text: 5.3 Experiment: Maximum-Likelihood Classiﬁcation From the initial set of results it was clear that using the response amplitude at the second harmonic frequency gives the best results, however the classiﬁcation accu- racy clearly left something to be desired. We performed a second experiment with the same batch of Droid de- Figure 4: Comparison of second-harmonic curves for all vices (excluding one which was faulty2). devices. Each device is represented by three adjacent curves that have the same color and ﬁll pattern. taken. The measurements were performed by an An- droid application that we side-loaded onto each phone. The application requires only the RECORD AUDIO and MODIFY AUDIO SETTINGS permissions, and once launched it adjusts the sound volume to a ﬁxed, medium level and then plays a three-second sound at each fre- quency, recording the middle one second back on a sep- Figure 5: Four overlaid frequency response curves at a arate Java thread. That one-second recording is used as different location. the vector R in subsequent analysis. Figure 4 juxtaposes the processed data obtained from all 16 devices. The similarity of the data from the same In the second experiment, we measured response am- device across locations is evident, as are the differences plitudes at 13 different frequencies, from 100Hz to between different devices. 1300Hz at 100Hz increments. We measured the response of each device at the same three locations we used in the previous experiment, however we ran the measurement Distance Metric four times at each location. Figure 5 illustrates why the Test Location1 A B B’ B” simple Euclidean distance-based classiﬁcation approach 2 68.8% 100% 62.5% 56.3% was not as precise as we wanted: at some frequencies, 3 43.8% 75% 50% 37.5% on some surfaces, there is simply too much variation— which results in a large penalty when calculating dis- tances. Other frequencies offer much smaller tolerances Table 3: Performance of simple L2 distance-based clas- and can be used to identify the device more accurately. siﬁcation of data from locations 2 and 3 (location 1 data Fortunately, the maximum-likelihood estimation method were used for the learning step). When ﬁrst harmonic is designed to deal with just this type of situation. (Distance A), and second harmonic (Distance B) feed- back are compared, second harmonic is almost always Our new scheme is based on the following simple more reliable and performs better than ﬁrst harmonic. setup (all frequency response measurements are at the Using the second harmonic’s ﬁrst and second derivatives second harmonic frequency—this will be implied for the only appears to have a negative overall impact. rest of this section): for each device we assume that its response at a certain frequency is a normally distributed random variable. Given enough samples, we can esti- In order to estimate the amount of information that can mate the mean and variance of the distribution. Later, be derived from this type of signal metric we designed a when we encounter a device, we can calculate the like- simple algorithm which uses one location’s measurement lihood that it matches any of the known devices in our from each device to “learn” its ﬁngerprint, and then tries database—and select the one that maximizes the likeli- to match the data from the remaining two locations to the hood. Formally, based on the training data from device device for which the learned ﬁngerprint is the closest. If D , at each frequency f we estimate the mean of the re- k i a data set is matched to the device it originated from, we sponse µ(k,i) as well as the variance σ2(k,i). Then we count it as a correct detection (Table 3). select the device D that was most likely to produce kmax 5 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: section

text: Experiment:

text: Maximum-Likelihood

text: Classiﬁcation

text: From

text: the

text: initial

text: set

text: of

text: results

text: it

text: was

text: clear

text: that

text: using

text: the

text: response

text: amplitude

text: at

text: the

text: second

text: harmonic

text: frequency

text: gives

text: the

text: best

text: results,

text: however

text: the

text: classiﬁcation

text: accu-

text: racy

text: clearly

text: left

text: something

text: to

text: be

text: desired.

text: We

text: performed

text: a

text: second

text: experiment

text: with

text: the

text: same

text: batch

text: of

text: Droid

text: de-

text: Figure

text: Comparison

text: of

text: second-harmonic

text: curves

text: for

text: all

text: vices

text: (excluding

text: one

text: which

text: was

text: devices.

text: Each

text: device

text: is

text: represented

text: by

text: three

text: adjacent

text: curves

text: that

text: have

text: the

text: same

text: color

text: and

text: ﬁll

text: pattern.

text: taken.

text: The

text: measurements

text: were

text: performed

text: by

text: an

text: An-

text: droid

text: application

text: that

text: we

text: side-loaded

text: onto

text: each

text: phone.

text: The

text: application

text: requires

text: only

text: the

text: RECORD

text: AUDIO

text: and

text: MODIFY

text: AUDIO

text: SETTINGS

text: permissions,

text: and

text: once

text: launched

text: it

text: adjusts

text: the

text: sound

text: volume

text: to

text: a

text: ﬁxed,

text: medium

text: level

text: and

text: then

text: plays

text: a

text: three-second

text: sound

text: at

text: each

text: fre-

text: quency,

text: recording

text: the

text: middle

text: one

text: second

text: back

text: on

text: a

text: sep-

text: Figure

text: Four

text: overlaid

text: frequency

text: response

text: curves

text: at

text: a

text: arate

text: Java

text: thread.

text: That

text: one-second

text: recording

text: is

text: used

text: as

text: different

text: location.

text: the

text: vector

text: R

text: in

text: subsequent

text: analysis.

text: Figure

text: juxtaposes

text: the

text: processed

text: data

text: obtained

text: from

text: all

text: devices.

text: The

text: similarity

text: of

text: the

text: data

text: from

text: the

text: same

text: In

text: the

text: second

text: experiment,

text: we

text: measured

text: response

text: am-

text: device

text: across

text: locations

text: is

text: evident,

text: as

text: are

text: the

text: differences

text: plitudes

text: at

text: different

text: frequencies,

text: from

text: to

text: between

text: different

text: devices.

text: at

text: increments.

text: We

text: measured

text: the

text: response

text: of

text: each

text: device

text: at

text: the

text: same

text: three

text: locations

text: we

text: used

text: in

text: the

text: previous

text: experiment,

text: however

text: we

text: ran

text: the

text: measurement

text: Distance

text: Metric

text: four

text: times

text: at

text: each

text: location.

text: Figure

text: illustrates

text: why

text: the

text: Test

text: A

text: B

text: B’

text: B”

text: simple

text: Euclidean

text: distance-based

text: classiﬁcation

text: approach

text: was

text: not

text: as

text: precise

text: as

text: we

text: wanted:

text: at

text: some

text: frequencies,

text: on

text: some

text: surfaces,

text: there

text: is

text: simply

text: too

text: much

text: variation—

text: which

text: results

text: in

text: a

text: large

text: penalty

text: when

text: calculating

text: dis-

text: tances.

text: Other

text: frequencies

text: offer

text: much

text: smaller

text: tolerances

text: Table

text: Performance

text: of

text: simple

text: distance-based

text: clas-

text: and

text: can

text: be

text: used

text: to

text: identify

text: the

text: device

text: more

text: accurately.

text: siﬁcation

text: of

text: data

text: from

text: locations

text: and

text: (location

text: data

text: Fortunately,

text: the

text: maximum-likelihood

text: estimation

text: method

text: were

text: used

text: for

text: the

text: learning

text: step).

text: When

text: ﬁrst

text: harmonic

text: is

text: designed

text: to

text: deal

text: with

text: just

text: this

text: type

text: of

text: situation.

text: (Distance

text: A),

text: and

text: second

text: harmonic

text: (Distance

text: B)

text: feed-

text: back

text: are

text: compared,

text: second

text: harmonic

text: is

text: almost

text: always

text: Our

text: new

text: scheme

text: is

text: based

text: on

text: the

text: following

text: simple

text: more

text: reliable

text: and

text: performs

text: better

text: than

text: ﬁrst

text: harmonic.

