embedding of $SL_2$ into larger matrix groups that decreases coefficient size?

Are there injections $\phi:SL_2(\mathbb{Z})\to M^{n\times n}(\mathbb{Z})$ that decrease the size of the coefficients? For the application I have in mind, coefficient growth is exponential in the word length w.r.t. a generating set, e.g. the maximum entry of words of length $n$ in the generators $$ R=\left(\begin{array}{cc}1&1\\0&1\\\end{array}\right),\quad L=\left(\begin{array}{cc}1&0\\1&1\\\end{array}\right), $$ of the free monoid $SL_2(\mathbb{N})$ is the $n$th Fibonacci number.

Is there some way to homomorphically reduce the rate of growth or coefficient size by embedding into larger matrix groups? I.e. somehow smear out the coefficients among more matrix entries?

Bitstrings under concatenation can use $n$ bits to represent a word of length $n$ in $\{0,1\}$ (I assume this is minimal), and words of length $n$ in $\{R, L\}$ can be capped at around $4n\log_2\left(\frac{1+\sqrt{5}}{2\sqrt{5}}\right)=2.77n$ bits, so I suppose my question is whether or not there is any arithmetic/matrix-y representation with growth in between.

I'm guessing the answer is no...