This　paper　proposes　novel　rate-compatible　protograph　LDPC　(RCP-LDPC)　codes　to　correct　memory　cell　errors　and　mitigate　the　raw　bit　error　rate　(BER)　diversity　for　spin-torque　transfer　magnetic　random　access　memory　(STT-MRAM).　Since　the　STT-MRAM　channel　is　asymmetric,　we　first　apply　an　independent　and　identically　distributed　(i.i.d.)　channel　adapter　to　force　the　channel　to　be　symmetric.　We　then　propose　to　use　the　protograph　extrinsic　information　transfer　(PEXIT)　algorithm,　the　asymptotic　weight　distribution　(AWE)　analysis,　as　well　as　the　actual　error　rate　performance　as　the　combined　guideline　for　designing　protograph　LDPC　codes　with　short　codeword　lengths　for　STT-MRAM.　By　further　applying　a　code　extension　approach,　we　construct　novel　RCP-LDPC　codes　that　can　work　with　a　single　encoder/decoder.　Simulation　results　show　that　our　proposed　RCP-LDPC　codes　outperform　the　well-known　rate-compatible　AR4JA　protograph　codes.