text: setup

text: (all

text: frequency

text: response

text: measurements

text: are

text: at

text: the

text: Using

text: the

text: second

text: harmonic’s

text: ﬁrst

text: and

text: second

text: derivatives

text: second

text: harmonic

text: frequency—this

text: will

text: be

text: implied

text: for

text: the

text: only

text: appears

text: to

text: have

text: a

text: negative

text: overall

text: impact.

text: rest

text: of

text: this

text: section):

text: for

text: each

text: device

text: we

text: assume

text: that

text: its

text: response

text: at

text: a

text: certain

text: frequency

text: is

text: a

text: normally

text: distributed

text: random

text: variable.

text: Given

text: enough

text: samples,

text: we

text: can

text: esti-

text: In

text: order

text: to

text: estimate

text: the

text: amount

text: of

text: information

text: that

text: can

text: mate

text: the

text: mean

text: and

text: variance

text: of

text: the

text: distribution.

text: Later,

text: be

text: derived

text: from

text: this

text: type

text: of

text: signal

text: metric

text: we

text: designed

text: a

text: when

text: we

text: encounter

text: a

text: device,

text: we

text: can

text: calculate

text: the

text: like-

text: simple

text: algorithm

text: which

text: uses

text: one

text: location’s

text: measurement

text: lihood

text: that

text: it

text: matches

text: any

text: of

text: the

text: known

text: devices

text: in

text: our

text: from

text: each

text: device

text: to

text: “learn”

text: its

text: ﬁngerprint,

text: and

text: then

text: tries

text: database—and

text: select

text: the

text: one

text: that

text: maximizes

text: the

text: likeli-

text: to

text: match

text: the

text: data

text: from

text: the

text: remaining

text: two

text: locations

text: to

text: the

text: hood.

text: Formally,

text: based

text: on

text: the

text: training

text: data

text: from

text: device

text: device

text: for

text: which

text: the

text: learned

text: ﬁngerprint

text: is

text: the

text: closest.

text: If

text: D

text: at

text: each

text: frequency

text: f

text: we

text: estimate

text: the

text: mean

text: of

text: the

text: re-

text: k

text: i

text: a

text: data

text: set

text: is

text: matched

text: to

text: the

text: device

text: it

text: originated

text: from,

text: we

text: sponse

text: µ(k,i)

text: as

text: well

text: as

text: the

text: variance

text: Then

text: we

text: count

text: it

text: as

text: a

text: correct

text: detection

text: (Table

text: select

text: the

text: device

text: D

text: that

text: was

text: most

text: likely

text: to

text: produce

text: kmax

text: ##LTFigure##

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text: Theory of the evolution of magnetic order in Fe Te compounds with increasing 1+y interstitial iron Samuel Ducatman,1,2 Rafael M. Fernandes,2 and Natalia B. Perkins1,2 1Department of Physics, University of Wisconsin-Madison, Madison, WI 53706, USA 2School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA We examine the inﬂuence of the excess of interstitial Fe on the magnetic properties of Fe Te 1+y compounds. Because in iron chalcogenides the correlations are stronger than in the iron arsenides, we assume in our model that some of the Fe orbitals give rise to localized magnetic moments. These moments interact with each other via exchange interactions as well as phonon-mediated biquadratic interactions that favor a collinear double-stripe state, corresponding to the ordering ± ± vectors ( π/2, π/2). The remaining Fe orbitals are assumed to be itinerant, giving rise to the ﬁrst-principle derived Fermi surface displaying nesting features at momenta (π, 0) / (0, π). Increasing the amount of itinerant electrons due to excess Fe, y, leads to changes in the Fermi surface and to the suppression of its nesting properties. As a result, due to the Hund’s coupling between the itinerant and localized moments, increasing y leads to modiﬁcations in the local moments’ exchange interactions via the multi-orbital generalization of the long-range Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. By numerically computing the RKKY corrections and minimizing the resulting eﬀective exchange Hamiltonian, we ﬁnd, in general, that the excess electrons introduced in the system change the classical magnetic ground state from a double-stripe state to an incommensurate spiral, consistent with the experimental observations. We show that these results can be understood as a result of the suppression of magnetic spectral weight of the itinerant electrons at momenta (π, 0) / (0, π), combined with the transfer of broad magnetic spectral weight from large to small momenta, promoted by the introduction of excess Fe. I. INTRODUCTION it is clear that the local picture alone cannot describe the magnetic properties of Fe Te, as it requires y- 1+y dependent exchange couplings. Alternatively, this prop- Fe Te chalcogenides are the parent compounds of 1+y erty indicates that itinerant electrons are also important the simplest family of iron-based superconductors.1–13 to describe the magnetism of these materials, suggest- Both the electronic and magnetic properties of Fe Te 1+y ing that hybrid models with coupled localized-itinerant compounds show strong sensitivity to the amount of non- moments are a suitable starting point.20–26 stoichiometric Fe ions. For small values of y, the low- In this paper, we argue that the evolution of the mag- temperature crystal structure is monoclinic (P21/m), netic interactions due to y-dependent charge doping is the and the magnetic order is described by the commensu- ± ± key to understand the experimentally observed magnetic rate propagation vector Q = ( π/2, π/2), correspond- phase diagram of Fe Te. Our study is based on the 1+y ing to a double-stripe pattern. This is remarkably dif- assumption that, in these particular iron chalcogenides, ferent from FeAs-based parent compounds, which dis- some of the Fe orbitals are almost localized while the play an antiferromagnetic order described by Q = (π, 0) other orbitals remain itinerant. This idea is supported by or Q = (0, π), corresponding to single-stripe patterns, recent dynamical mean-ﬁeld theory studies of FeTe sys- and a crystal structure with orthorhombic symmetry tems27,28 showing that the Hund’s coupling can promote (Pmmn). Experimentally, it is observed that by increas- an orbital-selective localization already in the paramag- ing the amount of interstitial Fe, the magnetic structure netic phase.29 Here, we demonstrate that the change in of Fe Te becomes an incommensurate spiral. The in- 1+y magnetic properties observed in the Fe Te compounds 1+y commensurate ordering manifests itself as a shift in the can be reasonably well captured by an eﬀective model in elastic neutron scattering peak with respect to the Q = ± ± which localized spins acquire a long-range RKKY-type ( π/2, π/2) positions. According to neutron scattering − − interaction,30–32 in addition to the J J J Heisen- experiments,1,5–7,12,13 in the range of 0.11 < y < 0.16 1 2 3 berg super-exchange14 and biquadratic couplings.33–37 the shift is approximately along the diagonal directions − − − − We note that in Refs.33,34 the biquadratic term was cal- Q = (π/2 δ, π/2 δ) or Q = ( π/2 + δ, π/2 δ); culated on a purely electronic basis, and obtained model however, δ does not vary smoothly with y. gave a good agreement with experimentally measured There have been several theoretical attempts to un- spin-wave spectrum in several Fe-based materials. An- derstand the magnetic properties of Fe Te compounds other possible origin of the biquadratic terms is due to 1+y within the localized spin scenario,14–18 since these mate- the magnetoelastic coupling.35–37 rials are known to be more strongly correlated than their The former, mediated by the multi-orbital spin sus- arsenide counterparts.19 Although the magnetic order at ceptibility of the itinerant electrons, is sensitive to the both low and at high levels of Fe excess can be success- addition of excess Fe y, rendering the magnetic ground − − fully described by a J J J super-exchange model, state of the local spins change as a function of y. In par- 1 2 3

text: Theory

label: title

text: of

text: the

text: evolution

text: of

text: magnetic

text: order

text: in

text: Fe

text: Te

text: compounds

text: with

text: increasing

text: interstitial

text: iron

text: Samuel

label: author

text: Rafael

text: M.

