Effects of spin density wave quantization on the electrical transport in epitaxial Cr thin films

We present measurements of the electrical resistivity $\rho$ in epitaxial Cr films of different thicknesses grown on MgO (100) substrates, as a function of temperature $T$. The $\rho \left(T\right)$ curves display hysteretic behavior in a certain temperature range, which is film thickness dependent. The hysteresis are related to the confinement of quantized incommensurate spin density waves (ISDW) in the film thickness. Our important finding is to experimentally show that the temperature $T_{\text{mid}}$ where the ISDW changes from $N$ to $N+1$ nodes decreases as the film thickness increases. Identifying $T_{\text{mid}}$ with a first-order transition between ISDW states with $N$ and $N+1$ nodes, and using a Landau approach to the free energy of the ISDW together with Monte Carlo simulations, we show that the system at high temperatures explores all available modes for the ISDW, freezing out in one particular mode at a transition temperature that indeed decreases with film thickness $L$. The detailed dependence of $T_{\text{mid}}\left(L\right)$ seems to depend rather strongly on the boundary conditions at the Cr film interfaces.

Figure 1

Resistivity as a function of temperature for a cooling-warming cycle for the 270 Å film. The inset shows the difference $({\rho }_c-{\rho }_w)/{\rho }_w$, expressed as a percentage.

Figure 2

Derivatives of the $\rho \left(T\right)$ curves shown in Fig. 1. The inset shows the difference of $\partial \rho /\partial T$ between the cooling and warming cycles.

Figure 3

Transition temperature as a function of film thickness. Red squares correspond to data in Ref. 9. The vertical bars are not error bars but the width of the hysteretic region (see text). The dashed lines are guides to the eye. Inset: Difference between the derivatives of $\rho \left(T\right)$ curves for cooling and warming cycles of films with increasing thicknesses. For clarity, the vertical zero of each curve (horizontal lines) has been arbitrarily shifted. The arrows indicate the temperature $T_{\text{mid}}$ (see text).

Figure 4

Available ISDW modes for different film thicknesses in the temperature region of interest. The temperature scale is the same for all panels.

Figure 5

Cuts of the two-dimensional free energy $F[{\Psi }_0,q\left(T\right)]$ at different temperatures and for a fixed value of ${\Psi }_0$. (a) $T/T_N=0.96$, (b) $T/T_N=0.74$, (c) $T/T_N=0.48$. Available modes correspond to a film of 760 Å.

Figure 6

Average $q$ as a function of temperature for different film thicknesses. The fluctuations in $q$ at high temperatures freeze to a fixed number of nodes at a temperature $T_{\text{mid}}$ which depends on the film thickness (arrows).

Figure 7

Freezing temperature from Fig. 6 as a function of film thickness. The dashed line is a guide to the eye.