behavior of the sublattice magnetization on approaching T_n

from below. In this sense we may use these data as a marker

in unraveling the more complex behavior of the ͑003͒ specu-

lar reﬂection as discussed in Secs. III C and III D.

scans transverse to the specular rod, taken within the vertical

scattering plane, the scattering remains essentially sharp over

the full range of temperatures studied save for those points in

the immediate vicinity of the phase transition.

In Fig. 5, we show the overall temperature dependence of

the specular scans in the 00l direction around the ͑003͒

͓

͔

point, on a displaced logarithmic scale, taken in cooling. The

continuous lines are ﬁts to the model cross section described

in detail in Sec. III. There is an evolution of rounded maxima

at elevated temperatures of a broad component into a two-

component sharp plus broad line shape around T_nwith the

sharp component being typically 100 times narrower than the

broad. This two length scale line shape is in turn dominated

by the sharp component at low temperatures.

D. Energy dependence

In order to tune the spectrometer for resonance at the

uranium M_IVedge, the sample was cooled well into the an-

tiferromagnetic state ͑40 K͒, the spectrometer set to the po-

sition of the ͑003͒ antiferromagnetic Bragg peak and the in-

cident x-ray energy scanned through energies close to the

resonant scattering peak which was found to lie at 3.728

keV. These data, shown in Fig. 2, are in agreement with

previous work on the isostructural compound UAs.^31–33

Careful scans at the points of antiferromagnetic reﬂection

were also taken through the resonant energy at a sample

temperature of 120.85 K. This temperature was chosen since

In contrast to the results from the specular reﬂection, scat-

tering from those domains with a propagation wave vector

lying in the surface plane of the sample „vis domains probed

at the ͑102͒ reﬂection with propagation wave vector parallel

to ͓100͔… exists only in the form of a sharp response in both

the h00 and 00l scan direction. Further, scattering is

͓

͔

͓

͔

present only up to temperatures of 120.90Ϯ0.05 K. On the

scale of the response seen around the ͑003͒ peak there is no

428

A. STUNAULT et al.

55

FIG. 8. Deﬁnition of the angles and geometry of the scattering

plane as used in the expression for the resolution function. The

surface is depicted by the heavy horizontal line in the ﬁgure. The

direction q_zis parallel to the surface normal and q_xis parallel to the

surface in the plane of the ﬁgure.

2

R ␣,␤͒ϭNexp Ϫ ␦␣/⌬␣͒ exp Ϫ ␦␤/⌬␤͒ .

͔

͑

͓

͑

͔

͓

͑

On transformation to the scattering plane coordinates

(q_x,q_z) one obtains

FIG. 7. Energy dependence of the resonant scattering at 120.85

K taken in the neighborhood of the ͑003͒ Bragg point scanning

through the sharp and broad components, open circles and closed

circles, respectively. The similarity with the results of Fig. 2 and

Refs. 31–33 ͑in line shape and center͒ indicates that the signal is

magnetic in origin for both components of the line shape. The error

bars at ͑003͒ are smaller than, or of the size of, the open circles.

R ␦q ,␦q ͒ϭNJ exp ϪA ͒exp ϪB ͒exp ϪC ͒

͑

x

z

xx

zz

xz

with the coefﬁcients A_xx, B_zz, C_xz, N, and J given in the

Appendix. The presence of a cross-coupling term (C_xz)

should be noted as this may give rise to extra width in the

essentially sharp transverse scans in the presence of broad

specular scans.

it exhibits the two-component line shape enabling us to

verify the resonant nature of the scattering for both compo-

nents under ﬁxed sample conditions. In Fig. 7 it may be seen

that the broad component ͑ﬁlled circles͒ does indeed show

the same resonant behavior in energy as the sharp component

͑open circles͒ giving conﬁdence that the two components of

the line shape are both magnetic. In order to identify the

individual components of the scattering ͑dipolar, quadrupo-

lar, etc.͒ it will be necessary to carry out a full polarization

study. Experiments carried out in UAs ͑Refs. 31–33 and 41͒

indicate that at the M_IVedge the dominant transition is elec-

tric dipole in nature and conﬁrm the magnetic origin of the

observed scattering. These conclusions may be transferred to

UP assuming the uranium electronic conﬁguration is similar.

Finally, it is important to note that at the low incident

energies used in this study ͑3.728 keV͒ the x-ray penetration

depth into the sample is limited by absorption and may sig-

niﬁcantly restrict the number of atomic layers probed per-

pendicular to the surface in the ordered phase. To account for

this the resolution function is convolved with a scattering

function which contains explicitly within it the x-ray pen-

etration depth ͑further discussion is given in the Appendix͒.

The resulting line shape at a given temperature is simulta-

neously ﬁtted to the data at both the specular ͑003͒ and off-

specular ͑102͒ reﬂections in the two orthogonal directions

parallel and perpendicular to the surface normal which lie in

the scattering plane with just one intensity scaling constant

which is ﬁxed by the response at low temperatures. The out

of scattering plane resolution is assumed sufﬁciently relaxed

that it integrates over the scattering proﬁle at all tempera-

tures.

E. Angular resolution

On account of the wide range of q space covered in the

scans and the changing widths at least some consideration of

the angular resolution must be taken. The resolution has

three principal components associated with the spectrometer;

the monochromaticity of the incident x-ray beam, the ﬁnite

divergence of the incident beam upon the sample, and the

ﬁnite acceptance angle of the detector. We exert no energy

selection of the scattered photons except that afforded by the

Bicron scintillation detector and the absorption in the sample

to detector ﬂight path both of which we take as energy inde-

pendent over the energy range of interest ͑few eV at 3.728

keV͒. The sample contributes to the resolution principally

through its mosaic spread. Taking the angles ␣ and ␤, de-

ﬁned in Fig. 8, as independent random variables distributed

about their mean value in Gaussian fashion gives an angular

resolution function of the form

III. INTERPRETATION OF THE RESULTS

A. X-ray resonant magnetic-scattering cross section

Before commencing an analysis of the data we brieﬂy

summarize our assumptions concerning both the nature of

the incident radiation and the x-ray resonant magnetic scat-

tering cross section itself. First, the synchrotron x-ray source

yields an inherently brilliant beam. The beam incident on the

sample may be considered as an incoherent bundle of el-

emental coherent rays with coherence volumes typically

0.1–1 ␮m in dimension. This large dimension of the coher-

ence volume is a primary result of the high brilliance of the

x-ray source and can exceed the typical size of the magneti-

cally ordered regions contributing to the diffracted signal

from the sample. In such cases it may be necessary to con-

55

NEAR-SURFACE EFFECTS AT THE . . .