text: and

label: paragraph

text: Natalia

text: B.

text: of

text: Physics,

text: University

text: of

text: Wisconsin-Madison,

text: Madison,

text: WI

text: USA

text: of

text: Physics

text: and

text: Astronomy,

text: University

text: of

text: Minnesota,

text: Minneapolis,

text: MN

text: USA

text: We

label: abstract

text: examine

text: the

text: inﬂuence

text: of

text: the

text: excess

text: of

text: interstitial

text: Fe

text: on

text: the

text: magnetic

text: properties

text: of

text: Fe

text: Te

text: compounds.

text: Because

text: in

text: iron

text: chalcogenides

text: the

text: correlations

text: are

text: stronger

text: than

text: in

text: the

text: iron

text: arsenides,

text: we

text: assume

text: in

text: our

text: model

text: that

text: some

text: of

text: the

text: Fe

text: orbitals

text: give

text: rise

text: to

text: localized

text: magnetic

text: moments.

text: These

text: moments

text: interact

text: with

text: each

text: other

text: via

text: exchange

text: interactions

text: as

text: well

text: as

text: phonon-mediated

text: biquadratic

text: interactions

text: that

text: favor

text: a

text: collinear

text: double-stripe

text: state,

text: corresponding

text: to

text: the

text: ordering

text: vectors

text: The

text: remaining

text: Fe

text: orbitals

text: are

text: assumed

text: to

text: be

text: itinerant,

text: giving

text: rise

text: to

text: the

text: ﬁrst-principle

text: derived

text: Fermi

text: surface

text: displaying

text: nesting

text: features

text: at

text: momenta

text: (π,

text: π).

text: Increasing

text: the

text: amount

text: of

text: itinerant

text: electrons

text: due

text: to

text: excess

text: Fe,

text: y,

text: leads

text: to

text: changes

text: in

text: the

text: Fermi

text: surface

text: and

text: to

text: the

text: suppression

text: of

text: its

text: nesting

text: properties.

text: As

text: a

text: result,

text: due

text: to

text: the

text: Hund’s

text: coupling

text: between

text: the

text: itinerant

text: and

text: localized

text: moments,

text: increasing

text: y

text: leads

text: to

text: modiﬁcations

text: in

text: the

text: local

text: moments’

text: exchange

text: interactions

text: via

text: the

text: multi-orbital

text: generalization

text: of

text: the

text: long-range

text: Ruderman-Kittel-Kasuya-Yosida

text: (RKKY)

text: interaction.

text: By

text: numerically

text: computing

text: the

text: RKKY

text: corrections

text: and

text: minimizing

text: the

text: resulting

text: eﬀective

text: exchange

text: Hamiltonian,

text: we

text: ﬁnd,

text: in

text: general,

text: that

text: the

text: excess

text: electrons

text: introduced

text: in

text: the

text: system

text: change

text: the

text: classical

text: magnetic

text: ground

text: state

text: from

text: a

text: double-stripe

text: state

text: to

text: an

text: incommensurate

text: spiral,

text: consistent

text: with

text: the

text: experimental

text: observations.

text: We

text: show

text: that

text: these

text: results

text: can

text: be

text: understood

text: as

text: a

text: result

text: of

text: the

text: suppression

text: of

text: magnetic

text: spectral

text: weight

text: of

text: the

text: itinerant

text: electrons

text: at

text: momenta

text: (π,

text: π),

text: combined

text: with

text: the

text: transfer

text: of

text: broad

text: magnetic

text: spectral

text: weight

text: from

text: large

text: to

text: small

text: momenta,

text: promoted

text: by

text: the

text: introduction

text: of

text: excess

text: Fe.

text: I.

text: INTRODUCTION

text: it

text: is

text: clear

text: that

text: the

text: local

text: picture

text: alone

text: cannot

text: describe

text: the

text: magnetic

text: properties

text: of

text: Fe

text: Te,

text: as

text: it

text: requires

text: y-

text: dependent

text: exchange

text: couplings.

text: Alternatively,

text: this

text: prop-

text: Fe

text: Te

text: chalcogenides

text: are

text: the

text: parent

text: compounds

text: of

text: erty

text: indicates

text: that

text: itinerant

text: electrons

text: are

text: also

text: important

text: the

text: simplest

text: family

text: of

text: iron-based

text: to

text: describe

text: the

text: magnetism

text: of

text: these

text: materials,

text: suggest-

text: Both

text: the

text: electronic

text: and

text: magnetic

text: properties

text: of

text: Fe

text: Te

text: ing

text: that

text: hybrid

text: models

text: with

text: coupled

text: localized-itinerant

text: compounds

text: show

text: strong

text: sensitivity

text: to

text: the

text: amount

text: of

text: non-

text: moments

text: are

text: a

text: suitable

text: starting

text: stoichiometric

text: Fe

text: ions.

text: For

text: small

text: values

text: of

text: y,

text: the

text: low-

text: In

text: this

text: paper,

text: we

text: argue

text: that

text: the

text: evolution

text: of

text: the

text: mag-

text: temperature

text: crystal

text: structure

text: is

text: monoclinic

text: netic

text: interactions

text: due

text: to

text: y-dependent

text: charge

text: doping

text: is

text: the

text: and

text: the

text: magnetic

text: order

text: is

text: described

text: by

text: the

text: commensu-

text: key

text: to

text: understand

text: the

text: experimentally

text: observed

text: magnetic

text: rate

text: propagation

text: vector

text: Q

text: correspond-

text: phase

text: diagram

text: of

text: Fe

text: Te.

text: Our

text: study

text: is

text: based

text: on

text: the

text: ing

text: to

text: a

text: double-stripe

text: pattern.

text: This

text: is

text: remarkably

text: dif-

text: assumption

text: that,

text: in

text: these

text: particular

text: iron

text: chalcogenides,

text: ferent

text: from

text: FeAs-based

text: parent

text: compounds,

text: which

text: dis-

text: some

text: of

text: the

text: Fe

text: orbitals

text: are

text: almost

text: localized

text: while

text: the

text: play

text: an

text: antiferromagnetic

text: order

text: described

text: by

text: Q

text: (π,

text: other

text: orbitals

text: remain

text: itinerant.

text: This

text: idea

text: is

text: supported

text: by

text: or

text: Q

text: π),

text: corresponding

text: to

text: single-stripe

text: patterns,

text: recent

text: dynamical

text: mean-ﬁeld

text: theory

text: studies

text: of

text: FeTe

text: sys-

text: and

text: a

text: crystal

text: structure

text: with

text: orthorhombic

text: symmetry

text: showing

text: that

text: the

text: Hund’s

text: coupling

text: can

text: promote

text: (Pmmn).

text: Experimentally,

text: it

text: is

text: observed

text: that

text: by

text: increas-

text: an

text: orbital-selective

text: localization

text: already

text: in

text: the

text: paramag-

text: ing

text: the

text: amount

text: of

text: interstitial

text: Fe,

text: the

text: magnetic

text: structure

text: netic

text: Here,

text: we

text: demonstrate

text: that

text: the

text: change

text: in

text: of

text: Fe

text: Te

text: becomes

text: an

text: incommensurate

text: spiral.

