At higher temperatures, will computers get faster? Evidently, one always wants cool a computer down as higher temperatures can damage core components.
However, is it an interplay between silicon, which at higher temperatures will release more electrons and the resistance of the metal components which will increase as temperature does? Or is this negligible in terms of overall computer performance?
Lets break your questions into sub-questions:
Faster computer:
The most common measure of computer's "speed" is its maximum clock frequency. This measure has never been an accurate one (Megahertz myth), but it became totally unimportant in recent years after multi-core processors became a standard. In today's computers, the top performance is determined by much more complex factors than just the maximum clock frequency (these factors include both HW and SW aspects).
Temperature's effect on clock frequency:
Said that, we still want to see how a temperature affects the clock frequency of the computer. Well, the answer is that it does not affect it in any appreciable way. The clock for the computer is (usually) derived from a crystal oscillator, which does not heat up at all. This means that oscillator's frequency is independent of the temperature. The signal produced by the oscillator is multiplied in frequency by PLLs. The PLLs' output frequency will not be affected by temperature (assuming that they were designed properly), but the level of noise in PLLs' clock signal will increase with temperature.
The above discussion leads to the following conclusion: the increase in temperature will not increase the frequency of the clock (by any appreciable amount), but can lead to a logical failure due to increased noise in clock signal.
Temperature's effect on maximum clock frequency:
The temperature has effectively no effect on the pre-defined frequency of the clock. However, maybe higher temperature allows for higher frequencies to be employed?
First of all you need to understand that modern computers do not have their clock rates pushed to the limit of technology. This question has already been asked here.
The above means that you can increase the frequency of your CPU above the one which was defined by default. However, it turns out that in this case the temperature is the limiting factor, not a benefit. Two reasons for this:
The first factor leads to a higher probability of logical failure at high temperatures (incorrect logical values being used). The second factor leads to a higher probability of physical failure at high temperatures (like permanent damage to a conductive wire).
Therefore, the temperature is the limiting factor of processors' maximum frequency. It is the reason why the most abusive overclocking of processors is performed while the processor is super-cooled.
Thermally excited carriers in silicon:
I believe that you were led to the wrong conclusions by the thought that silicon's resistivity reduces with temperature. It is not the case.
While the thermal generation rate is indeed increases with temperature, there is no much use to intrinsic silicon. The fact that the most of silicon used in the industry is doped means that the thermally excited carriers comprise a negligible fraction of the free carriers in silicon; therefore, even huge increase in thermal excitation rates will not affect the density of the free carriers appreciably. Check out this calculator and try to find at which temperatures the density of the thermally generated carriers approaches the usual doping concentrations (\$\geq 10^{16}cm^{-3}\$) - your processor will burn out long before the thermal generation will affect the conductivity of silicon.
Furthermore, the mobility of free carriers tend to decrease with temperature; therefore, instead of the increase in conductivity of silicon, you'll probably observe a decrease which will lead to a higher probability of logical failure.
Conclusion:
Temperature is the main limiting factors of computers' speeds.
Higher temperatures of processors also lead to the higher rates of Global Warming, which is very bad.
Advanced topics for interested readers:
The answers above, to my best knowledge, are completely correct for technologies down to 32nm. However, the picture may be different for Intel's 22nm finFET technology (I found no references for this newest process on the web), and it will certainly change as process technologies continue to scale down.
The usual approach for comparing the "speed" of transistors implemented using different technologies is to characterize the propagation delay of the minimal size inverter. Since this parameter depends on the driving circuit and the load of the inverter itself, the delay is calculated when few inverters are connected in a closed loop forming a Ring Oscillator.
If the propagation delay is increasing with temperature (slower logic), the device is said to operate in Normal Temperature Dependence Regime. However, depending on device's operating conditions, the propagation delay can decrease with temperature (faster logic), in which case the device is said to operate in Reverse Temperature Dependence Regime.
Even the most basic overview of the factors involved in the transition from Normal to Reverse temperature regimes is beyond the scope of a general answer, and requires pretty deep knowledge of semiconductors physics. This article is the simplest yet complete overview of these factors.
The bottom line of the above article (and other references I found on the web) is that Reverse temperature dependence should not be observed in currently employed technologies (except, maybe, for 22nm finFET, for which I found no data).
The primary type of switching element in typical computers is the metal oxide semiconductor field effect transistor. Such devices are less effective at passing current when hot than when cold. While there are some situations where such behavior can be a good thing (e.g. it improves the load-sharing ability of power MOSFETs) it also means that logic functions implemented with MOSFETs will take longer to switch at higher temperatures. Since reliable operation of a computer requires that all of the circuits that are supposed to switch in a given cycle manage to do so before the next cycle arrives, computers generally cannot operate as fast at high temperatures as they can at slow temperatures.
Furthermore, the amount of heat generated by a computer using complementary-MOSFET logic is in large measure proportional to the actual speed at which it is running. To prevent damage from overheating, a number of processors have circuitry which will automatically slow them down if temperatures exceed a certain threshold. This will of course severely curtail application performance, but having an application slow down may be better than having the processor completely cease operation either temporarily or permanently.
The answer is no.
Mainly because a computer is a clocked circuit. If the CPU, or the whole computer, is at a higher temperature, the clock circuit would not run faster. Thus the number of MIPS or FLOPS is the same, regardless of the temperature.
But, as seen in comments of your questions, the temperature could have an effect to the maximum clock rate that your CPU would support.
Computers run as fast as you clock them. Therefore, heating a computer without doing anything else different won't effect computational power until it is heated so far that it is damaged and the computational power goes to 0.
Running a computer uses electric power, which is dissipated in the computer as heat. The amount of electric power used is in part proportional to the clock speed. This means that the hotter the computer is, the slower you have to clock it to avoid reaching the critical point at which it can no longer function and possibly be damaged permanently.
This is why high-performance comuputers have temperature sensors. A external circuit clocks the computer as fast as possible, but to not exceed its maximum operating temperature. Therefore heating one of these units decreases computational power because the thermal management circuit will clock the computer slower since less electrical power is allowed before it hits its maximum operating temperature.
I remember seeing a commercial from Intel about this. They were showing off that their processor had this temperature sensing and clock adjusting circuit built-in. They showed two computers, one with their chip and one with a competitor's, running the same program at the same speed. Then they took the heat sinks off both processors. The one with the internal thermal management circuit slowed down. The other one kept going for a while, then quit completely when it overheated.