429

sider the scattered intensity as an incoherent sum of the in-

dividually diffracted elemental coherent rays rather than the

Fourier transform of a ͑two-point͒ magnetic correlation func-

tion spatially averaged over the sample. Although this dis-

tinction is usually not made, under the implicit assumption of

randomly phased scattering objects, it may become impor-

tant in the vicinity of a phase transition and for phase tran-

sitions of a discontinuous nature where the nucleation and

growth of phase-coherent islands of the ordered state may

occur. As a concrete example consider a material with a

magnetically ordered surface layer which supports a den-

dritic structure of magnetically transformed regions extend-

ing into a paramagnetic bulk. The dendrites will be phase

coherent since they originate from a common surface layer.

The resonant x-ray scattering amplitude refers to a coher-

ent two-photon process involving the excitation of an elec-

tron from a core state to an outer ͑magnetically polarized͒

energy level and its subsequent decay. In the ‘‘fast collision’’

approximation one assumes that the core hole liberated by

photon absorption does not have time to propagate before the

emission of the resonant photon and the concomitant reﬁlling

of the core level. This localizes the particle-hole pair ͑gener-

alized magnetic ﬂuctuation͒ and reduces the scattering cross

section to the Fourier transform in time of a two site inter-

ference function summed over the coherence volume. The

photon coherence length is normally sufﬁciently small that

the resonant x-ray-scattering technique yields an approxima-

tion to a two-site instantaneous interference function and en-

ables one to take a ‘‘snapshot’’ of the magnetic spatial cor-

relations. The precise nature of the two-point function is

determined from the conservation of angular momentum ͑or-

der and nature of transition: electric or magnetic; dipole,

quadrupole, mixed, etc.͒ and the harmonic order of the re-

ﬂection concerned.⁴²The resonant cross section coupled with

a brilliant photon beam permits us to resolve weak signals

and to deﬁne the diffraction proﬁle with precision, this per-

mits maximal separation of multicomponent line shapes.

sults. We note that intrinsic defects and inhomogeneities

which were present in the bulk of the sample at the point of

cleavage are not expected to be different from those below

the cleaved surface, hence it would be difﬁcult to invoke

them as a source of local magnetization by themselves. Con-

tamination from surface oxidation immediately prior to in-

sertion in the displex may play a role. In this regard, our only

control is to note that data taken on the same surface after a

storage time of six months in a desiccator (10^Ϫ2Torr͒ is

sensibly the same as that taken immediately after cleavage.

The further analysis given below does not depend upon the

microscopic origin of formation of magnetically ordered lay-

ers; we take this as a problem for future study.

As a preliminary model, we considered the specular scat-

tering from a set of magnetic layers of uniform ͑temperature

independent͒ extent parallel to the surface plane of the

sample and of varying number normal to the sample’s sur-

face. On this assumption all calculations became one dimen-

sional. The data were taken to represent the spatial Fourier

transform of the instantaneous correlations of a set of ferro-

magnetic sheets coupled alternatively up and down in the

direction normal to the surface of the sample. The model

parameters were simply the number of layers perpendicular

to the surface and their scattering moment. Results of this

model on the isostructural compound UAs have been

presented.¹⁶

In both this ﬁrst model and the one to be developed below

we make the approximation that the regions of antiferromag-

netic order are terminated abruptly in space. The second ap-

proximation made is to assume that each incident coherent

radiation volume scattering from the crystal samples the

same distribution of antiferromagnetic layers in space. And

ﬁnally that each incident coherent radiation volume is simi-

lar. Although these assumptions give rise to small oscilla-

tions in the computed cross section, they avoid additional

sets of parameters to represent the decay lengths, a volume

distribution function of antiferromagnetic layers and a distri-

bution function of incident coherence volumes, respectively.

The key result of our study, of an anisotropic near-surface

magnetism giving rise to a two-component line shape, should

not be affected by such considerations.

In this publication, we consider a slightly more sophisti-

cated model than that used in Ref. 16 based on antiferromag-

netically correlated planes of a temperature-dependent size.

As in the ﬁrst model, the number of antiferromagnetic planes

is determined by the diffraction proﬁle in the direction of the

ordering wave vector. However, the spatial extent of the

planes is now determined consistently from the scan trans-

verse to the ordering wave vector. Since experimental data

could only be taken along one of the two principal axes

which lie transverse to the ordering wave vector we invoke

the crystalline symmetry and infer a similar dimension for

the unmeasured axis. Explicitly, in an ordered region the

symmetry is tetragonal with the tetragonal axis parallel to the

ordering wave vector. If, considering a specular scan by way

of example, N_zis the number of antiferromagnetically cor-

related layers along the tetragonal axis and N_xis the mea-

sured dimension ͑in lattice units͒ of the layers from the trans-

verse scan, we set N_yϭN_xin all calculations. The full

scattering surface is computed as a function of N_x, N_y,

N_z, and the sublattice polarization at each temperature and

B. Model cross section

In the overview of the experimental data, given in Sec.

II C, Fig. 4 demonstrates the strongly anisotropic response

for scans parallel and perpendicular to the surface normal

around the ͑003͒ reﬂection. In particular, the specular scans

have an extra scattering intensity in the region of, and above

the transition temperature. This extra intensity is characteris-

tically ‘‘broad’’ for scans parallel to the surface and ‘‘sharp’’

in the transverse cuts which suggests the formation of ex-

tended sheets of ferromagnetically aligned moments parallel

to the sample surface which themselves are antiferromagneti-

cally coupled, only a few layers deep, normal to the surface.

In order to analyze this data we take the idea of formation of

near-surface antiferromagnetic layers literally and introduce

a simple model of layer formation in real space. The reason

for the nucleation of the phase transition in the near-surface

layers is unknown. Conceivably the cleavage of the sample

induces an anisotropic strain in the near-surface region

which, through the magnetovolume interaction,^43,44favors

near surface magnetism for one particular domain orienta-

tion. Alternatively, one may argue that the very presence of

the surface allows for relaxation of stress and that through

the magnetoelastic coupling a near-surface magnetization re-

430

A. STUNAULT et al.

55

convolved with the instrumental resolution function. From

the resultant surface one projects out that cut relevant to the

particular scan under consideration. The scans in orthogonal

directions are coupled through the cross term C_xzinherent in

the resolution function ͑the only way to avoid such a term is

to arrange ␣ϭ␤ and ⌬␣ϭ⌬␤) and in this way the line

shape in the q_xdirection of the specular scans is broadened

due to the ﬁnite extent of the antiferromagnetic diffracting

region in the z ͑surface normal͒ direction.

