Dennard scaling

Dennard scaling, also known as MOSFET scaling, is a scaling law which states roughly that as transistors get smaller, their power density stays constant, so that the power use stays in proportion with area; both voltage and current scale (downward) with length. The law, originally formulated for MOSFETs, is based on a 1974 paper co-authored by Robert H. Dennard, after whom it is named. 
Derivation.
Dennard observes that transistor dimensions could be scaled by -30% (0.7x) every technology generation, thus reducing their area by 50%. This would reduce circuit delays by 30% (0.7x) and therefore increase operating frequency by about 40% (1.4x). Finally, to keep the electric field constant, voltage is reduced by 30%, reducing energy by 65% and power (at 1.4x frequency) by 50%. Therefore, in every technology generation, if the transistor density doubles, the circuit becomes 40% faster, and power consumption (with twice the number of transistors) stays the same.
Relation with Moore's law and computing performance.
Moore's law says that the number of transistors doubles approximately every two years. Combined with Dennard scaling, this means that performance per watt grows even faster, doubling about every 18 months (1.5 years). This trend is sometimes referred to as Koomey's law. The rate of doubling was originally suggested by Koomey to be 1.57 years, but more recent estimates suggest this is slowing.
Breakdown of Dennard scaling around 2006.
The dynamic (switching) power consumption of CMOS circuits is proportional to frequency.
Historically, the transistor power reduction afforded by Dennard scaling allowed manufacturers to drastically raise clock frequencies from one generation to the next without significantly increasing overall circuit power consumption.
Since around 2005–2007 Dennard scaling appears to have broken down. As of 2016, transistor counts in integrated circuits are still growing, but the resulting improvements in performance are more gradual than the speed-ups resulting from significant frequency increases. The primary reason cited for the breakdown is that at small sizes, current leakage poses greater challenges and also causes the chip to heat up, which creates a threat of thermal runaway and therefore further increases energy costs.
The breakdown of Dennard scaling and resulting inability to increase clock frequencies significantly has caused most CPU manufacturers to focus on multicore processors as an alternative way to improve performance. An increased core count benefits many (though by no means all) workloads, but the increase in active switching elements from having multiple cores still results in increased overall power consumption and thus worsens CPU power dissipation issues. The end result is that only some fraction of an integrated circuit can actually be active at any given point in time without violating power constraints. The remaining (inactive) area is referred to as dark silicon.