The entropy functional introduced by Colding and Minicozzi plays a fundamental role in the analysis of mean curvature flow. However, unlike the hypersurface case, relatively little about the entropy is known in the higher-codimension case. In this note, we use measure-theoretical techniques and rigidity results for self-shrinkers to prove a compactness theorem for a family of self-shrinkers with low entropy. Based on this, we prove that there exists a $2$-dimensional complete, non-flat, and smooth embedded self-shrinker that minimizes the entropy among all $2$-dimensional non-flat self-shrinkers in $\mathbb R^N$ for any $N.$
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