Since we are unable to make reliable calculations of the

absolute measured intensity of the resonantly diffracted

beam, we scale the measured intensity at 77 K to the square

of the effective moment reported by neutron diffraction in

the literature at this temperature.^39,45,46Caution must be ex-

ercised in interpretation since the energy ranges, wave-vector

ranges and the coherence volumes covered by the respective

experimental resolution functions in the x-ray and neutron

experiments are very different. The scaling of the low-

temperature moment amounts to the assumption that at these

temperatures the diffracted intensity is dominated by the

static ͑long-time average͒ sublattice magnetization and the

presence of long-range order eliminates the problems posed

by the different coherence volumes of the two radiation

sources. At elevated temperatures, where such assumptions

break down, we are on more delicate ground. It must be held

in mind that the estimated value of magnetic moment is that

appropriate to the instantaneous two-point correlation func-

tion and this is not necessarily equal to the equilibrium long-

time average.

FIG. 9. Thermal evolution of the number of magnetic layers of

the bulk response deduced from the width ͑and intensity͒ of the

measured response in the directions parallel and perpendicular to

the ordering wave vector around the specular ͑003͒ Bragg point.

The layers are in units of the chemical unit cell. The estimated error

in the number of layers is of the order of 10%.

heating the scattered intensity starts to fall as evidenced in

the plots of integrated intensity ͑Fig. 3͒. Less obvious are the

changes in line shape. Since the diffracting objects remain

relatively large until very close to the transition the FWHM

is dominated by the angular resolution and only by looking

in detail at the tails of the scattering may the changing sizes

of the coherently diffracting regions be seen. Once the

changing size of the diffracting regions is taken into account

one may estimate the effective scattering moment ͑see also

discussion in Sec. IV C͒.

C. Application of the model to the specular scans

We now turn to the measured x-ray magnetic-scattering

response around the ͑003͒ antiferromagnetic Bragg point

taken on cooling. We divide the data into three temperature

regimes according to line shape. A ﬁrst low-temperature

span below 120.5 K, a second transitional span from 120.5 to

121.1 K and a ﬁnal high-temperature region above 121.1 K.

These regions are characterized by a single sharp proﬁle in

the specular scan, a classic two-component line shape and a

broad proﬁle respectively ͑see Sec. II C and Fig. 5͒.

Starting at low temperatures the magnetic peak is, over its

central portion, dominated by the angular resolution func-

tion. The specular and transverse cuts differ in width by a

factor of 2 in this regime and this is picked up by the relative

widths of the resolution function which scale, respectively,

as the sine and cosine of the half-scattering angle ͑for further

discussion see the Appendix͒. Away from the central posi-

tion the Lorentzian-like tails in the specular scans are domi-

nated by the ﬁnite penetration depth. In a simple manner this

may be understood as the exponential decay of the incident

and scattered beams due to absorption which yields, on Fou-

rier transformation, a Lorentzian form in reciprocal space. In

the transverse direction extended tails are also seen. These

arise from the ﬁnite domain size and/or the transverse coher-

ence length of the incident radiation. From the low-

temperature data we then set the effective angular resolution,

the effective penetration depth into the sample and obtain an

estimate of the domain size or transverse coherence length.

The numerical values of these parameters are given in the

Appendix under the discussion of the resolution function. On

Empirically, the scans start to become noticeably broader

at temperatures above 120.5 K, Fig. 5. This arises from two

effects. First, the size of the diffracting objects themselves

become smaller giving rise to a width in the sharp compo-

nent greater than that contributed by the angular resolution.

Secondly, the development of near-surface magnetism oc-

curs which gives rise to a broad component in the specular

proﬁle. The thermal evolution of the number of magnetically

ordered layers in the bulk magnetic phase is given for direc-

tions parallel and perpendicular to the ordering wave vector

in Fig. 9. Experimental transverse cuts through the scattering

surface away from the position of maximum intensity were

made and compared favorably with the calculated response

within the model function given. A typical pair of scans

taken in the vicinity of the phase transition and exhibiting the

two component line shape in the longitudinal direction are

given in Fig. 10. The model function is able to reproduce the

width and intensity in the directions both longitudinal and

transverse to the ordering wave vector. The broadening

raises a problem in the deﬁnition of T_n. Direct plots of in-

verse linewidth against temperature would imply the phase

transition is signaled by an onset of ‘‘long-range order’’

͑resolution limited scattering proﬁle͒ occurring below T_nas

determined from intensity measurements ͑see, for example,

Ref. 13͒. Within the model used the interpretation is rather of

55

NEAR-SURFACE EFFECTS AT THE . . .

431

FIG. 10. Data and ﬁts to the

model function in the two length

scale regime. The inset on the lon-

gitudinal scan ͑parallel to the or-

dering wave vector͒ shows the

sharp component on an enlarged

scale. The small oscillations in the

ﬁtted functions come from the

single component distributions

used for the surface and bulk lay-

ers as discussed in the text. The

maximum of the response occurs

just off the low-temperature Bragg

position in the specular direction;

for this reason the transverse cut is

less intense in its peak height than

the longitudinal cut. The displace-

ment has been put into the model

calculation.

120.5 K to a broad maximum, with a value close to the

free-ion moment, and starts to decay again above 121.1 K.

The abrupt fall at 121.1 K is followed by a plateau extending

to temperatures as high as 130 K, ͑Fig. 12͒. Further discus-

sion of the evolution of the ordered magnetic moment and its

interpretation within a simple Ginzburg-Landau model are

given in Sec. IV.

a falling size of domain on approaching T_n. Further prob-

lems occur with differences between the specular and off-

specular response, and we give a working deﬁnition of T_nin

Sec. III D below.

Above 120.5 K an additional broad feature associated

with near-surface magnetism arises in the scattering proﬁle.

The evolution of the number of magnetically ordered layers

in this near-surface magnetic phase is given for the directions

parallel and perpendicular to the ordering wave vector in Fig.

11. Again, as in Fig. 9, we note the extreme anisotropy of the

response parallel and perpendicular to the ordering wave

vector. Interestingly, the inferred moment associated with the

near-surface magnetism increases from the bulk value at

D. Application of the model to the off-specular scans

As observed in Sec. II C the scattering at the ͑102͒ reﬂec-

tion is much simpler than that around the ͑003͒ Bragg point.

