Counting discrete emission steps from intrinsic localized modes in a quasi-one-dimensional antiferromagnetic lattice

Intrinsic localized modes (ILMs) in a quasi-1D antiferromagnetic material ${\left({\mathrm{C}}_2{\mathrm{H}}_5\mathrm{N}{\mathrm{H}}_3\right)}_2\mathrm{Cu}{\mathrm{Cl}}_4$ are counted by using a novel nonlinear energy magnetometer. The ILMs are produced by driving the uniform spin wave mode unstable with an intense microwave pulse. Subsequently a subset of these ILMs become captured by and locked to a cw driver so that their properties can be examined at a later time with a tunable cw low power probe source. Four-wave mixing is used to enhance the emission signal from the few large amplitude ILMs over that associated with the many small amplitude plane wave modes. A discrete step structure observed in the emission signal is identified with individual ILMs becoming unlocked from the driver. At most driver power and frequency settings the resulting emission step structure appears uniformly distributed; however, sometimes, nearby in parameter space, families of emission steps are evident as the driver frequency or power is varied. Two different experimental methods give consistent results for counting individual ILMs. Because of the discreteness in the emission both the size of an ILM and its energy can be estimated from these experiments. For the uniformly distributed case each ILM extends over $\sim 42$ antiferromagnetic unit cells and has an energy value of $1.3\times {10}^{-12}\mathrm{J}$ while for the case with families the ILM length becomes $\sim 54$ antiferromagnetic unit cells with an energy of $1.5\times {10}^{-12}\mathrm{J}$. An unresolved puzzle is that the emission step height does not depend on experimental parameters the way classical numerical simulations suggest.

Figure 1

Lattice and spin structure of ${\left({\mathrm{C}}_2{\mathrm{H}}_5\mathrm{N}{\mathrm{H}}_3\right)}_2\mathrm{Cu}{\mathrm{Cl}}_4$. Circles denote ${\mathrm{Cu}}^{2+}$ ions and arrows indicate spin configuration in the antiferromagnetic state. Only ${\mathrm{Cu}}^{2+}$ ions are shown in this layered, face centered, orthorhombic compound. The easy, second easy, and hard spin axes are labeled the $a$, $b$, and $c$ crystal directions, respectively.

Figure 2

Spin-wave dispersion curve of the antiferromagnet ${\left({\mathrm{C}}_2{\mathrm{H}}_5\mathrm{N}{\mathrm{H}}_3\right)}_2\mathrm{Cu}{\mathrm{Cl}}_4$. (a) Upper and lower branches along the $c$ axis. The inset shows the uniform mode spin motion for the lower AFMR mode, which has a net ac magnetization only along the $c$ axis. The stored energy density for this excitation is described in Appendix . The axes are identified in Fig. 1. (b) Expanded view of the lower branch, near the zone center with dispersion curves now along all crystal axes. The AFMR frequencies for different sample shapes are indicated by the solid dots.

Figure 3

Frequency diagram for the four-wave mixing experiment. Vertical arrows represent transverse ac magnetizations. The $f_2$ driver and $f_3$ probe produce transverse magnetizations $M_2$ and $M_3$ inside the sample. The sample then generates the new magnetization at frequency $(2f_2-f_3)$ by the third-order nonlinearity, as indicated by the downward arrow. The microwave signal from this magnetization is observed in emission.

Figure 4

Experimental setup for the ILM locked state measurement. Three microwave sources are used: a high power pulse pump source for the initial excitation $\left(f_1\right)$, a source for locking $\left(f_2\right)$, which is followed by a switch (SW) and a middle power amplifier (AMP), and a low power probe source $\left(f_3\right)$. The high power pulse microwave driver $f_1$ excites the sample, which is immersed in $1.2\mathrm{K}$ liquid helium. The middle power $f_2$ cw driver is employed to lock ILMs. Here one branch of the microwave signal goes to the sample via the directional coupler. Since the reflected $f_2$ signal is often larger than the maximum linear input of the spectrum analyzer, the other branch of the $f_2$ driver is fed into the spectrum analyzer to cancel it, thus avoiding a spurious mixing signal inside the spectrum analyzer. The $f_3$ probe is used both in absorption and in emission measurements.

Figure 5

Time-dependent absorption spectrum showing the break up of the uniform mode induced by a strong microwave pulse at frequency $f_1$. Here $f_1=1.29\mathrm{GHz}$, the input power at the cryostat is $52\mathrm{W}$, and the pulse length is $3\mu \mathrm{s}$. The power of the cw driver $f_2$ (dotted line) is typically 1000 times smaller than $f_1$.

Figure 6

Time-dependent emission signal showing both locking and releasing of ILMs. (a) Single on-off-on driver sequence. (b) Double on-off-on sequence. The square waves on the lower part of panels show the on-off pattern of the $f_2$ driver. The horizontal dotted line indicates the frequency position of the $f_2$ driver. Shortly after the $f_1$ pump is pulsed at $t=20\mu \mathrm{s}$, the emission signal appears. The locked ILMs can be maintained as long as $f_2$ is on. Once turned off, most of ILMs are unlocked and no longer emit coherently. When $f_2$ is again turned on, only some ILMs are relocked.

