Spin-wave excitations in the SDW state of doped iron pnictides

TL;DR

This paper studies how spin-wave excitations in doped iron pnictides' SDW state vary with doping, revealing sensitivity patterns, anisotropy changes, and spectral weight shifts consistent with neutron scattering experiments.

Contribution

It provides a detailed analysis of doping-dependent spin-wave excitations in a five-orbital model, highlighting new anisotropy and spectral weight behaviors.

Findings

Excitations along ($, 0$) are highly doping-sensitive.

Anisotropy around ($, 0$) grows with hole doping at low energy.

Spectral weight shifts to low energy with increased doping.

Abstract

We investigate the spin-wave excitations in the spin-density wave state of doped iron pnictides within a five-orbital model. We find that the excitations along ($\pi, 0$)$\rightarrow$($\pi, \pi$) are very sensitive to the doping whereas they do not exhibit a similar sensitivity along ($0, 0$) $\rightarrow$ ($\pi, 0$). Secondly, anisotropy in the excitations around ($\pi, 0$) with an elliptical shape grows on moving towards the hole-doped region for low energy, whereas it decreases for the high-energy excitations on the contrary. Thirdly, spin-wave spectral weight shifts towards the low-energy region on moving away from zero doping. We find these features to be in qualitative agreement with the inelastic neutron-scattering measurements for the doped pnictides.

Equations

$${\hat{\mathcal{H}}}_{\sigma \mathbf{k}}=\sum _{\mathbf{k}\sigma }{\Psi }_{\mathbf{k}\sigma }^{†}\left(\begin{array}{cc}{\hat{\epsilon }}_{\mathbf{k}}+\hat{N}& \mathrm{sgn}\bar{\sigma }\hat{\Delta }\\ \mathrm{sgn}\bar{\sigma }\hat{\Delta }& {\hat{\epsilon }}_{\mathbf{k}+\mathbf{Q}}+\hat{N}\end{array}\right){\Psi }_{\mathbf{k}\sigma }.$$

$${\hat{\bar{\chi }}}_{\mathrm{RPA}}(\mathbf{q},i{\omega }_n)=(\hat{1}-\hat{U}{)}^{-1}\hat{\bar{\chi }}(\mathbf{q},i{\omega }_n),$$

$$\hat{\bar{\chi }}(\mathbf{q},i{\omega }_n)=\left(\begin{array}{cc}\hat{\bar{\chi }}(\mathbf{q},\mathbf{q},i{\omega }_n)& \hat{\bar{\chi }}(\mathbf{q},\mathbf{q}+\mathbf{Q},i{\omega }_n)\\ \hat{\bar{\chi }}(\mathbf{q}+\mathbf{Q},\mathbf{q},i{\omega }_n)& \hat{\bar{\chi }}(\mathbf{q}+\mathbf{Q},\mathbf{q}+\mathbf{Q},i{\omega }_n)\end{array}\right),$$

$${\bar{\chi }}_{\alpha \beta ,\mu \nu }={\chi }_{\alpha \beta ,\mu \nu }^{+-}+{\chi }_{\bar{\alpha }\beta ,\bar{\mu }\nu }^{+-}+{\chi }_{\alpha \bar{\beta },\mu \bar{\nu }}^{+-}+{\chi }_{\bar{\alpha }\bar{\beta },\bar{\mu }\bar{\nu }}^{+-},$$

$${\bar{\chi }}^{ps}(\mathbf{q},i{\omega }_n)=\sum _{\alpha \mu }{\bar{\chi }}_{\alpha \alpha ,\mu \mu }(\mathbf{q},\mathbf{q},i{\omega }_n).$$

$${\chi }_{\alpha \beta ,\mu \nu }^{+-}(\mathbf{q},{\mathbf{q}}^{\prime },i{\omega }_n)$$