It is also much weaker. This loss of intensity at low tempera-

tures is accounted for by the proposed resolution function

given the assumption of an equal domain population. The

typical difference in intensity and line shape of the ͑003͒ and

FIG. 11. Thermal evolution of the number of near-surface mag-

netic layers deduced from the width ͑and intensity͒ of the measured

response in the directions parallel and perpendicular to the ordering

wave vector around the specular ͑003͒ Bragg point. The layers are

in units of the chemical unit cell. At and above 130 K there exists a

weak intensity over and above that seen at 200 K; however the

uncertainty in the line shape is too great to allow an unambiguous

analysis.

FIG. 12. Thermal evolution of the effective near-surface sublat-

tice magnetization per uranium atom deduced from the model scat-

tering function introduced in Sec. III. The peaking of the moment in

the vicinity of the phase transition may be understood in terms of an

extended Ginzburg-Landau model of Sec. IV. The solid line is the

expected temperature variation of the Ginzburg-Landau surface

layer moment using parameters derived in Sec. III. The estimated

error in the value of the effective moment is of the order of 10%.

432

A. STUNAULT et al.

55

FIG. 13. Contrasting intensi-

ties and line shapes shown for

scans taken in the specular ͑003͒

and off-specular ͑102͒ reﬂections

in the antiferromagnetic phase.

The longitudinal scan is along the

direction lying parallel to the sub-

lattice magnetization in the given

domain. This is the ͓00l͔ direction

for the specular ͑003͒ reﬂection

and the h00 direction for the

͓

͔

͑102͒ off-specular reﬂection. The

transverse scan directions lie in

the scattering plane and are or-

thogonal to the given longitudinal

directions. The solid line is a re-

sult of a ﬁt given by the model

described in the text with one

scaling parameter for the vertical

axis.

͑102͒ reﬂections in the ordered phase is shown in Fig. 13. In

each case the longitudinal scan is parallel to the magnetic

propagation vector ͓͑001͔ for the ͑003͒ reﬂection and ͓100͔

for the ͑102͒ reﬂection͒ and the continuous line is the result

of a ﬁt to the model. The transverse scans are perpendicular

to the ordering wave vector and in the scattering plane. At

both reﬂections the data presented has been taken on cooling;

as noted in Sec. II B there is no observable hysteresis in up

and down temperature sweeps. In the analysis of the specular

and off-specular scans there is just one scaling constant for

all scans. The sublattice moment, sizes of diffracting regions

and domain fraction are then determined consistently from

all data at a given temperature. Since the data for the two

domain orientations were taken on different heating and

cooling cycles this procedure supposes that the experimen-

tally sampled domain fraction is approximately independent

of sample history. This was conﬁrmed by the near reproduc-

ibility of data taken on the same sample surface in two sepa-

rate experiments and is a result of the averaging given by the

large incident beam size ͑of order one millimeter͒ in com-

parison with a typical domain size ͑of order micrometer͒. As

illustrated in Fig. 14͑a͒, the fraction of off-specular domains

goes to zero at a temperature of 120.9 K as does the sublat-

tice moment ͓Fig. 14͑b͔͒. We assign this temperature T_n͑of

the bulk͒. In contrast, the specular reﬂections which give

evidence of continued bulk order up to temperatures of 121.1

K are taken to be extraordinary. The difference is illustrated

in Fig. 15 where at 120.92Ϯ0.01 K, for scans along the

ordering wave vector, a sharp response is observed for the

specular ͑solid circles͒ in contrast with the off-specular ͑open

circles͒ where no observable response is seen. This then sug-

gests that the different domains, specular and off-specular,

have signiﬁcantly different physical behavior above 120.9 K,

an issue which is discussed further in Sec. IV C.

ordered phase antiferromagnetic domains occur with ap-

proximately equal probability. The bulk sublattice magneti-

zation evolves in a smooth manner to around 120.8 K,

whereafter the specular and off-specular reﬂections have

quite different behaviors as detailed in Secs. III C and III D.

Above 120.5 K, one has evidence of a near-surface sublattice

magnetization with an effective moment substantially differ-

ent from that observed in the bulk which occurs exclusively

for that domain having its ordering wave vector parallel to

FIG. 14. Top panel: The domain fraction as extracted from the

ﬁts to the model of Sec. III. The open crosses and circles represent

the specular and off-specular fractions, respectively. The dashed

line is a guide to the eye. Bottom panel: The overall temperature

dependence of the effective moment per uranium atom ͑scaled to

the low-temperature value͒ for the specular and off-specular scans

given by the crosses and circles, respectively. The solid line is a ﬁt

to a ﬁrst-order Ginzburg-Landau model of the form f_bgiven in Sec.

IV. DISCUSSION

A. Overview

From the model interpretation of the observed diffraction

proﬁles the following picture of the antiferromagnetic phase

transition in UP arises: starting at low temperatures in the

*

IV. The parameters used are T_cϭ120.8 K, T ϭ121.1 K, and

*

m ϭ0.64␮ per uranium atom.

B

55

NEAR-SURFACE EFFECTS AT THE . . .

433

pronounced the more disordered the surface in the sense that

one may imagine blocks of material able to relax with mini-

mal constraint both parallel and perpendicular to the average

sample surface. A minimization of the free energy of the

coupled order-parameter system of the magnetic and strain

degrees of freedom is then required. In the classic case, ex-

plored in detail by Bean and Rodell,^43,44such a minimization

is well known to shift the transition temperature relative to

that occurring in the strain free but magnetically or otherwise

stressed conﬁguration. In particular, such a minimization will

always reduce the coefﬁcient of the quartic term in the free

energy through the magnetoelastic coupling effect. The as-

sumed positive quartic term which gives the mode cross cou-

pling depresses the phase transition temperature from its in-

dependent mode mean-ﬁeld value. The depression of the

temperature of the phase transition has been calculated for

example, in a quantitative manner, for a series of incipient

and weakly magnetic metals.^47–50The reduction of the effec-

tive coefﬁcient of the quartic term through the magnetoelas-

tic effect, reducing as it does the mode coupling, leads to an

enhancement of the renormalized phase transition tempera-

ture in the stress-free regime. This is different from an en-

hancement of T_nwhich is equally possible on the application

of hydrostatic or uniaxial stress where the appropriate free

energy to be minimized is not that of a stress-free system.