Figure 7

At $2\mathrm{ms}$ after the $f_1$ pulse a snapshot of the simultaneous emission (solid curve) and absorption (dotted curve) spectra. For this time scale, the frequency resolution is $\sim 100\mathrm{kHz}$, and the raw emission spectra are shown. (a) $f_2=1.34\mathrm{GHz}$ at a power of $51\mathrm{mW}$. (b) $f_2=1.32\mathrm{GHz}$ at a power of $240\mathrm{mW}$. Note the AFMR frequency is pulled slightly to lower frequencies by $f_2$ at this power level. In case (b) the small number of locked ILMs is not apparent in absorption. A number of features are seen in emission. The strongest and the second strongest peaks on either side of the driver are associated with the locked ILMs, resulting in a sideband pair in both figures. The third strongest peak at the AFMR frequency is emission from the uniform mode. The other half of the sideband pair appears as a small shoulder in (a) and as a small peak in (b).

Figure 8

Complex time-dependent emission output for a near cubic shaped sample. Square root of the time dependent emission output as a function of time for different values of the cw $f_2$ power level. The 2.3% increments between curves vary the $f_2$ power from $32.4\text{to}81.3\mathrm{mW}$. The driver frequency $f_2$ is fixed at $1.35\mathrm{GHz}$. The AFMR frequency of this sample is $1.395\mathrm{GHz}$ somewhat higher than that of rod shaped samples $\left(1.375\mathrm{GHz}\right)$. Although some emission plateaus can be identified, the overall structures are very complex.

Figure 9

Square root of the time-dependent emission output versus time for a $c$-axis directed thin plate sample. (a) For fixed driving frequency $f_2=1.33\mathrm{GHz}$ as a function of the cw $f_2$ power level from $52.5\text{to}105\mathrm{mW}$. (b) For fixed $f_2$ power level $77.6\mathrm{mW}$, as a function of its frequency from $1.325\text{to}1.335\mathrm{GHz}$. (c) For fixed frequency $f_2=1.335\mathrm{GHz}$ as a function of the cw power level from $34.7\text{to}87.1\mathrm{mW}$. (d) For fixed $f_2$ power level of $55\mathrm{mW}$ as a function of its frequency from $1.33\text{to}1.34\mathrm{GHz}$. The power increment between curves is 2.3%. The frequency interval between curves is $250\mathrm{KHz}$.

Figure 10

Characterizing the time dependence of the four-wave emission signals. The dotted traces in Figs. 9a, 9b are singled out for analysis. (a) Dotted curve in Fig. 9a for conditions $f_2=1.33\mathrm{GHz}$ at $83.2\mathrm{mW}$; time constant of the dotted $\text{curve}=1.18\mathrm{ms}$, step $\text{height}=0.26{\mathrm{nW}}^{1∕2}$. (b) Dotted curve in Fig. 9b for conditions $f_2=1.33123\mathrm{GHz}$ at $77.6\mathrm{mW}$, time constant of the dotted $\text{curve}=1.18\mathrm{ms}$, step $\text{height}=0.26{\mathrm{nW}}^{1∕2}$. In both frames the steps are superimposed on an exponential time-dependent background.

Figure 11

Emission decay curve dependence on the $f_2$ driver delay time. After $\sim 3\times {10}^6$ $f_2$ periods the driver is turned off and after a brief delay again turned on. Solid curve, driver unchanged. The dotted, and dot-dashed curves are for 10 and $50\mu \mathrm{s}$ delay times. Unlocked ILMs lose phase coherence, shift in frequency so their emission signal quickly disappears. When the driver is turned on again, a subset of the ILMs oscillating around the driver frequency are relocked. Different ILM states can be produced without changing the $f_2$ driver frequency or its power.

Figure 12

(a), (b) Square root of the emission power observed at $t=4\mathrm{ms}$ as a function of the driver delay time at $2\mathrm{ms}$ for the $c$-axis directed plate sample. The $f_2$ driver frequency and power for the solid curves are (a) $1.335\mathrm{GHz}$, $93.3\mathrm{mW}$; (b) $1.32\mathrm{GHz}$, $191\mathrm{mW}$. The distance between the horizontal dotted lines is $0.26{\mathrm{nW}}^{1∕2}$. The dot-dash curves in (a) and (b) are for slightly higher power: (a) $102\mathrm{mW}$ and (b) $200\mathrm{mW}$. (c) Results for a $c$-axis directed rod sample measured at $1.31\mathrm{GHz}$, $97.7\mathrm{mW}$. The distance between these horizontal dotted lines is $0.18{\mathrm{nW}}^{1∕2}$. The dot-dash curve is for the higher power, $110\mathrm{mW}$.

Figure 13

(a) Illustration of the classical anharmonic oscillator resonance response curve for soft anharmonicity. The frequency axis is measured with respect to the higher AFMR frequency so that as $\Delta f=f_{\mathrm{AFMR}}-f_l$ increases a single step can occur. (b) The proposed step emission pattern can occur when the locked ILM frequency depends weakly on the number of ILMs in the 1D lattice.

Figure 14

Step height as a function of the normalized frequency difference between the AFMR and the $f_2$ driver as determined from numerical simulations. The sum of the square of the transverse component (SSTC) for an ILM eigenvector (proportional to the square root of the emission) versus the normalized frequency gap is shown. The region between the two vertical arrows corresponds to the range of the $f_2$ frequency scanning experiment presented in Fig. 9b. These simulations predict a 12% increase in step height for this fractional increase while the experiment shows less than a 4% change.