Re: Why resistance of silicon and germanium decreases with increase of temperature?

I guess the question you are asking is, "Why does the resistivity of
silicon and germanium get lower as temperature increases when the
resistance of something like a metal gets higher?"

What this really boils down to is what makes silicon different from
something like iron?

If we are very simple-minded we can classify materials into three
different groups: Metals, insulators and semiconductors.

Metals are things like iron, copper, aluminium and lead that you might come
across every day. Metals share a number of properties. They typically
conduct electricity and heat well, and are often shiny in appearance.

Insulators are things like glass, plastics, wood and rubber. These are
materials which are rarely shiny, do not conduct electricity and are
good thermal insulators too.

Semiconductors are not something that you come across every day, even
if they are a vital part of modern daily life. Nearly every
electronic/electric device has some semiconductor material in it. The
reason is directly related to your question. A semiconductor can be
either conducting or not, depending on what is happening to it (hence
the name - "semi-conductor".

What a semiconductor needs to be conductive (less resistive) is
some excitement. This excitement has to be in the form of energy. Any
form will actually do. This is why, when you heat up a semiconductor, it
becomes excited and starts to conduct electricity better. Another way
to excite it would be to shine a light on it. Light also gives a
semiconductor energy and makes it more conductive. The final way to
excite a semiconductor is to pump electricity into it. Often it
doesn't take much electricity to make a semiconductor highly
conductive. It is this last sort of excitation that makes it so useful
as a switch. A little bit of electricity can enable a semiconductor to
switch on a much larger amount.

Anyway, that was the simple explanation. Now let us think about what
is happening on a microscopic level. You can imagine that the
electrons in a semiconductor are normally bound quite closely to the
silicon or germanium atoms in a crystal (silicon and germanium are
both naturally crystalline). The fact that they are bound to the atoms
means that they cannot move and conduct electricity. However, when
they get energy they are no longer bound strongly to an individual
atom, but can hop from atom to atom. This hopping allows them conduct
electricity since each electron is charged with one unit of electrical
charge. A flow of charge is an electrical current.

Now let us quickly come back to metals. A metal is like an excited
semiconductor where the electrons are not bound to the atoms, but can
easily move around in the material. However, when you heat a metal up,
the electrons start to bounce off each other and the atoms themselves,
since there are so many of them in close proximity. This causes them
to get in each other's way and they cannot move as freely. This is the
cause of the increase in resistance in something like a copper wire
when you heat it up.

I haven't given any references with this explanation, but any good
textbook on physics or electric circuits at your local library will
explain this in as much detail as you like. Also try your favourite
search engine with search strings such as

"temperature coefficient of resistance"
"resistivity of silicon"
"negative temperature coefficient of resistance"
"diode thermometer"

there are many many links with many many interesting pieces of
information.

One final thing (in case I haven't already made this point): 
Semiconductors are probably the most versatile materials known to 
humankind. You can make switches, microprocessors, temperature sensors, 
satellite receivers, light sensors, high-power lasers, variable 
capacitors, LEDs, single photon detectors and much more out of 
semiconducting materials - all related to this increase in conductivity 
when you put energy into the material.