Eventually, in the case of extreme reduction of the quartic

coefﬁcient from a positive to a negative value one may imag-

ine driving the phase transition ﬁrst order introducing the

possibility of passing through a tricritical point. Thus we

have an intrinsic mechanism for ͑near͒ surface ordering

above the bulk phase transition for systems in which the

coupling of ﬂuctuations and internal stresses plays a signiﬁ-

cant role ͑for a related discussion see, e.g., Ref. 1͒. Clearly,

coupling to any other parameter in a similar fashion will

work; strain just seems most obvious. Alternatively, one may

suppose that sample dependent stresses, for example, in-

duced by cutting or other surface treatments, yield an en-

hancement of T_nin the near-surface region in an analogous

manner to the application of hydrostatic or uniaxial stress.

However it is not clear that internal stresses would necessar-

ily always act in the correct direction and be of sufﬁcient

magnitude such as to be able to give the measured enhance-

ment of T_n. Further, the volumetric distribution of stress in

the near surface, and hence T_n, would have to be such as to

match the observed evolution of scattering intensity. The ex-

plicit ﬁrst-order nature of an idealized transition and the dis-

ruption of translational invariance by the surface are our next

concern.

FIG. 15. Longitudinal scans along the ordering wave vector in

*

the temperature regime between T_cand T for both the specular

and off-specular directions. Only the specular reﬂection ͑ﬁlled

circles͒ shows any response in this temperature interval. This re-

sponse appears to be nucleated by the presence of near-surface

magnetic layers which occur exclusively for domains orientated

with their ordering wave vector along the free surface normal.

the surface normal. Both the near surface and bulk polarized

regions are of similar lateral dimensions, giving an image of

continuous blocks of antiferromagnetic layers extending

down from the surface into the bulk. Below 120.9 K these

blocks are interspersed by those of the other domains. The

unusual temperature dependence of the near-surface magne-

tization, its existence to temperatures as high as 130 K and

the peaking in intensity occurring in the vicinity of the bulk

phase transition lead us to consider a general analysis of

surface magnetism at phase transitions of a discontinuous

nature.

We discuss a mean-ﬁeld Ginzburg-Landau theory and

consider separately two effects, both of which indicate the

possibility of having a surface ordering occurring above the

bulk phase transition. The simple Ginzburg-Landau approach

is open to the criticism that it is known to yield incorrect

predictions near critical points; however, in many cases it

does yield insight into the qualitative nature of the phase

diagram and may give some guidance to the underlying

physical effects. It is in this spirit that we use it here. The

ﬁrst approach is to continue to ignore the surface explicitly

in as much as the breaking of translational invariance is ne-

glected, while emphasis is put upon the magnetoelastic inter-

action and relative strains occurring in the surface and bulk

layers. The second approach is to focus precisely upon the

perturbations to the order parameter induced by the breaking

the spatial translational invariance. In the concluding section

we comment brieﬂy on how the breaking of translational

invariance by point and line defects may play a role in the

more general case.

C. Surfaces with the breaking of translational invariance

We consider a scalar order parameter to represent the

sublattice magnetization and set up a coordinate system with

the sample being supposed semi-inﬁnite in the x,y plane and

having the z axis running from 0 at the sample free surface to

inﬁnity into the bulk.⁵¹The scalar nature of the order param-

eter does not admit the general vector form of a magnetic

problem, but in our case of extreme uniaxial anisotropy in a

given domain is taken as a starting point for discussion with

the caveat that the true vector nature of the problem may

eventually lead to signiﬁcant differences.^47–50The inclusion

B. Surfaces ignoring the breaking of translational invariance

First, we ignore the explicit breaking of translational in-

variance imposed by the surface. The effect of the free sur-

face is to allow lattice relaxation; this effect will be more

434

A. STUNAULT et al.

55

of the surface breaks the translational invariance of the prob-

lem and we anticipate having to add two terms to the free

energy. The ﬁrst is to take care of the spatial gradient of the

order parameter perpendicular to the surface mϭm(z) im-

plied by the lack of translational invariance. The second is a

term to mimic the effect of the open surface, essentially to

signal out the surface layer. This is taken in the spirit of

simple Ginzburg-Landau theory to be proportional to the

square of the surface magnetization and is labeled f₁. We

consider a free energy of the following form:

To expand upon these descriptive ideas we search for the

functional form of m(z) which renders the free-energy sta-

tionary; one derives the Euler-Lagrange equation:

d²m

df_b

dm

c

ϭ

͑1͒

ͩ ͪ

dz²

and the boundary condition

df₁

dm

ϭc

͑2͒

dz

ϱ

evaluated at zϭ0.

Fϭ

dz f ϩf ͒,

͑

b s

͵

0

Equation ͑1͒ is integrable ͑cf. Newton’s law͒ to give

where the free energy density has been arbitrarily broken

into two pieces for ease of calculation. The term f_bis chosen

to yield a ﬁrst-order transition with the coefﬁcients a, b, and

d positive. To accommodate the gradients introduced by the

surface the sublattice magnetization density is generalized to

mϭm(z) giving

2

c dm

ϭf_bm͒ϭconst .

͑3͒

͑

ͩ ͪ

2

dz

We are primarily interested in the occurrence of near-surface

magnetization when the bulk is paramagnetic, this gives the

boundary condition that at large distances from the surface

m(z) tends to zero. As may be seen from Eq. ͑1͒, a mono-

tonic f_bas a function of m, characteristic of a continuous

phase transition above T_c, cannot yield an extended region

of near-surface magnetism. At most one may arrive at a de-

caying magnetization starting from an assumed magnetic

surface layer given by a negative value of a_s. Such a model

has been explored and is incompatible with our data sets.

Extended or delocalized near-surface magnetism existing in

the bulk paramagnetic phase implies f_bexhibiting a dip as a

function of m to yield a point of inﬂection in m(z) ͓given by

Eq. ͑1͔͒ coupled with a negative value of a_s.

a

b

d

f_bϭ m²Ϫ m⁴ϩ m⁶,

2

4

6

with f_sthe surface-induced contribution of explicit form

2

c dm

f_sϭ

ϩf ␦ z͒,

͑

1

ͩ ͪ

2

dz

a_s

f₁ϭ m²,

2

where aϭa(T) is the primary temperature-dependent param-

eter of the theory. The bulk free energy has two characteris-

tic temperatures of interest. The ﬁrst is contained in the sup-

posed linear variation of a(T) about T_c, a(T)

ϭa₁(TϪT_c); at and below T_cthe stable state is one of ﬁnite

sublattice polarization. The second temperature comes from

the point of instability where the free energy passes with its

minima at ﬁnite sublattice magnetization through zero at

Explicitly solving Eqs. ͑1͒ and ͑2͒ for temperatures above

T_c:

