Thanks　to　its　superior　features　of　non-volatility,　fast　read/write　speed,　high　endurance,　and　low　power　consumption,　spin-torque　transfer　magnetic　random　access　memory　(STT-MRAM)　has　become　a　promising　candidate　for　the　next　generation　non-volatile　memories　(NVMs)　and　storage　class　memories　(SCMs).　However,　it　has　been　found　that　the　write　errors　and　read　errors　caused　by　thermal　fluctuation　and　process　variation　severely　degrade　the　reliability　of　STT-MRAM.　Moreover,　process　imperfection　also　causes　a　diversity　of　the　raw　bit　error　rate　(BER)　among　different　dies　of　STT-MRAM.　In　this　paper,　we　propose　the　design　of　novel　rate-compatible　protograph　low-density　parity-check　(RCP-LDPC)　codes　to　correct　memory　cell　errors　and　mitigate　the　raw　BER　diversity　of　STT-MRAM.　In　particular,　to　deal　with　the　asymmetric　property　of　the　STT-MRAM　channel,　we　first　apply　an　independent　and　identically　distributed　(i.i.d.)　channel　adapter　to　symmetrize　the　STT-MRAM　channel.　We　then　present　a　modified　protograph　extrinsic　information　transfer　(P-EXIT)　algorithm　for　the　symmetrized　STT-MRAM　channel.　We　further　propose　a　combined　guideline,　including　the　modified　P-EXIT　algorithm,　the　asymptotic　weight　enumerator　(AWE)　analysis,　as　well　as　the　actual　error　rate　performance,　for　designing　protograph　LDPC　codes　with　short　information　word　lengths　for　STT-MRAM.　By　further　applying　a　code　extension　approach,　we　design　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　as　well　as　the　fixed-rate　quasi-cyclic　(QC)　LDPC　codes　in　terms　of　both　the　error　rate　performance　and　the　convergence　speed　over　the　STT-MRAM　channel.