Far-from-equilibrium growth of magnetic thin films with Blume-Capel impurities

We investigate the irreversible growth of $(2+1)$-dimensional magnetic thin films. The spin variable can adopt three states $\left(s_i=\pm 1,0\right)$, and the system is in contact with a thermal bath of temperature $T$. The deposition process depends on the change of the configuration energy, which, by analogy to the Blume-Capel Hamiltonian in equilibrium systems, depends on Ising-like couplings between neighboring spins $\left(J\right)$ and has a crystal field $\left(D\right)$ term that controls the density of nonmagnetic impurities $\left(s_i=0\right)$. Once deposited, particles are not allowed to flip, diffuse, or detach. By means of extensive Monte Carlo simulations, we obtain the phase diagram in the crystal field vs temperature parameter space. We show clear evidence of the existence of a tricritical point located at $D_t/J=1.145\left(10\right)$ and $k_B{T}_t/J=0.425\left(10\right)$, which separates a first-order transition curve at lower temperatures from a critical second-order transition curve at higher temperatures, in analogy with the previously studied equilibrium Blume-Capel model. Furthermore, we show that, along the second-order transition curve, the critical behavior of the irreversible growth model can be described by means of the critical exponents of the two-dimensional Ising model under equilibrium conditions. Therefore, our findings provide a link between well-known theoretical equilibrium models and nonequilibrium growth processes that are of great interest for many experimental applications, as well as a paradigmatic topic of study in current statistical physics.

Figure 1

(a) Average absolute value of the magnetization $\langle \left|m\right|\rangle$, (b) susceptibility $\chi$, and (c) Binder cumulant $U$ versus temperature, corresponding to different choices of the crystal field $D$, as indicated. Data obtained by using samples of linear size $L=12$. The temperature and the crystal field $D$ are measured in units of $J/k_B$ and $J$, respectively.

Figure 2

(a) Average absolute value of the magnetization $\langle \left|m\right|\rangle$, (b) susceptibility $\chi$, and (c) Binder cumulant $U$ versus crystal field $D$. (d) shows in detail the behavior of the cumulant within the relevant temperature interval, namely $0.40\le k_{B}T/J\le 0.50$. Data obtained for different choices of the temperature $T$ and by using samples of linear size $L=12$.

Figure 3

(a) Effective phase diagram of the MEM-BC model obtained by using samples of linear size $L=12$. Circles and squares indicate lines of second- and first-order effective transitions, respectively, which meet each other at the effective tricritical point marked with a star. (b) Characteristic snapshots representing growth regimes dominated by different mixtures of impurities in red (gray), up spins in black and down spins in green (light gray).

Figure 4

Probability distributions of the magnetization $P_L\left(m\right)$ [(a), (c)] and the density of impurities $P_L\left(\mathrm{impurities}\right)$ for different values of $T$ and $D$. (a)–(b) For $D/J=0.8$, the effective second-order transition occurs at $k_B{T}_c^{\mathrm{eff}}/J=0.60\left(1\right)$. (c)–(d) For $k_{B}T/J=0.3$, the effective first-order transition occurs at $D_c^{\mathrm{eff}}/J=1.27\left(1\right)$. Data obtained by using samples of linear size $L=12$.

Figure 5

(a) Average density of impurities, (b) fluctuations in the impurity density (i.e., susceptibility for impurities, ${\chi }_{\mathrm{impurity}})$, and (c) Binder cumulant of impurities, $U_{\mathrm{impurity}}$, as functions of the crystal field, $D$. Data obtained for different values of the temperature $T$ and by using samples of linear size $L=12$.

Figure 6

(a) Average absolute value of the magnetization, $\langle \left|m\right|\rangle$, (b) susceptibility, $k_{B}T\chi$, and (c) Binder cumulant, $U$, versus the temperature, obtained for $D=0$ and samples of different size, as indicated. The inset in (b) shows the extrapolation of the finite-size effective critical temperatures corresponding to the peaks of $\chi$ $(T_{{\chi }_{\max }}$ versus $L^{-1/\nu }$ with $\nu =1)$, which yields $k_B{T}_c/J=0.647\left(15\right)$ in the $L\to \infty$ limit.

Figure 7

(a) Scaling plots of the average absolute value of the magnetization, $\langle \left|m\right|\rangle L^{\beta /\nu }$ versus $|T-T_c|L^{1/\nu }$, obtained by setting $\beta =1/8,\nu =1$, and the critical temperature $k_B{T}_c/J=0.647$, following the best fit to the data shown in the inset to Fig. 6. (b) Scaling plots of the Binder cumulant. Both panels show data corresponding to $D=0$ and obtained for lattices of different size, as indicated.

Figure 8

Phase diagram of the MEM-BC model obtained for samples of different linear size $L$, as indicated. Solid and open symbols indicate lines of second- and first-order effective transitions, respectively, which meet at the effective tricritical point. The inset displays a close-up view, which shows that the tricritical point is essentially independent of the sample size.