Ϫ1

*

TϪT

TϪT_c

2

*

m z͒ ϭm

1ϩ

ͱ

sinh 2z␬ϩAS͒

,

͑

ͩ

ͪ

ͫ

ͬ

*

T ϪT_c

where the inverse correlation length is given in the normal

manner as

*

TϭT . We label the magnetization density at this point

*

m . Between T and T_ca metastable state pertains in which

temperature interval the system may pass discontinuously

from a paramagnetic to ordered state.

a

␬ϭ

ͱ

c

The model works in the following manner. At a given

temperature above T the stimulating term f₁␦(z) raises the

and

*

surface layer to an ͑arbitrary͒ magnetization. Physically, this

may represent the presence of surface disorder or other ef-

fects as considered in Sec. IV B. Provided this surface mag-

2

*

m

*

TϪT ͒/ T ϪT ͒ Ϫ1

͓͑

͑

͔

ͩ

ͪ

c

m 0͒

͑

ͩ

ͪ

ASϭasinh

,

*

netization is greater than m , the system will support a spa-

ͱ _{TϪT ͒/ T ϪT ͒}

*

͑

*

͑

c

tially extended decay of magnetization density into the bulk.

Roughly speaking this new length scale arises in the follow-

ing manner. Starting from the surface where the free energy

has a large negative contribution, the minimum in f_btraps

the system at ﬁnite magnetization density over a distance

given by the energy cost incurred by the gradient term con-

tained in f_s. This allows for a delocalized near-surface mag-

with m(0) being the surface magnetization.

*

At temperatures above T solving Eqs. ͑2͒ and ͑3͒ one

obtains

T_sϪT

_{m 0͒ϭm}*ͱ_1ϩ

,

͑

ͱ

*

T ϪT_c

*

netism on a length scale which diverges at T , even though

where the effective surface transition temperature

the stimulating term is strictly localized at the surface. The

*

divergence at T corresponds to a bulk transition in the

a_s²

metastable regime with the surface layer playing the role of a

nucleation center.

*

T_sϭT ϩ

.

a₁c

55

NEAR-SURFACE EFFECTS AT THE . . .

435

FIG. 17. Detailed temperature dependences of the near-surface

moment ͑open squares͒, the specular bulk response ͑open circles͒,

and the off-specular response ͑ﬁlled triangles͒. The solid line is

from a ﬁrst-order Ginzburg-Landau model of the form f_bused in

Fig. 14 with the same parameters. The deviations of the bulk re-

sponses from the solid line are discussed in Sec. IV C. The near-

surface magnetic moment falls to merge with the bulk value starting

around T_c, below which temperature a more sophisticated model is

required.

*

FIG. 16. Proﬁle of m(z) calculated in the vicinity of T in terms

of the reduced length scale ͑the correlation length͒. Comparison

with the experimental data indicates a correlation length in the re-

gion of 7 Å.

The possibility to have a plateaulike region in m(z) at a

given temperature depends on the quantity AS being nega-

tive and the spatial extent is given in magnitude as AS times

the correlation length. From the explicit equation for AS this

*

is seen to imply m(0) being greater than m . Further, the

*

phased diffracting objects is less than this theoretical maxi-

mum, the moment assigned would have to be increased to

yield the observed intensity in agreement with the Ginzburg-

Landau estimate. For the specular ͑003͒ scans above T_c

͑open circles in Fig. 17͒ there is an ever decreasing moment

closer T is to T the more negative AS becomes and hence

the larger the region of delocalized or near-surface magneti-

zation. This then mimics the behavior found in our analysis

of the data. The parameters T_cand T we estimate from the

behavior of the bulk ͑003͒ and ͑102͒ magnetization density

*

up to T . To satisfy the extrapolated Ginzburg-Landau value

to be 120.8 and 121.1 K, respectively. This leaves the scaling

*

of moment one would infer that the nucleation probability,

and hence the number of transformed regions, is falling rap-

parameter m , the value to which the sublattice magnetiza-

*

tion jumps at T in those regions which transform. This is

*

idly on approaching T due to an increasing free energy

estimated to be approximately 0.64␮ per uranium atom

B

barrier.

from the temperature dependence of the magnetic moment

Although formally outside our starting Ginzburg-Landau

model, an increase in near-surface moment on approaching

derived in Secs. III C and III D. A proﬁle of m(z), calculated

*

in the vicinity of T , is sketched in Fig. 16. From the spatial

*

T

from above may be mimicked by assuming a temperature

extent of the near-surface layers estimated within the model

dependence of the Ginzburg-Landau stiffness parameter c.

Taking the effective surface moment to be limited to the

free-ion value, and c to vary as (TϪT_c) with a value of

*

of Sec. III B, which is approximately 25 Å near T , one

obtains the correlation length from this ﬁgure to be around 7

Å which corresponds well with that estimated by indepen-

*

m

from the calculation in the ordered phase the solid line is

dent neutron-scattering measurements for temperatures

37

obtained in Fig. 12. This displays a characteristic upturn in

the vicinity of the bulk phase transition and an estimated

T_sof 130 K in rough agreement with the observation. As the

parameter c becomes small the moments become decoupled.

This near criticality of the stiffness in addition to that of the

susceptibility appears to be an important aspect of strongly

correlated systems.⁵²On further cooling below T_cthe sur-

face moment falls whereafter it merges with the bulk.

In a more general framework one may obtain a delocal-

ized order parameter in the paramagnetic state whenever the

free energy admits a dip at ﬁnite magnetization, provided

there is some term analogous to f₁locally to surmount the

free-energy barrier and provide the nucleation center. This

forcing term may be static or dynamic; importantly it does

not have to exhibit a long-range ﬁeld itself, the length scale

comes, as in the case of spontaneous ﬂuctuations, from the

functional minimization of the free energy. Examples of

static forcing terms are point, linear, or planar defects and

*

above T .