text: The

text: in-

text: magnetic

text: properties

text: observed

text: in

text: the

text: Fe

text: Te

text: compounds

text: commensurate

text: ordering

text: manifests

text: itself

text: as

text: a

text: shift

text: in

text: the

text: can

text: be

text: reasonably

text: well

text: captured

text: by

text: an

text: eﬀective

text: model

text: in

text: elastic

text: neutron

text: scattering

text: peak

text: with

text: respect

text: to

text: the

text: Q

text: which

text: localized

text: spins

text: acquire

text: a

text: long-range

text: RKKY-type

text: positions.

text: According

text: to

text: neutron

text: scattering

text: in

text: addition

text: to

text: the

text: J

text: J

text: J

text: Heisen-

text: in

text: the

text: range

text: of

text: y

text: berg

text: and

text: biquadratic

text: the

text: shift

text: is

text: approximately

text: along

text: the

text: diagonal

text: directions

text: We

text: note

text: that

text: in

text: the

text: biquadratic

text: term

text: was

text: cal-

text: Q

text: δ,

text: δ)

text: or

text: Q

text: δ,

text: δ);

text: culated

text: on

text: a

text: purely

text: electronic

text: basis,

text: and

text: obtained

text: model

text: however,

text: δ

text: does

text: not

text: vary

text: smoothly

text: with

text: y.

text: gave

text: a

text: good

text: agreement

text: with

text: experimentally

text: measured

text: There

text: have

text: been

text: several

text: theoretical

text: attempts

text: to

text: un-

text: spin-wave

text: spectrum

text: in

text: several

text: Fe-based

text: materials.

text: An-

text: derstand

text: the

text: magnetic

text: properties

text: of

text: Fe

text: Te

text: compounds

text: other

text: possible

text: origin

text: of

text: the

text: biquadratic

text: terms

text: is

text: due

text: to

text: within

text: the

text: localized

text: spin

text: since

text: these

text: mate-

text: the

text: magnetoelastic

text: rials

text: are

text: known

text: to

text: be

text: more

text: strongly

text: correlated

text: than

text: their

text: The

text: former,

text: mediated

text: by

text: the

text: multi-orbital

text: spin

text: sus-

text: arsenide

text: Although

text: the

text: magnetic

text: order

text: at

text: ceptibility

text: of

text: the

text: itinerant

text: electrons,

text: is

text: sensitive

text: to

text: the

text: both

text: low

text: and

text: at

text: high

text: levels

text: of

text: Fe

text: excess

text: can

text: be

text: success-

text: addition

text: of

text: excess

text: Fe

text: y,

text: rendering

text: the

text: magnetic

text: ground

text: fully

text: described

text: by

text: a

text: J

text: J

text: J

text: super-exchange

text: model,

text: state

text: of

text: the

text: local

text: spins

text: change

text: as

text: a

text: function

text: of

text: y.

text: In

text: par-


text: 2 − ticular, we ﬁnd that the shift in the chemical potential 3z2 r2 orbitals form local moments with S = 1 due promoted by the excess electrons changes the Fermi sur- to the Hund’s coupling. The remaining 3d electrons have face in a way that suppresses the (π, 0) / (0, π) peaks of itinerant character and can be controlled by the concen- the itinerant spin susceptibility of the parent compound, tration of excess Fe as y increases. A full derivation of promoting at the same time peaks at small-momentum such an eﬀective model from the microscopic Hamilto- values. As a result, the eﬀective exchange interaction J nian is an intricate problem beyond the scope of this 2 is suppressed, J is enhanced, and J changes sign, fa- work. 3 1 voring a spiral incommensurate state, in contrast to the In most of the paper, we assume that each excess Fe double-stripe state of the stoichiometric compound. atom contributes eight electrons,39 but our main conclu- The outline of the paper is as follows. In Sec. II, we sions do not change if one considers that less electrons are present an eﬀective super-exchange model describing lo- introduced by each Fe.40,41 To account for the changes in calized magnetic moments in the multi-band correlated the Fermi surface promoted by these excess Fe atoms, electron sea and argue that this is a suitable minimal mi- we employ a rigid band approximation in which the dop- croscopic model to describe the magnetism of Fe Te ing of excess electrons shifts the chemical potential from 1+y compounds. In Sec. III, we study the evolution of the µ = 0 to positive values. Fermi surface of FeTe with increasing level of the Fe Thus, the microscopic Hamiltonian we use is an ef- 1+y excess using the tight-binding model (TBM) originally fective double-exchange model describing localized mag- proposed by Ma et al14, which had been later also used netic moments in the multi-band correlated electron sea, by Wang et al38 to explain further resutls in FeTe . which can be written as 1+y We show that while the Fermi surface at small y has both small hole pockets at the Γ- and M-points and el- H = H + H + H . (1) liptical electron pockets at the X- and Y -points, at large e S σS y all pockets are electron-like. In Sec. IV, we study spin ﬂuctuations in the correlated multi-band electron The ﬁrst term describes the interacting itinerant elec- system and compute the Pauli susceptibility within the trons: random phase approximation (RPA). We ﬁnd that at small y the itinerant spin susceptibility peaks at (π, 0) H = H + H . (2) and (0, π) due to the weak nesting between the hole and e 0 int the electron pockets connected by these momenta. The y- dependent charge doping suppresses the (π, 0) and (0, π) The non-interacting part H is based on the ﬁve-orbital 0 peaks, but it leads to an increase of the spin ﬂuctuation TBM (cid:88) (cid:16) (cid:17) in the central part of the Brillouin zone. In Sec. V, we compute the RKKY interactions. We ﬁrst perform † a qualitative computation of the RKKY interactions us- H = tab c c + H.C. , (3) 0 kσ kaσ kbσ ing a simple phenomenological model, and then perform k,a,b,σ a quantitative analysis based on the realistic RPA sus- † ceptibility obtained in Sec. IV. In Sec. VI, the classical where c denotes the creation operator for an elec- kσa phase diagram of the eﬀective spin model is presented. tron of momentum k with spin σ in the orbital a, and In agreement with experimental ﬁndings, the computed tab are the tight-binding matrix elements. Here, we are kσ phase diagram displays a transition, above a certain level interested only in the contribution coming from the xz, of Fe excess, from a commensurate double-stripe phase, yz, and xy orbitals, as explained above. Since the other ± characterized by the wave-vectors Q = (π/2, π/2) or orbitals do not contribute to the Fermi surface, the static ± Q = ( π/2, π/2), to an incommensurate spiral (IC) spin susceptibility, which will give rise to the RKKY in- phase characterized by the wave-vector Q = (q, q). We teractions, is very similar regardless of whether we con- conclude with a summary in Sec. VII. The paper has sider a projected three-orbital model or the actual ﬁve- two appendices. Appendix A contains the derivation orbital tight-binding model. of the biquadratic exchange couplings arising from the The interaction (cid:88)part consists of four terms:42–44 magneto-elastic coupling. Appendix B provides the ex- plicit expression of the classical energy of the eﬀective † † H =U c (cid:88)↑c ↑c ↓c ↓ super-exchange model describing FeTe1+y. int ia ia ia ia II. THE MODEL i,a (cid:48) (cid:88) † † + U ciaσciaσcibσ(cid:48)cibσ(cid:48) (cid:54) (cid:48) To capture the fact that in Fe Te correlations lead to i,a=b,σ,σ 1+y † † diﬀerent levels of itineracy in distinct Fe orbitals,27,28 we + JH(cid:88) ciaσcibσ(cid:48)ciaσ(cid:48)cibσ consider a semi-phenomenological “hybrid” model con- (cid:54) (cid:48) i,a=b,σ,σ taining both localized and itinerant moments – similar (cid:48) † † in spirit to the models of Refs.22–24. In parti−cular, we + J (cid:54) cia↑cia↓cib↓cib↑ (4) assume that the electrons occupying the x2 y2 and i,a=b

text: ticular,

text: we

text: ﬁnd

text: that

text: the

text: shift

text: in

text: the

text: chemical

text: potential

text: orbitals

text: form

text: local

text: moments

text: with

text: S

text: due

text: promoted

text: by

text: the

text: excess

text: electrons

text: changes

text: the

text: Fermi

text: sur-

text: to

text: the

text: Hund’s

text: coupling.