$${\mathcal{S}}_{\mathbf{q}}^i=\sum _{\mathbf{k}}\sum _{\sigma {\sigma }^{\prime }}\sum _{\mu {\mu }^{\prime }}{d}_{\mu \sigma }^{†}(\mathbf{k}+\mathbf{q})E_{\mu {\mu }^{\prime }}{\sigma }_{\sigma {\sigma }^{\prime }}^i{d}_{{\mu }^{\prime }{\sigma }^{\prime }}(\mathbf{k}).$$

$${\chi }_{\alpha \beta ,\mu \nu }^{+-}(\mathbf{q},\mathbf{q},i{\omega }_n)$$

$${\mathcal{H}}_0=\sum _{\mathbf{k}}\sum _{\mu ,\nu }\sum _{\sigma }{\epsilon }_{\mathbf{k}}^{\mu \nu }{d}_{\mathbf{k}\mu \sigma }^{†}{d}_{\mathbf{k}\nu \sigma }+\text{H.c.},$$

$${\mathcal{H}}_{int}$$

$$2{\Delta }_{\mu \mu }$$

$$2{\Delta }_{\mu \nu }$$

$$2N_{\mu \mu }$$

$$2N_{\mu \nu }$$

$$n_{\mu \nu }=\sum _{\mathbf{k}\sigma }⟨c_{\mathbf{k}\mu \sigma }^{†}{c}_{\mathbf{k}\nu \sigma }⟩,m_{\mu \nu }=\sum _{\mathbf{k}\sigma }⟨c_{\mathbf{k}\bar{\mu }\sigma }^{†}{c}_{\mathbf{k}\nu \sigma }⟩.$$

Full text

I Introduction

Iron pnictides exhibit a very rich temperature-doping phase diagram, where doping of either electrons or holes suppresses the long-range collinear magnetic order giving way to the sign-changing $s$-wave superconductivity (SC). Magnetic order in some of these materials is stabilized over a range of dopings $x_{e}\lesssim$ 0.06 and $x_{h}\lesssim$ 0.3 in the electron- and hole-doped regions, respectively.dai ; avci Despite the competing SC and magnetic long-range order on further doping, superconducting state retains the spin fluctuations responsible for the pairing in a manner similar to that in the high-$T_{c}$ cuprates.wang ; terashimaa Experimentally, the role of spin fluctuations in mediating SC can be probed with the help of the inelastic neutron scattering (INS) a powerful experimental tool.dai2

Origin of magnetic order in these materials lies in the Fermi surface (FS) instability because of a good nesting present between the Fermi pockets. According to the angle-resolved photoemission spectroscopy (ARPES) as well as the band structure calculations, FSs consist of concentric hole pockets around $\Gamma$ and elliptical electron pocket at X.mazin ; singh ; haule ; yi ; kondo ; yi1 ; brouet ; kordyuk Nesting between these two sets of pockets leads to the ($\pi$, 0) SDW state or collinear magnetic order. When electrons or holes are doped, FSs can be modified in a significant manner thereby altering the nature of nesting, and that can have a significant impact on the nature of the SDW state as well as on the spin fluctuations responsible for the SC.wang1

A remarkably high-energy scale of the excitations have been observed using INS in the SDW state of the parent compound, which are highly dispersive as well as sharp.ewings ; diallo ; zhao1 ; harriger ; ewings1 They can extend up to $\sim$200 meV corresponding to the zone-boundary modes ${\bf q}=(\pi,\pi)$. There exists an in-plane anisotropy,zhao1 which persists even in the nematic phase.lu Excitations have also been studied extensively for various dopings though in the superconducting state. For instance, the spin excitation is manifested as a resonance in the superconducting state maier ; zhang because of it’s dependence on the BCS coherence factors for the wavevector equal to the nesting vector and the opposite signs of the superconducting gap on the electron and hole pockets. Spin-wave dispersion similar to that of parent compound BaFe2As2 has been observed along the high-symmetry directions for the superconducting Ba0.67K0.33Fe2As2 with hole doping $x\approx 0.33$ though with a significant zone-boundary softening.wang1 ; horigane At the same time, magnetic-exchange coupling is reduced by $10\%$. High-energy spin excitations are suppressed and magnetic spectral weight is shifted to low energies. The excitations are gapped below $\sim$ 50 meV for the electron doped BaFe1.7Ni0.3As2 while those of high-energy are largely unaffected. It has been further suggested that both the low- and high-energy excitations may be associated with the superconductivity.wang1