Application of this formalism to our results is shown by

the solid line in Fig. 14͑b͒. Overall, the form of f_baccounts

reasonably well for the observed temperature dependence of

the magnetic moment inferred in Secs. III C and III D. In

detail, Fig. 17 indicates that the sublattice moment of the

off-specular ͑102͒ domain ͑ﬁlled triangles in ﬁgure͒ jumps at

a value close to the T_cof the Ginzburg-Landau model. This

suggests that the off-specular domains are not signiﬁcantly

nucleated above this temperature ͑120.8 K͒. Also notable is

that until temperatures below 120.5 K the estimated moment

is lower than the theoretical curve. This may be accounted

for by the analysis ͑Sec. III͒ since the size of the diffracting

regions in the dimension transverse to the wave vector of

propagation is smaller than the beam coherence area at these

temperatures and we ͑arbitrarily͒ set the number of such dif-

fracting objects to the ratio of the beam coherence area to

their size ͑its maximum value͒. If the number of randomly

436

A. STUNAULT et al.

55

spontaneous thermodynamic ﬂuctuations or mobile defects

provide dynamic ones. Since in our experiments we only see

the two-component line shape for the specular scans of those

domains having the ordering wave vector parallel to the sur-

face normal we assign the forcing term in our discussion to a

planar defect in some way associated with the surface. Such

equilibrium models have nothing to say on dynamic forma-

tion of magnetic surface layers and, as emphasized, x-ray

measurements cannot at this stage eliminate such processes.

We further note that in the vicinity of a phase transition we

may anticipate modiﬁcations in line shape arising from

changes in the long-range nature of the magnetic two-point

correlation function⁴corresponding within a Ginzburg-

Landau formulation to temperature-induced modulations of

the stiffness coefﬁcient (c) above. These variations may give

rise to two-component line shapes even for continuous tran-

sitions in inﬁnite media.

nucleated sites. The lifetime of the extended spatial order is

anticipated to be greater than or of the magnitude of its

stimulus ͑see discussion in Sec. IV͒.

Our primary experimental result is the existence of a two-

component line shape for the specular ͑003͒ reﬂection in the

vicinity of the paramagnetic to antiferromagnetic phase tran-

sition in UP. While appreciating that since we measure a

spatial Fourier transform we cannot prove that the broader of

the two components is uniquely associated with the near-

surface region, the analysis given makes it plausible. It is

possible, through a detailed analysis of the scattering, to

identify this antiferromagnetic near-surface phase at tem-

peratures as high as 10 K above T_n. The coexistence of this

phase with the incipient bulk phase is deemed responsible for

the observed two-component proﬁle in the vicinity of the

bulk phase transition. The building blocks of the antiferro-

magnetic structure are ferromagnetically aligned sheets of

sublattice magnetization of temperature-dependent spatial

extent which are themselves antiferromagnetically coupled.

The second observation is that the bulk phase transition

propagates as a two-stage process for the domains involved

in the ͑003͒ specular reﬂection. There is an initial growth of

antiferromagnetic order into the bulk of the sample from the

near-surface layers associated with this specular reﬂection

followed by the simultaneous growth of the two orthogonal

domains, which have their moments lying in the surface

plane, marking the start of the bulk phase transition. At tem-

V. CONCLUSION

Before summarizing our results we pause to stress a dif-

ference of detail apparent between the interpretation of our

results and those previously reported. In the latter, the two

length scale diffraction proﬁle is usually taken to exist only

at temperatures above the ͑nominal͒ bulk transition identify-

ing the anomalously sharp response with ͑a pretransition of͒

the near-surface layer and the broad response with the nor-

mal thermodynamic ﬂuctuations. Indeed the temperature at

which the sharp linewidth becomes resolution limited is of-

ten taken as a deﬁnition of T_n. UP and UAs are experimen-

tally distinct from other materials studied to date such as

Ho,⁵UO₂,²and NpAs ͑Ref. 3͒ in having a highly anisotropic

line shape. In our data the manifestly broad response along

the specular direction has been shown to be associated with

the near surface and the two length scale diffraction proﬁles

with a region of coexistence of surface and bulk ͑ordered͒

magnetism.

The model introduced in Sec. IV may be generalized in

several ways. First, as already noted, if the transition is dis-

continuous terms other than the free surface may act to pro-

vide the necessary nucleation center for delocalized magne-

tism. Secondly, in the vicinity of a continuous phase

transition strong ﬂuctuations renormalize the free

energy,^49,50,53,54and may provide the necessary change in

free energy to enable the local formation of delocalized near-

surface magnetism. Finally, we note that a spatially modu-

lated order parameter may be susceptible to the formation of

similar pseudo-long-range order even for a continuous phase

transition in the presence of point defects. All such mecha-

nisms would yield, in the region immediately above T_c, ﬁ-

nite but long-range order ͑the sharp component͒ which in-

creases in spatial extent with lowering temperature. Within

the framework outlined in Sec. IV the sharp length scale

would be expected to diverge as some increasing multiple of

the thermodynamic correlation length, ␨_c, on approaching

*

peratures just above T the transverse extent of the near-

surface layers involved in the specular ͑003͒ reﬂection falls

dramatically from its high-temperature ͑resolution limited͒

value, Fig. 11, and then picks up during the growth stage of

long-range order. It remains an interesting question to under-

stand this dissolving of initial ͑pseudo͒ long-range order of

the near-surface layers on approaching the bulk phase tran-

sition and its regrowth. At low temperatures the measured

domains, located in the near-surface volume, appear to be

approximately equally populated in agreement with supple-

mentary neutron-scattering investigations on the bulk of the

same crystal.³⁷It would clearly be of great interest to make

detailed experiments both on a ͑polished/etched͒ smooth sur-

face in addition to the reported experiments which were per-

formed on a cleaved surface and on surfaces exhibiting dif-

ferent crystallographic planes. Given the cubic rocksalt

crystal structure and the natural cleavage planes the above

proposed transition ͑Sec. IV C͒ may commonly take place in

such materials.

In addition to the above discussed precursor phase,

neutron-scattering data indicates the formation, in the bulk,

of ͑dynamic͒ antiferromagnetically correlated sheets, just a

couple of layers deep, for all three domains at temperatures

above and extending just into the bulk phase transition. We

suggest that these may arise in the normal way from sponta-

neous ﬂuctuations in the free energy ͑not large enough to

surmount the barrier͒ with a spatial magnitude given by the

thermodynamic correlation length. One thereby identiﬁes, in

general, three length scales in the discontinuous phase tran-

sition. These length scales, in ascending order, are those of

thermodynamic correlations, the near-surface correlations,

and the bulk correlations. The presence of highly anisotropic

layers and a short thermodynamic correlation length facili-

tates the separation of such contributions, one or more of

T_c; that is it will diverge faster than ␨ as observed. Further,

c

the intensity associated with the second length scale will

diverge with an exponent characteristically faster than that of

the thermodynamic ﬂuctuations, since the observed intensity

of the second length scale would be driven both by an in-

creasing correlated moment and an increasing number of

55

NEAR-SURFACE EFFECTS AT THE . . .