text: The

text: remaining

text: electrons

text: have

text: face

text: in

text: a

text: way

text: that

text: suppresses

text: the

text: (π,

text: π)

text: peaks

text: of

text: itinerant

text: character

text: and

text: can

text: be

text: controlled

text: by

text: the

text: concen-

text: the

text: itinerant

text: spin

text: susceptibility

text: of

text: the

text: parent

text: compound,

text: tration

text: of

text: excess

text: Fe

text: as

text: y

text: increases.

text: A

text: full

text: derivation

text: of

text: promoting

text: at

text: the

text: same

text: time

text: peaks

text: at

text: small-momentum

text: such

text: an

text: eﬀective

text: model

text: from

text: the

text: microscopic

text: Hamilto-

text: values.

text: As

text: a

text: result,

text: the

text: eﬀective

text: exchange

text: interaction

text: J

text: nian

text: is

text: an

text: intricate

text: problem

text: beyond

text: the

text: scope

text: of

text: this

text: is

text: suppressed,

text: J

text: is

text: enhanced,

text: and

text: J

text: changes

text: sign,

text: fa-

text: work.

text: voring

text: a

text: spiral

text: incommensurate

text: state,

text: in

text: contrast

text: to

text: the

text: In

text: most

text: of

text: the

text: paper,

text: we

text: assume

text: that

text: each

text: excess

text: Fe

text: double-stripe

text: state

text: of

text: the

text: stoichiometric

text: compound.

text: atom

text: contributes

text: eight

text: but

text: our

text: main

text: conclu-

text: The

text: outline

text: of

text: the

text: paper

text: is

text: as

text: follows.

text: In

text: Sec.

text: II,

text: we

text: sions

text: do

text: not

text: change

text: if

text: one

text: considers

text: that

text: less

text: electrons

text: are

text: present

text: an

text: eﬀective

text: super-exchange

text: model

text: describing

text: lo-

text: introduced

text: by

text: each

text: To

text: account

text: for

text: the

text: changes

text: in

text: calized

text: magnetic

text: moments

text: in

text: the

text: multi-band

text: correlated

text: the

text: Fermi

text: surface

text: promoted

text: by

text: these

text: excess

text: Fe

text: atoms,

text: electron

text: sea

text: and

text: argue

text: that

text: this

text: is

text: a

text: suitable

text: minimal

text: mi-

text: we

text: employ

text: a

text: rigid

text: band

text: approximation

text: in

text: which

text: the

text: dop-

text: croscopic

text: model

text: to

text: describe

text: the

text: magnetism

text: of

text: Fe

text: Te

text: ing

text: of

text: excess

text: electrons

text: shifts

text: the

text: chemical

text: potential

text: from

text: compounds.

text: In

text: Sec.

text: III,

text: we

text: study

text: the

text: evolution

text: of

text: the

text: µ

text: to

text: positive

text: values.

text: Fermi

text: surface

text: of

text: FeTe

text: with

text: increasing

text: level

text: of

text: the

text: Fe

text: Thus,

text: the

text: microscopic

text: Hamiltonian

text: we

text: use

text: is

text: an

text: ef-

text: excess

text: using

text: the

text: tight-binding

text: model

text: (TBM)

text: originally

text: fective

text: double-exchange

text: model

text: describing

text: localized

text: mag-

text: proposed

text: by

text: Ma

text: et

text: which

text: had

text: been

text: later

text: also

text: used

text: netic

text: moments

text: in

text: the

text: multi-band

text: correlated

text: electron

text: sea,

text: by

text: Wang

text: et

text: to

text: explain

text: further

text: resutls

text: in

text: FeTe

text: which

text: can

text: be

text: written

text: as

text: We

text: show

text: that

text: while

text: the

text: Fermi

text: surface

text: at

text: small

text: y

text: has

text: both

text: small

text: hole

text: pockets

text: at

text: the

text: Γ-

text: and

text: M-points

text: and

text: el-

text: H

text: H

text: H

text: H

text: liptical

text: electron

text: pockets

text: at

text: the

text: X-

text: and

text: Y

text: -points,

text: at

text: large

text: e

text: S

text: σS

text: y

text: all

text: pockets

text: are

text: electron-like.

text: In

text: Sec.

text: IV,

text: we

text: study

text: spin

text: ﬂuctuations

text: in

text: the

text: correlated

text: multi-band

text: electron

text: The

text: ﬁrst

text: term

text: describes

text: the

text: interacting

text: itinerant

text: elec-

text: system

text: and

text: compute

text: the

text: Pauli

text: susceptibility

text: within

text: the

text: trons:

text: random

text: phase

text: approximation

text: (RPA).

text: We

text: ﬁnd

text: that

text: at

text: small

text: y

text: the

text: itinerant

text: spin

text: susceptibility

text: peaks

text: at

text: (π,

text: H

text: H

text: H

text: and

text: π)

text: due

text: to

text: the

text: weak

text: nesting

text: between

text: the

text: hole

text: and

text: e

text: int

text: the

text: electron

text: pockets

text: connected

text: by

text: these

text: momenta.

text: The

text: y-

text: dependent

text: charge

text: doping

text: suppresses

text: the

text: (π,

text: and

text: π)

text: The

text: non-interacting

text: part

text: H

text: is

text: based

text: on

text: the

text: ﬁve-orbital

text: peaks,

text: but

text: it

text: leads

text: to

text: an

text: increase

text: of

text: the

text: spin

text: ﬂuctuation

text: TBM

text: in

text: the

text: central

text: part

text: of

text: the

text: Brillouin

text: zone.

text: In

text: Sec.

text: V,

text: we

text: compute

text: the

text: RKKY

text: interactions.

text: We

text: ﬁrst

text: perform

text: a

text: qualitative

text: computation

text: of

text: the

text: RKKY

text: interactions

text: us-

text: H

text: tab

text: c

text: c

text: H.C.

text: kσ

text: kaσ

text: kbσ

text: ing

text: a

text: simple

text: phenomenological

text: model,

text: and

text: then

text: perform

text: k,a,b,σ

text: a

text: quantitative

text: analysis

text: based

text: on

text: the

text: realistic

text: RPA

text: sus-

text: ceptibility

text: obtained

text: in

text: Sec.

text: IV.

text: In

text: Sec.

text: VI,

text: the

text: classical

text: where

text: c

text: denotes

text: the

text: creation

text: operator

text: for

text: an

text: elec-

text: kσa

text: phase

text: diagram

text: of

text: the

text: eﬀective

text: spin

text: model

text: is

text: presented.

