Bifurcations of Coupled Electron-Phonon Modes in an Antiferromagnet Subjected to a Magnetic Field

We report on a new effect caused by the electron-phonon coupling in a stoichiometric rare-earth antiferromagnetic crystal subjected to an external magnetic field, namely, the appearance of a nonzero gap in the spectrum of electronic excitations in an arbitrarily small field. The effect was registered in the low-temperature far-infrared (terahertz) reflection spectra of an easy-axis antiferromagnet ${\mathrm{PrFe}}_3{({\mathrm{BO}}_3)}_4$ in magnetic fields ${\mathbf{B}}_{\text{ext}}\parallel c$. Both paramagnetic and magnetically ordered phases (including a spin-flop one) were studied in magnetic fields up to 30 T, and two bifurcation points were observed. We show that the field behavior of the coupled modes can be successfully explained and modeled on the basis of the equation derived in the framework of the theory of coupled electron-phonon modes, with the same field-independent electron-phonon interaction constant $|W|=14.8{\mathrm{cm}}^{-1}$.

Figure 1

The $\pi$-polarized far-infrared reflection spectra of ${\mathrm{PrFe}}_3{({\mathrm{BO}}_3)}_4$ in external magnetic fields ${\mathbf{B}}_{\text{ext}}\parallel c$ up to 8 T and the corresponding reflection intensity maps [dark-brown (dark-green) color corresponds to the reflectance close to unity (zero)] as functions of the frequency and magnetic field at (a),(b) $T=1.5\mathrm{K}$ and (c),(d) $T=40\mathrm{K}>T_N=32\mathrm{K}$. The spin-flop transition in the field $B_{\mathrm{SF}}\approx 4.5\mathrm{T}$ is clearly observed in the spectra taken at 1.5 K. Blue solid lines in (b) show the field behavior of the phonon and two electronic branches [see Eq. (1)] in the absence of the electron-phonon coupling. Dotted lines represent the results of calculations according to Eqs. (1) and (2), with the following set of parameters: (a)–(d) $|W|=14.8{\mathrm{cm}}^{-1}$, ${\omega }_{\mathrm{ph}}=40{\mathrm{cm}}^{-1}$, $E=49{\mathrm{cm}}^{-1}$, $g_0⟨{\mathrm{\Gamma }}_1|J_Z|{\mathrm{\Gamma }}_2⟩=1.9$, and (a),(b) $B_{\mathrm{int}}=10.5\mathrm{T}$ for $B_{\mathrm{ext}}<B_{\mathrm{SF}}$ (for details, see the text).

Figure 2

Scheme illustrating the formation of an effective magnetic field at the two praseodymium subsystems in different regions of the $B\text{−}T$ phase diagram in the case ${\mathbf{B}}_{\text{ext}}\parallel c$. (a) AFM phase, $B_{\mathrm{ext}}=0$; (b) AFM phase, $B_{\mathrm{ext}}<B_{\mathrm{SF}}$; (c) spin-flop phase, $B_{\mathrm{ext}}\ge B_{\mathrm{SF}}$.

Figure 3

Graphical solution of Eq. (3). Roots of the equation are found as abscissas of the points where the straight line crosses branches (0), (1), and (2). The branch (2) disappears in the case $x_{\mathrm{el},1}=x_{\mathrm{el},2}$. $x_{\mathrm{el}}\equiv {{\omega }_{\mathrm{el}}}^2=E^2+4{{\mu }_B}^2{g_0}^2{|⟨{\mathrm{\Gamma }}_1|J_Z|{\mathrm{\Gamma }}_2⟩|}^2{B_{\mathrm{int}}}^2$. For other notations, see the text.

Figure 4

Unpolarized far-infrared reflection intensity maps for ${\mathrm{PrFe}}_3{({\mathrm{BO}}_3)}_4$ in external fields ${\mathbf{B}}_{\text{ext}}\parallel c$ up to 30 T, at $T=1.5\mathrm{K}$. Dotted lines represent results of calculations according to Eqs. (1) and (2), with the following set of parameters: $|W|=14.8{\mathrm{cm}}^{-1}$, ${\omega }_{\mathrm{ph}}=40{\mathrm{cm}}^{-1}$, $E=49{\mathrm{cm}}^{-1}$, and $g_0$$|⟨{\mathrm{\Gamma }}_1|J_z|{\mathrm{\Gamma }}_2⟩|=1.9$.