C ϭ Ϫ2sin ␤͒cos ␤͒/ ⌬␣͒²ϩ2sin ␣͒cos ␣͒/ ⌬␤͒

437

2

which may become merged into another in the presence of

more isotropic and longer ranging correlation lengths. In a

more general way it may be useful to consider a distribution

of active length scales in a given problem, here we approxi-

mate this distribution at each temperature by a singular value

characteristic of each of three regimes. An analysis of the

scattering proﬁles occurring in these more general cases is

dealt with elsewhere.⁴²

͑

͕

͖

xz

ϫ ␦q /„k sin 2⌰͒… ␦q /„k sin 2⌰͒… ,

͔͓

͓

͑

͔

x

0

z

0

Jϭ k²sin ␣ϩ␤͒

͔

,

Ϫ1

͓

͑

0

where ␪ is the Bragg angle.

The outgoing angular divergence is given approximately by

the gap of the slits in front of the detector. This gives

⌬␤ϭ0.002. From the low-temperature scans we set

⌬␣ϭ0.0004. The wave vector k₀is given by the resonant

energy and the angles ␣ and ␤ from the conservation of

momentum. The effective penetration depth is in the region

of 1200 Å and the transverse coherence/domain size is of the

order of 1 ␮m.

At the low incident energies used in this study the x-ray

penetration depth into the sample is limited. Since the attenu-

ation depends on the actual scattering path length, that is, the

angles ␣ and ␤ and the active scattering depth we compute it

directly in the effective scattering function S(q_x,q_z) rather

than in the resolution function. It is of interest that the expo-

nential decay of beam intensity with path length gives an

identical form ͑with opposite sign͒ to the resolution as that

obtained by considering a rough surface with an exponen-

tially depleting number of layers.⁵⁵The two effects may

work one against the other to yield spurious estimates of

both surface roughness and absorption coefﬁcients. We work

in large scattering angle geometry and ascribe our measured

effects to an effective absorption depth.

ACKNOWLEDGMENTS

We thank P. Burlet for making his UP sample available to

us for this study and M. Altarelli and F. de Bergevin for

illuminating discussions. S.L. and N.B. would like to thank

the European Union for the HCM awards which made this

work possible including work carried out under Grant No.

CHRXCT93-0135. Work performed at Brookhaven National

Laboratories is supported by the US-DOE under Contract

No. DE-AC02-76CD00016.

APPENDIX: ANGULAR RESOLUTION

Following the nomenclature of Fig. 8 we express the ex-

perimental resolution function in terms of the angular vari-

ables ␣ and ␤ and their deviations ⌬␣, ⌬␤. Three principal

components are associated with the spectrometer; the mono-

chromaticity of the incident x-ray beam, the ﬁnite divergence

of the incident beam upon the sample, and the ﬁnite accep-

tance angle of the detector. We assume that the incoming and

outgoing angular deviations are uncorrelated and no energy

selection of the scattered photons over the energy range of

interest ͑few eV at the resonant energy of 3.728 keV͒. The

sample contributes to the resolution principally through its

mosaic spread and possibly by its ﬁnite size for a small in-

cident beam. Taking the incident monochromaticity, angular

divergence, and sample mosaic together we deﬁne an effec-

tive incident angular divergence ⌬␣ in the vertical plane. In

the same plane for open detector slits at a distance l from the

sample and an illuminated sample of length s we have an

outgoing divergence approximately given by ⌬␤

The scattering amplitude f appropriate to the model func-

tion of Sec. III and from which S(q_x,q_z) is constructed is

taken to have the following form:

fϭ

M r͒e^iq•rϪ␮z

ϩ

M

r͒e^iq•rϪ␮z

͑

B

͑

͚

s

r

2

_ϭͱ _{⌬␣ ϩ (s/l) sin␤}

*

. For an incident beam of height

͔

͖

͓

͕

where M_Sand M_Bare the effective surface and bulk sublat-

tice magnetization densities, respectively. The position vec-

tor rϭ(x,y,z) has directions x and y in the sample surface

plane and z running from 0 at the sample free surface to

inﬁnity in the bulk. The scattering wave vector is equal to

q and ␮ is the effective inverse penetration depth given by

w smaller than the sample length, s is to be replaced by

w/sin␣. We take ␣ and ␤ distributed about their mean value

in Gaussian fashion yielding an angular resolution function

2

R ␣,␤͒ϭN exp Ϫ ␦␣/⌬␣͒ exp Ϫ ␦␤/⌬␤͒

,

͑

͓

͑

͔

͓

͑

͔

with a normalization

Ϫ1

Nϭ ␲⌬␣⌬␤͒

,

1

͑

␮ϭ␮

ϩ

.

ͩ

ͪ

0

sin␣ sin␤

for the case where the slits are sufﬁciently wide open not to

cut the outgoing beam. On transformation to the (q_x,q_z)

coordinate system

Corrections to the simple Fourier-transform relationship be-

tween the object and the diffraction proﬁle used in the analy-

sis are generated by the nonplanar nature of the diffracted

wavelets. Such effects are anticipated to become important

when the dimension of the object giving rise to coherent

diffraction exceeds the ﬁrst Fresnel zone as subtended at the

detector.⁵⁶In the experimental geometry this would corre-

spond to a coherent scattering object approximately 30 ␮m

in dimension. This is much longer than the typical incident

beam coherence length, excluding this as a source of error in

the analysis.

R ␦q ,␦q ͒ϭNJ exp ϪA ͒exp ϪB ͒exp ϪC ͒,

͑

x

z

xx

zz

xz

with

2

A ϭ cos²␤͒/ ⌬␣͒²ϩcos²␣͒/ ⌬␤͒

͑

͕

͖

xx

2

ϫ ␦q /„k sin 2⌰͒… ,

͓

͑

͔

x

0

2

B ϭ sin²␤͒/ ⌬␣͒

͑

͕

zz

2

ϩsin²␣͒/ ⌬␤͒ ␦q /„k sin 2⌰͒… ,

͑

͓

͑

͔

͖

z

0

438

A. STUNAULT et al.

55

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Article Doi

Near-surface effects at the antiferromagnetic phase transition in uranium phosphide

DOI: 10.1103/PhysRevB.55.423

Source and publish data:

Authors:

Article abstract of DOI:10.1103/PhysRevB.55.423

Full text of DOI:10.1103/PhysRevB.55.423

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Organofluorine sulfur-containing compounds: V. Joint pyrolysis with chlorine or bromine of polyfluoroarenethioles, polyfluorohetarenethioles, and their derivatives