text: tron

text: of

text: momentum

text: k

text: with

text: spin

text: σ

text: in

text: the

text: orbital

text: a,

text: and

text: In

text: agreement

text: with

text: experimental

text: ﬁndings,

text: the

text: computed

text: tab

text: are

text: the

text: tight-binding

text: matrix

text: elements.

text: Here,

text: we

text: are

text: kσ

text: phase

text: diagram

text: displays

text: a

text: transition,

text: above

text: a

text: certain

text: level

text: interested

text: only

text: in

text: the

text: contribution

text: coming

text: from

text: the

text: xz,

text: of

text: Fe

text: excess,

text: from

text: a

text: commensurate

text: double-stripe

text: phase,

text: yz,

text: and

text: xy

text: orbitals,

text: as

text: explained

text: above.

text: Since

text: the

text: other

text: characterized

text: by

text: the

text: wave-vectors

text: Q

text: or

text: orbitals

text: do

text: not

text: contribute

text: to

text: the

text: Fermi

text: surface,

text: the

text: static

text: Q

text: to

text: an

text: incommensurate

text: spiral

text: (IC)

text: spin

text: susceptibility,

text: which

text: will

text: give

text: rise

text: to

text: the

text: RKKY

text: in-

text: phase

text: characterized

text: by

text: the

text: wave-vector

text: Q

text: (q,

text: q).

text: We

text: teractions,

text: is

text: very

text: similar

text: regardless

text: of

text: whether

text: we

text: con-

text: conclude

text: with

text: a

text: summary

text: in

text: Sec.

text: VII.

text: The

text: paper

text: has

text: sider

text: a

text: projected

text: three-orbital

text: model

text: or

text: the

text: actual

text: ﬁve-

text: two

text: appendices.

text: Appendix

text: A

text: contains

text: the

text: derivation

text: orbital

text: tight-binding

text: model.

text: of

text: the

text: biquadratic

text: exchange

text: couplings

text: arising

text: from

text: the

text: The

text: interaction

text: consists

text: of

text: four

text: magneto-elastic

text: coupling.

text: Appendix

text: B

text: provides

text: the

text: ex-

text: plicit

text: expression

text: of

text: the

text: classical

text: energy

text: of

text: the

text: eﬀective

text: H

text: =U

text: c

text: ↑c

text: ↓c

text: super-exchange

text: model

text: describing

text: int

text: ia

text: ia

text: ia

text: ia

text: II.

text: THE

text: MODEL

text: i,a

text: U

text: To

text: capture

text: the

text: fact

text: that

text: in

text: Fe

text: Te

text: correlations

text: lead

text: to

text: i,a=b,σ,σ

text: diﬀerent

text: levels

text: of

text: itineracy

text: in

text: distinct

text: Fe

text: we

text: consider

text: a

text: semi-phenomenological

text: “hybrid”