For the SDW state, various theoretical and experimental studies have focused largely on the spin excitations in the parent compound. Two different types of itinerant models excitonicbrydon and orbitalkaneshita ; nimisha ; knolle ; kovacic have often been employed to understand the spin-wave excitations as well their damping. The description of various characteristics of excitations has been challenging in view of the fact that the observed magnetic moments are small while the excitations are sharp and dispersing upto 200meV. Nonetheless, several characteristics have been captured within the five-orbital models.kaneshita ; kovacic In particular, it has been shown recently that a large Hund’s coupling plays an essential role in describing different features such as sharpness, anisotropy around X, and spin-wave spectral function.dheeraj Another important factor that has a significant impact on the aforementioned features is the doping induced modification in the bandstructure, which has not attracted much attention.

In this paper, we examine various aspects of the spin-wave excitations in the ($\pi,0$) SDW state of doped iron pnictides within the doping range $-0.4\lesssim x\lesssim 0.05$. For this, we consider the five-orbital tight-binding model of Ikeda et al.ikeda The model with a rigid bandshift is known to exhibit ($\pi,0$) SDW state within the doping range $0\lesssim x_{h}\lesssim 0.18$ for intraorbital Coulomb interaction $U\sim 0.9$eV,schmiedt and the range can be expected to increase for relatively larger interaction parameters. The reconstructed FSs agrees qualitatively with the ARPES measurements.yi2 The quasiparticle interference obtained with the reconstructed bands within the model has reproduced several features of the local density of state modulation in the doped state.chuang ; dheeraj2 We use this model to highlight three important consequences of doping on the spin-wave excitations. (i) The excitations along ($\pi,0$)$\rightarrow$($\pi,\pi$) are very sensitive to the doping and a similar sensitivity is absent along (0,0)$$\rightarrow($\pi,0$). (ii) Anisotropy around ($\pi,0$) in the form of elliptical structure increases on moving towards the hole-doped region for the low-energy excitations, whereas it decreases for high-energy excitations on the contrary. (iii) Spin-wave spectral weight shifts towards the low-energy region on moving away from zero doping.

II Transverse-spin fluctuations

In order to investigate the doping dependence of spin-wave excitations, we consider the following mean-field Hamiltonian in the ($\pi,0$) SDW state

[TABLE]

Here, $\Psi^{\dagger}_{{\bf k}\sigma}=(d^{\dagger}_{{{\bf k}}1\sigma},....,d^{\dagger}_{{{\bf k}}5\sigma},{d}^{\dagger}_{{{\bf k}}\bar{1}\sigma},....,{d}^{\dagger}_{{\bf k}\bar{5}\sigma})$ with $\bar{d}^{\dagger}_{{{\bf k}}\bar{\mu}\sigma}$ = $d^{\dagger}_{{{\bf k}+{\bf Q}}\mu\sigma}$. $d^{\dagger}_{{{\bf k}}i\sigma}$ ($d_{{{\bf k}}1\sigma}$) is the electron creation (destruction) operator for the momentum ${{\bf k}}$ in the orbital $i$ with spin $\sigma$. $\hat{\varepsilon}_{{\bf k}}$ is the hopping matrix corresponding to the five-orbital model. The elements of the matrices $\hat{\Delta}$ and $\hat{N}$ as described in the Appendix are dependent on the onsite interaction parameters, orbital magnetizations, and charge densities. The Hamiltonian matrix is diagonalized and various order parameters are obtained in a self-consistent manner using the eigenvalues and eigenvectors. Then, the eigenvalues and eigenvectors corresponding to the self-consistent SDW state are used to calculate the spin-wave excitations.

