I have noticed that a lot of smart phones say that they won't operate under -4 degrees F (-20 degrees C). Can anyone explain to me what happens when the phones get cold that stops them from operating?
-4 F is -20C, which is a standard low limit for chips and electrical components. Some of that is just because it is very hard to test chips at low temperature, but there are real issues you can run into, which include:
Batteries degrade at low temperatures, depending on their chemistry.
The battery output voltage is lower, meaning you need more current to get the same power
The battery internal resistance can increase. The added resistance can heat up the board, but it also wastes power and makes the battery output voltage less stable, as it will change with current draw.
The heat caused by the extra resistance can potentially damage the battery, since you are heating up the inside while the outside is cold, creating a thermal gradient which adds mechanical stress.
Thermal cycling of parts can become worse. Things break when you make them cold and heat them up because of thermal expansion. I believe this issue is worse at lower temperatures, possible related to metals becoming brittle when they are very cold.
Chips can draw more current at low temperatures. This issue compounds the other two, since more current becomes more heat, which increasing thermal cycling.
Chip timing changes. Digital circuits have special timing rules to ensure that all signals are in the right place at the right time. Lowering the temperature changes all that and can create a race condition.
For most of these devices it's the display...
LCDs don't like the cold.
Typically, standard LCD character and graphics modules provide a temperature range of 0°C to +50°C. However, several display manufacturers offer extreme temperature models with operating temperatures of -40°C to +80 or +85°C. There also is a wide selection of standard versions that range from -20°C to +70°
The newer OLED types do have a much better temperature tolerance though, -40°C to +80°C.
Batteries dislike cold.
Generally all batteries lose capacity and current in the very cold. (However, using them often warms them up.). Lithiums have a particular problem with being charged in the very cold.
Also, devices are concerned about condensation occurring inside the device from humid air entering the headphone jack etc.
Crystal oscillators may not start up; or the crystal resonant frequency, which has a temperature coefficient, may be outside the guaranteed Automatic Frequency Control (afc) range, needed to ensure the data packets start on expected time slots even after some hours of operation and phase slipping.
To add my 2 cents on all the great answers (which would apply not only to electronic devices but generally all electric devices) - the temperature drop results in change of material resistance (namely for metals they become less resistant), while this might seem a minor thing, in industrial equipment this is one of the items accounted for. Electronic devices would suffer most at that because many microchips rely on resistors between some of their lines to be of specific value, if that value changes the microchip may start misbehaving or completely shutdown.
Analog circuits can also have problems at low temperatures. Resistance changes across temperature, and so do transistor threshold voltages and transconductances. If a reference voltage or current goes out of spec, it can affect other analog circuits that depend on it (like an ADC or a charge pump).
When you're simulating a design and (later) characterizing and testing hardware, you have to pick a lower bound for temperature. There may not be any actual problems if you go a little below that temperature, but the manufacturer can only guarantee correct operation if you stay within the tested limits.
That being said, for smartphones the battery and display are probably the bigger concerns, as the other answers show.
When you get into minus figures things start to slow down, there is something called ABSOLUTE ZERO which is 0 degrees Kelvin or minus 273 C. @ 0 Kelvin nothing moves, including protons and electrons, it basically freezes electricity (not sure what happens with photons). Eventually chip speeds will slow down but not at the same rate so the sync is lost.
My 2 cents is that there is a "Diode Equation" (google it!) that includes current, voltage and temperature. So a semiconductors behaviour depends on temperature. Useful in a digital thermometer, but must be counteracted by adding trace elements and other magic by the manufacturers of ordinary chips to make them work "straight". But this works only within a limited temp. range depending on budget and usage. So I guess it would be possible to make a phone that works well only between -220C and -150C, for example.