text: model

text: con-

text: i,a=b,σ,σ

text: taining

text: both

text: localized

text: and

text: itinerant

text: moments

text: similar

text: in

text: spirit

text: to

text: the

text: models

text: of

text: In

text: parti−cular,

text: we

text: J

text: cia↑cia↓cib↓cib↑

text: assume

text: that

text: the

text: electrons

text: occupying

text: the

text: and

text: i,a=b


text: 3 3 (a) 3 (b) 3 (c) 2 2 2 1 1 1 qy 0 qy 0 qy 0 1 1 1 (cid:45) (cid:45) (cid:45) 2 2 2 (cid:45) (cid:45) (cid:45) 3 3 3 (cid:45) (cid:45) (cid:45) 3 2 1 0 1 2 3 3 2 1 0 1 2 3 3 2 1 0 1 2 3 (cid:45) (cid:45) (cid:45) qx (cid:45) (cid:45) (cid:45) qx (cid:45) (cid:45) (cid:45) qx FIG. 1: Fermi surfaces of the TBM describing the itinerant electrons obtained for diﬀerent values of the chemical potential µ: (a) µ = 0 eV, (b) µ = 0.4 eV, (c) µ = 0.8 eV. The hole-like and the electron-like pockets are shown by yellow (light) and purple (dark) regions, correspondingly. The primary contribution to the electron pockets is from the xz and yz orbitals, whereas the hole pocket at the corner of the Brillouin zone is mainly of xy character. (cid:88) (cid:48) where U is the intra-orbital Coulomb repulsion, U is 1 (cid:48)− · † the inter-orbital(cid:48) Coulomb repulsion, JH is the Hund’s σja = N ei(k k) Rjckaστσσ(cid:48)ck(cid:48)aσ(cid:48) , (7) coupli(cid:48)ng, an−d J is the pa(cid:48) ir-hopping term. Hereafter, we k,k(cid:48),σ,σ(cid:48) set U = U 2J and J = J to ensure the invariance H H of the Hamiltonian under rotations in orbital space. where τ (cid:48) are the Pauli matrices and N is the num- σσ The second term in Eq.(1) describes the interaction ber of lattice sites. For our investigations, the sign of between the local(cid:18)ized spins: (cid:19) JH is not important, as shown in the following. Due (cid:88) to the interaction described by Eq.(6), the itinerant elec- trons mediate additional exchange couplings between the · − K · localized moments: (cid:88) H = J S S ij (S S )2 , (5) S ij i j S2 i j ij · H = JRKKYS S . (8) RKKY ij i j where J , J , and J are super-exchange couplings be- ij 1 2 3 tween ﬁrst-, second-, and third-nearest neighbors. In this work, we use the values of J , J , and J ob- The couplings JRKKY between localized spins on lat- 1 2 3 ij tained from the ﬁrst-principles electronic-structure cal- tice sites R and R , known as RKKY interactions, can i j culations.14 K denote generalized non-Heisenberg ex- be obtained by integrating out the itinerant degrees of ij change couplings between ﬁrst and second neighbors and freedom. In particular, JRKKY are determined by the ij ring-exchange. These couplings are predominantly deter- static spin susceptibility χ(q, ω = 0) = χ(q) of the multi- mined by the magneto-elastic coupling, and their deriva- band conduction electron sea: tions are shown in Appendix A. In particular, here we (cid:88) consider the ﬁrst- and second-neighbors biquadratic cou- − − plings K−1 and K2, as well as a “diagonal” ring-exchange JiRjKKY = JH2 χ(Ri Rj) = (9) K = K . − − · diag 2 J2 eı(Ri Rj) qχ(q). H The third term (cid:88) q · Taking into account both Heisenberg and RKKY inter- HσS = JH σja Sj (6) actions, the eﬀective low-energy Hamiltonian can then be j,a written as (cid:88) (cid:88) anddetshceriibteinsetrhaentcoeulepcltirnognbs eσtjwae.eHn−etrhee, Slojcaarliezethde−slpoicnasl,izSejd, Heﬀ = JiejﬀSiSj − KSi2j (Si · Sj)2, (10) spins of electrons on orbitals x2 y2 and 3z2 r2, and ij ij σ are the spins of itinerant electrons on orbitals a = ja xy, yz, xz deﬁned as where, for convenience, we deﬁned Jeﬀ = J +JRKKY. ij ij ij ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: 10 concentration y, which aﬀects indirectly the eﬀective ex- 0.20 change couplings Jeﬀ via the RKKY interaction. ij In Figs. 8 (a)-(c), we draw the real-space spin con- ﬁgurations presented in the phase diagram of Fig. 7. Among all possible states, in the parameter space pre- 0.15 sented in the phase diagram, only three states are re- alized: the stripe phase characterized by ϕ1 = 0, ϕ2 = Double Stripe π, ϕ = π, q = 0, q = 0 shown in Fig. 8 (a), the dou- 3 x y ble stripe phase characterized by ϕ1 = 0, ϕ2 = 0, ϕ3 = K1K20.10Stripe π, q = π/2, q = π/2 shown in Fig. 8 (b) and the in- x y commensurate spiral (labeled as IC) state shown in Fig. − − 8 (c) characterized by ϕ = π/2 δ, ϕ = π 2δ, ϕ = − − 1 − 2 3 π/2 δ, q = π/2 δ, q = π/2 δ. Note that in Fig. 8 x y 0.05 IC Spiral (c) we show the IC state with an exaggerated canting an- gle. To clarify the structure of these diﬀerent orderings, we also take the Fourier transforms of the ground state spin conﬁguration obtained by the minimization and then 0.00 compute the corresponding spin structure factor. As ex- 0.02 0.04 0.06 0.08 0.10 0.12 0.14 pected, the structure factor exhibits peaks at the fol- y lowing ordering wave-vectors: Q = (0, π) for the stripe phase, Q = (π/2, π/2) for the bicollinear double-stripe − − FIG. 7: (Color online) Magnetic phase diagram of the ef- phase, and Q = (π/2 δ, π/2 δ) for the IC phase. fective low-energy model (10) as a function of Fe excess The structure of the phase diagram (see Fig. 7) can y and the ratio between the ﬁrst- and second-neighbor bi- be summarized as follows. For small values of y, there quadratic exchanges, K1 , computed with (in meV/S2 units) is a thin strip of the single-stripe phase. This phase is − K2 Jeﬀ(y) = 3.4 + JRKKY(y), Jeﬀ(y) = 11.6 + JRKKY(y), stabilized by a strong J2eﬀ coupling (see Fig. 6 (b)) and J1eﬀ(y) = 15.1 + JRK1KY(y). The2RKKY interaction2s JRKKY quickly disappears because Jeﬀ decreases rapidly with 3 3 ij 2 are shown in Fig. . We set K2 = 3. increasing y. We believe that the stripe phase has not been observed in Fe Te because all known compounds 1+y belonging to this family are, actually, nonstoichiometric sistent with our phase diagram presented in Fig. 7 if and have a small amount of interstitial Fe signiﬁcantly (cid:39) one takes K1 0.1, which seems to be a realistic ratio, larger than the boundary value of y = 0.014 below which K2 since the magneto-elastic coupling favors the biquadratic the stripe phase is stable. exchange between second neighbors (see Appendix A for The central region in the phase diagram (y > 0.014) is more details). We emphasize that, because our model occupied by the bicollinear double-stripe state. This is contains assumptions about the bare values of J and ij the state which is experimentally observed in the Fe Te 1+y the amount of electrons introduced by each excess Fe, the compound with a low level of excess Fe. The stability of precise value of y for which the transition takes place is this phase over a wide range of parameters is explained beyond our scope. Yet, the general tendency of a double- by the smallness of the eﬀective nearest-neighbor cou- stripe to IC transition for increasing charge doping en- pling Jeﬀ and relatively strong strength of the third- 1 coded in the phase diagram of Fig. 7 is robust and con- neighbor coupling Jeﬀ. The biquadratic couplings K 3 ij sistent with the experimental observations. among which the dominating role is played by K and − 2 VII. CONCLUSIONS K = K (see Appendix A), also play an impor- diag 2 tant role in stabilizing this state: except in the region near y = 0.075, a ﬁnite value of the nearest-neighbor bi- In summary, we have studied the evolution of the mag- quadratic coupling K is necessary to stabilize the double netic order in Fe Te as function of y. Starting with a 1 1+y stripe phase over the incommensurate spiral state. The model containing both localized spins and itinerant elec- region near y = 0.075 is rather peculiar, as there the trons, we derived an eﬀective superexchange Hamiltonian eﬀective nearest neighbor coupling Jeﬀ is equal to zero to describe the magnetic properties of Fe Te which 1 1+y or is very small compared with the other interactions, contains both the long-range RKKY-type spin-spin in- making the bicollinear double stripe state the most sta- teraction mediated by the itinerant electrons and the ble one even in the absence of the biquadratic exchange. biquadratic interactions due to magneto-elastic eﬀects. The rest of the phase diagram is occupied by an IC phase Thus, y-dependent exchange interactions naturally arise (see, Fig. 8 (c)), which is the (q, q) spiral state experi- in our model due to changes in the low-energy itinerant mentally observed for suﬃciently large y. electronic states promoted by charge doping. In the experiment performed in Ref.11, the phase tran- After calculating the classical phase diagram of the sition from the bicollinear double stripe to the IC phase y-dependent eﬀective superexchange model, we showed ≈ was observed approximately for y 0.11. This is con- that Fe Te has a general tendency for a double-stripe 1+y ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: concentration

text: y,

text: which

text: aﬀects

text: indirectly

text: the

text: eﬀective

text: ex-

text: change

text: couplings

text: Jeﬀ

text: via

text: the

text: RKKY

text: interaction.

text: ij

text: In

text: Figs.

text: (a)-(c),

text: we

text: draw

text: the

text: real-space

text: spin

text: con-

text: ﬁgurations

text: presented

text: in

text: the

text: phase

text: diagram

text: of

text: Fig.

text: Among

text: all

text: possible

text: states,

text: in

text: the

text: parameter

text: space

text: pre-

text: sented

text: in

text: the

text: phase

text: diagram,

text: only

text: three

text: states

text: are

text: re-

text: alized:

text: the

text: stripe

text: phase

text: characterized

text: by

text: ϕ1

text: ϕ2

text: Double

text: Stripe

text: π,

text: ϕ

text: π,

text: q

text: q

text: shown

text: in

text: Fig.

text: (a),

text: the

text: dou-

text: x

text: y

text: ble

text: stripe

text: phase

text: characterized

text: by

text: ϕ1

text: ϕ2

text: ϕ3

text: π,

text: q

text: q

text: shown

text: in

text: Fig.

text: (b)

text: and

text: the

text: in-

text: x

text: y

text: commensurate

text: spiral

text: (labeled

text: as

text: IC)

text: state

text: shown

text: in

text: Fig.

text: (c)

text: characterized

text: by

text: ϕ

text: δ,

text: ϕ

text: π

text: ϕ

text: δ,

text: q

text: δ,

text: q

text: δ.

text: Note

text: that

text: in

text: Fig.

text: x

text: y

text: IC

text: Spiral

text: (c)

text: we

text: show

text: the

text: IC

text: state

text: with

text: an

text: exaggerated

text: canting

text: an-

text: gle.

text: To

text: clarify

text: the

text: structure

text: of

text: these

text: diﬀerent

text: orderings,

text: we

text: also

text: take

text: the

text: Fourier

text: transforms

text: of

text: the

text: ground

text: state

text: spin

text: conﬁguration

text: obtained

text: by

text: the

text: minimization

text: and

text: then

text: compute

text: the

text: corresponding

text: spin

text: structure

text: factor.

text: As

text: ex-

text: pected,

text: the

text: structure

text: factor

text: exhibits

text: peaks

text: at

text: the

text: fol-

text: y

text: lowing

text: ordering

text: wave-vectors:

text: Q

text: π)

text: for

text: the

text: stripe

text: phase,

text: Q

text: for

text: the

text: bicollinear

text: double-stripe

text: FIG.

text: (Color

text: online)

text: Magnetic

text: phase

text: diagram

text: of

text: the

text: ef-

text: phase,

text: and

text: Q

text: δ,

text: δ)

text: for

text: the

text: IC

text: phase.

