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storage capacity, but instead make it harder to achieve a given storage capacity. In order to deal with these kinds of limiting factors, resistance distribution tighten- ing techniques have been developed based on write-and- verify procedures. These iterative techniques consist of applying programming pulses and verifying that a spec- ified precision criterion is met, along the lines of what is currently done in Flash memories[231]. These methods have been used to demonstrate 16 level PCM[60], 4 level PCM[60, 61, 162, 211], and to tighten the distribution of SET state resistances for binary PCM[168]. 

The effect of a write-and-verify technique is to re- shape the conditional probability density function, as shown in Figure 35. This can be obtained by succes- sively refining[232] the write procedure until the verify step finds the resistance value within the desired range around the nominal resistance target. When properly used, a write-and-verify algorithm produces tighter re- sistance distributions (compare Figure 24 to Figure 25), and therefore allows the packing of more MLC levels into the same resistance range. This increase in the number of levels obtained with write-and-verify reflects an ac- tual increase in the information-theoretic storage capac- ity. There exists an intricate tradeoff between storage capacity and the average number of write-and-verify iter- ations. In particular, References [228,229] show that, for a simple cell model affected by write noise, the achiev- able storage capacity tends to increase logarithmically with the number of write-and-verify iterations[233, 234]. This logarithmic increase is expected to hold even for more realistic cell models at a sufficiently large number of write iterations. 

Among the factors representing a fundamental limita- tion to the storage capacity, resistance drift plays an im- portant role. Short term drift manifests itself as a slow but steady increase of the resistivity of the amorphous material (see Reference [51] and references therein). The resistance drift has been shown to follow a power law, 

where R ( t ) denotes the resistance at time t , R$_{0}$ de- notes the initial resistance at time t$_{0}$ , and ν is a drift coefficient[90]. Typical values for ν for thin amorphous GST layers are on the order of 0.05-0.1[51]. This phe- 

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nomenon has been explained in terms of structural re- laxation in the amorphous material influencing a Poole or Poole-Frenkel conduction[49, 90], and in terms of ki- netics of electrically active defects in the amorphous GST material[91]. The drift process is fairly predictable in the case of thin films of amorphous GST, which would sug- gest that a few cells of known state in a block that evolved in time together might serve to identify the needed shifts in threshold bias. However, short term drift appears to be a random process, which can be expected to vary from cell to cell[89, 91, 211, 235]. By introducing yet another source of unpredictability, this short-term drift reduces the effective storage capacity of PCM. The phenomenon can be perceived as a broadening of the resistance dis- tributions over time. Figure 36 compares the cumulative distributions for four resistance levels measured immedi- ately after programming to the distributions after pro- grammed cells have been drifting, both at room temper- ature and then at elevated temperature[211]. 

A number of techniques have been proposed for cop- ing with drift. These include changing the write target resistances to take into account the expected broadening of the resistance distributions due to drift[211], and com- pensation techniques at read time, where pulses are used upon readout to return the device to its initial as-written resistance (presumably without accidentally reprogram- ming the cell)[236]. We remark that these topics are the subject of current active research in the PCM research community. 

Although little or no published literature yet exists on coding techniques designed expressly for PCM, the suc- cess that these techniques have had in established mem- ory and storage technologies would imply that coding will 

## B. Role of coding 

