Spin-density fluctuations and the fluctuation-dissipation theorem in $3d$ ferromagnetic metals

Spatial and time scales of spin-density fluctuations (SDFs) were analyzed in $3d$ ferromagnets using ab initio linear-response calculations of complete wave-vector and energy dependence of the dynamic spin susceptibility tensor. We demonstrate that SDFs are spread continuously over the entire Brillouin zone and while the majority of them reside within the $3d$ bandwidth, a significant amount comes from much higher energies. A validity of the adiabatic approximation in spin dynamics is discussed. The SDF spectrum is shown to have two main constituents: a minor low-energy spin-wave contribution and a much larger high-energy component from more localized excitations. Using the fluctuation-dissipation theorem, the on-site spin correlator and the related effective fluctuating moment were properly evaluated and their universal dependence on the $3d$ band population is further discussed.

Figure 1

Small wave-vector transverse SDF for Fe, Co, and Ni calculated using method I. Top: transverse spectral function. The inset shows the low-energy part of the plot. Bottom: bare transverse spectral function. We used $\mathbf{q}=(0,0,0.125)2\pi /a$. Vertical axis units are ${\hslash }^2/\text{eV}$. Well-defined spin-wave excitations exist at low energies.

Figure 2

Small wave-vector longitudinal SDF for Fe, Co, and Ni calculated using method I. Top: longitudinal spectral function. Bottom: bare longitudinal spectral function. We used $\mathbf{q}=(0,0,0.125)2\pi /a$. Vertical axis units are ${\hslash }^2/\text{eV}$. Low-$\mathbf{q}$ longitudinal SDF are significantly smaller than the transverse one.

Figure 3

On-site transverse SDF spectrum for Fe, Co, Ni calculated using method I. Top: density of transverse SDF. The inset shows the low-energy part of the plot for Fe (the red curve denoted as “full”) compared with the partial-$\mathbf{q}$ density of transverse SDF [see Eq. (7), different curves are denoted by the value of the ${\mathrm{\Omega }}_{\mathbf{q}}/{\mathrm{\Omega }}_{\text{BZ}}$ ratio]. Bottom: bare density of transverse SDF. Vertical axis units are ${\hslash }^2/\text{eV}$. Transverse SDF in $3d$ ferromagnets show a generic two-peak structure.

Figure 4

On-site transverse SDF spectrum for Fe, Co, Ni calculated using method II. Top: density of transverse SDF. Bottom: bare density of transverse SDF. Vertical axis units are ${\hslash }^2/\text{eV}$. Spectra calculated using method II are in a good agreement with the results obtained using method I.

Figure 5

Local moment sum rule [see Eq. (5)] evaluated using method I. Red, blue, and green curves correspond to Fe, Co, and Ni, respectively. Horizontal dashed lines denote the LSDA value of the local moment. Full and dotted lines denote $m_{\text{a}}\left(\omega \right)$ and $m_{\text{a},0}\left(\omega \right)$, respectively. Vertical axis units are ${\mu }_B$. For all materials the sum rule is satisfied by including SDFs up to energy of the order of $W_{\text{el}}$.

Figure 6

Local moment sum rule [see Eq. (5)] evaluated using method II. Red, blue, and green curves correspond to Fe, Co and Ni, respectively. Horizontal dashed lines denote the LSDA value of the local moment. Full and dotted line denote $m_{\text{a}}\left(\omega \right)$ and $m_{\text{a},0}\left(\omega \right)$, respectively. Vertical axis units are ${\mu }_B$. method II produces results similar to method I even at high energies.

Figure 7

Energy dependence of the number of on-site transverse SDFs for Fe, Co, and Ni calculated using method I. Top: number of on-site transverse SDFs. Bottom: bare number of on-site transverse SDFs. Vertical axis units are ${\hslash }^2$. The inset shows the energy dependence of the adiabaticity parameter $\alpha$ defined as twice the ratio of $n^l\left(\omega \right)$ to $n^t\left(\omega \right)$.

Figure 8

The full effective fluctuating magnetic moment $\left(m_{\text{eff}}\right)$ calculated using method I as a function of the number of $3d$ electrons (red). Note that $m_{\text{eff}}$ does not contain contribution from the equilibrium local moment. The blue curve denotes the bare $m_{\text{eff}}$ evaluated using the Kohn-Sham susceptibility. The inset shows $m_{\text{eff}}$ as a function of $n_{3d}=\text{min}\left(n_{3d}^e,n_{3d}^h\right)$, where $n_{3d}^e$ and $n_{3d}^h$ is the number of $3d$ electrons and holes, respectively. The line in the inset is the linear fit of the data. The effective fluctuating moment is independent on the presence of local moment and is determined solely by the $3d$ band population.

Figure 9

Difference between the number and the bare number of transverse (top) and longitudinal (bottom) SDFs calculated using method I. Vertical axis units are ${\hslash }^2$. Transverse SDFs up to energy of the order of $W_{\text{el}}$ should be explicitly included in electronic structure calculations.