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The growing memory demands of modern applications have driven the adoption of far memory technologies in data centers to provide cost-effective, high-capacity memory solutions. However, far memory presents new performance challenges because its access latencies are significantly longer and more variable than local DRAM. For applications to achieve acceptable performance on far memory, a high degree of memory-level parallelism (MLP) is needed to tolerate the long access latency. While modern out-of-order processors are capable of exploiting a certain degree of MLP, they are constrained by resource limitations and hardware complexity. The key obstacle is the synchronous memory access semantics of traditional load/store instructions, which occupy critical hardware resources for a long time. The longer far memory latencies exacerbate this limitation. This paper proposes a set of Asynchronous Memory Access Instructions (AMI) and its supporting function unit, Asynchronous Memory Access Unit (AMU), inside a contemporary Out-of-Order Core. AMI separates memory request issuing from response handling to reduce resource occupation. Additionally, AMU architecture supports up to several hundreds of asynchronous memory requests through re-purposing a portion of L2 Cache as scratchpad memory (SPM) to provide sufficient temporal storage. Together with a coroutine-based programming framework, this scheme can achieve significantly higher MLP for hiding far memory latencies. Evaluation with a cycle-accurate simulation shows AMI achieves 2.42x speedup on average for memory-bound benchmarks with 1us additional far memory latency. Over 130 outstanding requests are supported with 26.86x speedup for GUPS (random access) with 5 us latency. These demonstrate how the techniques tackle far memory performance impacts through explicit MLP expression and latency adaptation.