The transverse-spin susceptibility within the random-phase approximation can be obtained as

[TABLE]

where $\hat{{\bf 1}}$ is a $2n^{2}\times 2n^{2}$ identity matrix with $n=5$. The elements of block-diagonal matrix ${U}_{\mu_{1}\mu_{2};\mu_{3}\mu_{4}}$ = $U,\,U-2J,\,J$ and $J$ for $\mu_{1}=\mu_{2}=\mu_{3}=\mu_{4},\,\mu_{1}=\mu_{2}\neq\mu_{3}=\mu_{4},\,\mu_{1}=\mu_{2}\neq\mu_{3}=\mu_{4}$ and $\mu_{1}=\mu_{4}\neq\mu_{2}=\mu_{3}$, respectively. They vanish otherwise. $U$ and $J$ are the intraorbital and interorbital Coulomb interactions. The bare-level susceptibility matrix is

[TABLE]

where the elements in the ordered state is given by

[TABLE]

where Umklapp processes are included. Physical transverse-spin susceptibility corresponding to the spin operators to be defined below is

[TABLE]

The transverse-spin susceptibility is defined as

[TABLE]

where components of the spin operator are

[TABLE]

$E$ is a 5$\times$5 identity matrix belonging to the orbital bases. Thus the susceptibility when ${\bf q}^{\prime}={\bf q}$ is given in terms of Green’s function as

[TABLE]

In the following, analytic continuation $i\omega_{n}\rightarrow\omega+i\eta$ with $\eta=2$meV is used throughout. Interaction parameters $U$ and $J$ are set to be 1.1eV and 0.25$U$ so as to obtain magnetic moment $m\sim 1$ for zero doping motivated by the observed magnetic moments in 122 series of pnictides.huang Unit of energy is set to be eV unless stated otherwise.

III results

FSs obtained in the unordered state for various dopings (a) $n=5.6$, (b) $n=5.8$, (c) $n=6.0$, and (d) $n=6.05$ are shown in Fig. 1. As can be seen, the nesting deteriorates fast on moving away from zero doping because of the electron and hole pockets shrink and expand on moving from the electron-doped to hole-doped region, respectively. Fig. 2 shows the reconstructed FSs in the SDW state for the respective dopings. It is evident that the reconstructed structure and topology of the FSs are very sensitive to the doping when compared to the unordered state.

Fig. 3 shows Im$\bar{\chi}^{sp}_{\rm RPA}$ obtained in the SDW state for various dopings (a) 5.6, (b) 5.7, (c) 5.8 ,(d) 5.9, (e) 6.0, and (f) 6.05. Note that the red color also represents those value of Im$\bar{\chi}^{sp}_{\rm RPA}$ that exceed 200. The excitations are heavily damped throughout particularly along $\Gamma$-X-M ((0, 0)$\rightarrow$($\pi,0$)$\rightarrow$($\pi,\pi$)) for the electron doping $n=6.05$, which is not surprising because there is a fast reduction in the net magnetization, and hence of the magnetic-exchange gap on electron doping. That is reflected in the particle-hole continuum extending down to low energy in the bare susceptibility (Fig. 4). The nesting between the electron pocket and the hole pocket in the unordered state is optimal in the vicinity of $n\approx 6.0$. Therefore, moving away from this band filling is expected to lead to a reduced magnetic-exchange gap according to the nesting-based scenario though the reduction may not necessarily be symmetrical about zero doping.

Spin-wave damping reduces quickly on moving towards the hole-doped region and the excitations become optimally sharp and well-defined for $n\approx 5.9$ in the hole-doped region. In an earlier work, maximum $T_{c}$ for the SDW state has also been shown to occur within the model for small hole doping. On doping holes further, the spin-wave excitations get softened rapidly along X-M especially in the region close to M. Meanwhile, damping increases so that the excitations disappear finally for $n\approx 5.75$ in a large part of X-M. However, they remain largely unaffected along $\Gamma$-X.

