  Gradual switching between multiple resistance levels is desirable for analog
in-memory computing using resistive random-access memory (RRAM). However, the
filamentary switching of $HfO_x$-based conventional RRAM often yields only two
stable memory states instead of gradual switching between multiple resistance
states. Here, we demonstrate that a thermal barrier of $Ge_2Sb_2Te_5$ (GST)
between $HfO_x$ and the bottom electrode (TiN) enables wider and weaker
filaments, by promoting heat spreading laterally inside the $HfO_x$. Scanning
thermal microscopy suggests that $HfO_x+GST$ devices have a wider heating
region than control devices with only $HfO_x$, indicating the formation of a
wider filament. Such wider filaments can have multiple stable conduction paths,
resulting in a memory device with more gradual and linear switching. The
thermally-enhanced $HfO_x+GST$ devices also have higher on/off ratio ($>10^3$)
than control devices ($<10^2$), and a median set voltage lower by approximately
1 V (~35%), with a corresponding reduction of the switching power. Our
$HfO_x+GST$ RRAM shows 2x gradual switching range using fast (~ns) identical
pulse trains with amplitude less than 2 V.