text: fective

text: low-energy

text: model

text: as

text: a

text: function

text: of

text: Fe

text: excess

text: The

text: structure

text: of

text: the

text: phase

text: diagram

text: (see

text: Fig.

text: can

text: y

text: and

text: the

text: ratio

text: between

text: the

text: ﬁrst-

text: and

text: second-neighbor

text: bi-

text: be

text: summarized

text: as

text: follows.

text: For

text: small

text: values

text: of

text: y,

text: there

text: quadratic

text: exchanges,

text: computed

text: with

text: (in

text: units)

text: is

text: a

text: thin

text: strip

text: of

text: the

text: single-stripe

text: phase.

text: This

text: phase

text: is

text: Jeﬀ(y)

text: JRKKY(y),

text: Jeﬀ(y)

text: JRKKY(y),

text: stabilized

text: by

text: a

text: strong

text: coupling

text: (see

text: Fig.

text: (b))

text: and

text: JRKKY

text: quickly

text: disappears

text: because

text: Jeﬀ

text: decreases

text: rapidly

text: with

text: ij

text: are

text: shown

text: in

text: Fig.

text: We

text: set

text: increasing

text: y.

text: We

text: believe

text: that

text: the

text: stripe

text: phase

text: has

text: not

text: been

text: observed

text: in

text: Fe

text: Te

text: because

text: all

text: known

text: compounds

text: belonging

text: to

text: this

text: family

text: are,

text: actually,

text: nonstoichiometric

text: sistent

text: with

text: our

text: phase

text: diagram

text: presented

text: in

text: Fig.

text: if

text: and

text: have

text: a

text: small

text: amount

text: of

text: interstitial

text: Fe

text: signiﬁcantly

text: one

text: takes

text: which

text: seems

text: to

text: be

text: a

text: realistic

text: ratio,

text: larger

text: than

text: the

text: boundary

text: value

text: of

text: y

text: below

text: which

text: since

text: the

text: magneto-elastic

text: coupling

text: favors

text: the

text: biquadratic

text: the

text: stripe

text: phase

text: is

text: stable.

text: exchange

text: between

text: second

text: neighbors

text: (see

text: Appendix

text: A

text: for

text: The

text: central

text: region

text: in

text: the

text: phase

text: diagram

text: (y

text: is

text: more

text: details).

text: We

text: emphasize

text: that,

text: because

text: our

text: model

text: occupied

text: by

text: the

text: bicollinear

text: double-stripe

text: state.

text: This

text: is

text: contains

text: assumptions

text: about

text: the

text: bare

text: values

text: of

text: J

text: and

text: ij

text: the

text: state

text: which

text: is

text: experimentally

text: observed

text: in

text: the

text: Fe

text: Te

text: the

text: amount

text: of

text: electrons

text: introduced

text: by

text: each

text: excess

text: Fe,

text: the

text: compound

text: with

text: a

text: low

text: level

text: of

text: excess

text: Fe.

text: The

text: stability

text: of

text: precise

text: value

text: of

text: y

text: for

text: which

text: the

text: transition

text: takes

text: place

text: is

text: this

text: phase

text: over

text: a

text: wide

text: range

text: of

text: parameters

text: is

text: explained

text: beyond

text: our

text: scope.

text: Yet,

text: the

text: general

text: tendency

text: of

text: a

text: double-

text: by

text: the

text: smallness

text: of

text: the

text: eﬀective

text: nearest-neighbor

text: cou-

text: stripe

text: to

text: IC

text: transition

text: for

text: increasing

text: charge

text: doping

text: en-

text: pling

text: Jeﬀ

text: and

text: relatively

text: strong

text: strength

text: of

text: the

text: third-

text: coded

text: in

text: the

text: phase

text: diagram

text: of

text: Fig.

text: is

text: robust

text: and

text: con-

text: neighbor

text: coupling

text: Jeﬀ.

text: The

text: biquadratic

text: couplings

text: K

text: ij

text: sistent

text: with

text: the

text: experimental

text: observations.

text: among

text: which

text: the

text: dominating

text: role

text: is

text: played

text: by

text: K

text: and

text: VII.

text: CONCLUSIONS

text: K

text: K

text: (see

text: Appendix

text: A),

text: also

text: play

text: an

text: impor-

text: diag

text: tant

text: role

text: in

text: stabilizing

text: this

text: state:

text: except

text: in

text: the

text: region

text: near

text: y

text: a

text: ﬁnite

text: value

text: of

text: the

text: nearest-neighbor

text: bi-

text: In

text: summary,

text: we

text: have

text: studied

text: the

text: evolution

text: of

text: the

text: mag-

text: quadratic

text: coupling

text: K

text: is

text: necessary

text: to

text: stabilize

text: the

text: double

text: netic

text: order

text: in

text: Fe

text: Te

text: as

text: function

text: of

text: y.

text: Starting

text: with

text: a

text: stripe

text: phase

text: over

text: the

text: incommensurate

text: spiral

text: state.

text: The

text: model

text: containing

text: both

text: localized

text: spins

text: and

text: itinerant

text: elec-

text: region

text: near

text: y

text: is

text: rather

text: peculiar,

text: as

text: there

text: the

text: trons,

text: we

text: derived

text: an

text: eﬀective

text: superexchange

text: Hamiltonian

text: eﬀective

text: nearest

text: neighbor

text: coupling

text: Jeﬀ

text: is

text: equal

text: to

text: zero

text: to

text: describe

text: the

text: magnetic

text: properties

text: of

text: Fe

text: Te

text: which

text: or

text: is

text: very

text: small

text: compared

text: with

text: the

text: other

text: interactions,

text: contains

text: both

text: the

text: long-range

text: RKKY-type

text: spin-spin

text: in-

text: making

text: the

text: bicollinear

text: double

text: stripe

text: state

text: the

text: most

text: sta-

text: teraction

text: mediated

text: by

text: the

text: itinerant

text: electrons

text: and

text: the

text: ble

text: one

text: even

text: in

text: the

text: absence

text: of

text: the

text: biquadratic

text: exchange.

text: biquadratic

text: interactions

text: due

text: to

text: magneto-elastic

text: eﬀects.

text: The

text: rest

text: of

text: the

text: phase

text: diagram

text: is

text: occupied

text: by

text: an

text: IC

text: phase

text: Thus,

text: y-dependent

text: exchange

text: interactions

text: naturally

text: arise

text: (see,

text: Fig.

text: (c)),

text: which

text: is

text: the

text: (q,

text: q)

text: spiral

text: state

text: experi-

text: in

text: our

text: model

text: due

text: to

text: changes

text: in

text: the

text: low-energy

text: itinerant

text: mentally

text: observed

text: for

text: suﬃciently

text: large

text: y.

text: electronic

text: states

text: promoted

text: by

text: charge

text: doping.

text: In

text: the

text: experiment

text: performed

text: in

text: the

text: phase

text: tran-

text: After

text: calculating

text: the

text: classical

text: phase

text: diagram

text: of

text: the

text: sition

text: from

text: the

text: bicollinear

text: double

text: stripe

text: to

text: the

text: IC

text: phase

text: y-dependent

text: eﬀective

text: superexchange

text: model,

text: we

text: showed

text: was

text: observed

text: approximately

text: for

text: y

text: This

text: is

text: con-

text: that

text: Fe

text: Te

text: has

text: a

text: general

text: tendency

text: for

text: a

text: double-stripe

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