Magnetic moment grows on moving from the electron-doped region to the hole-doped region as the system approaches half filling (Fig. 5). This is expected to suppress the density of states at the Fermi level and the electron movement in the lattice leading to the departure from metallicity.medici ; lafuerza Despite that the FSs exist for the all the dopings considered here because of a small ratio $U$/$W\approx 0.25$ with $W$ as the electron bandwidth, and consequently reconstructed bandstructure plays a crucial role in the SDW excitations. For instance, the particle-hole continuum in the bare-spin susceptibility shifts towards low energy near M, which should play a very important role in the damping and disappearance of the spin-wave excitations along X-M in the hole-doped region.

Figs. 6, 7 and 8 show the constant energy cuts for the bandfilling $n=5.7,5.9\,\,{\rm and}\,\,6.05$, respectively. It is noted that the anisotropy in the spin-wave excitations around X is sensitive to both energy as well as doping. It diminishes on increasing the energy whereas grows on moving from the electron-doped region to the hole-doped region for the low-energy excitations $\sim$ 50meV. On the other hand, it decreases for the high-energy excitations $\gtrsim$ 100meV. The anisotropic behavior around X at low energy as well as square like shape of excitations around M at higher energy particularly when $n\approx 5.9$ is qualitatively similar to what is observed in the INS experiments.wang1 ; dai2

Fig. 9 shows the doping dependence of the spin-wave spectral function. There are two separate peaks for $n=6.05$, one positioned near very small energy and other one around 200meV. At zero doping, the peak near 200meV gets prominent. Whereas the position of the peak shifts rapidly towards the low-energy region on hole doping. A similar observation has been made in the paramagnetic and superconducting phases of doped pnictides. Moreover, the location of peak in the spin-wave spectral function for the optimal doping is near 200meV, which is also in accordance with the experiments.wang1

IV Conclusions and Discussions

In conclusions, we have investigated the spin-wave excitations in the SDW state of doped iron pnictides. We use a five-orbital model with realistic electronic structure and fixed set of interaction parameters corresponding to the magnetic moment $1\mu_{B}$ in the undoped case. We find that the excitations along ($\pi,0$)$\rightarrow$($\pi,\pi$) are very sharp and dispersive in a very small doping range centered around $n\approx 5.9$ lying in the hole-doped region and they get damped heavily on moving away from that doping on either side. Unlike along ($\pi,0$)$\rightarrow$($\pi,\pi$), the excitations along (0,0)$$\rightarrow($\pi,0$) do not show much variation with doping except in the electron-doped region, where they are damped heavily. Further, we find that the anisotropy around X decreases at high energy for any doping. Whereas it increases and decreases on hole doping for low- and high-energy excitations, respectively. We also find that the spin-wave spectral weight shifts towards lower energy on doping either holes or electrons, whereas it is peaked near 200meV for the undoped case.

In several doped pnictides, there is a phase transition from the high-temperature SDW state to the low-temperature superconducting state within a certain doping range. Some of the characteristics of the spin excitations such as anisotropy around ($\pi,0$) are expected to be retained across the phase transition in a manner similar to the phase transition from the paramagnetic to SDW state. We find several features such as anisotropy around ($\pi,0$), softening of zone-boundary modes, and shifting of the spectral weight on doping to the low-energy region being in qualitatively similar to those measured in the INS for the paramagnetic and superconducting phases.

We acknowledge the use of HPC clusters at HRI.

Appendix

The kinetic part of the model Hamiltonian that we consider is given by

[TABLE]

where $d_{{\bf k}\mu\sigma}^{\dagger}$ ($d_{{\bf k}\mu\sigma}$) is the electron creation (destruction) operators and $\varepsilon_{{\bf k}}^{\mu\nu}$ are the hopping elements from orbital $\mu$ to $\nu$ for the momentum ${\bf k}$.

The interaction term is given by

[TABLE]

It has the intra- and inter-orbital Coulomb interaction terms as first and second terms, respectively. The third and fourth term represents the Hund’s coupling and the pair hopping. The Hamiltonian possesses only two independent interaction parameters due to the rotational invariance condition $U=U^{\prime}-2J$.

Matrix elements of matrices $\hat{\Delta}$ and $\hat{N}$ in Eq. (1) are

[TABLE]

and

[TABLE]

where charge densities and magnetizations are given by

[TABLE]