
The first essential point to note is
that a modulation of an output signal obtained by modulating some input
always requires a change or modulation in some internal state of the
device. 


And changing something always takes some time.
Nothing happens instantaneously, changing something consumes some time. We thus may start by listing the
time consuming processes that we already
encountered. 

What kind of typical time constants in semiconductors did we encounter
so far? Think about it for a minute. Well, we had 


The minority carrier
life time t . It measures the average time that a
minority carrier "lives" before it recombines with a majority
carrier. It can be rather large for very clean indirect semiconductors
(ms), and rather small for indirect semiconductors (ns). The
numerical value of a minority carrier life time implies that you cannot change
the minority carrier concentration at a frequency much larger than 1/t.We have a first
limit to how fast you can change an internal state. 


The dielectric relaxation time t_{d}. It measures the average time
that majority carriers need to respond to
some disturbance of their distribution. It was rather small, typically in the
ps range and given by 




Those were the two fundamental
material related time constants that we encountered so far. But there are more
time constants which are not so directly obvious: 

First, we have the "trivial" electrical time constant t_{RC} inherent in any electrical
system, simply given by the R · C product. R
is the ohmic resistivity, and C the capacitance of the circuit
(part) considered. 


R and C need not be
actual resistors or capacitors intentionally included in the system, but unwanted,
nevertheless unavoidable, components. The resistivity of Al
metallization lines together with the parasitic capacitance of this line in a
Si integrated circuit. e.g., gives a t_{RC} of roughly 10^{–9}
s, and this value (per cm line length) is directly determined by the
product of the specific resistivity r of the
conducting material times the relative dielectric constant e_{r}of the dielectric separating individual
wires  it is thus a rather intrinsic material
property. 


The physical
meaning of t_{RC} is
clear: It is the time needed to charge or discharge the capacitors in the
system. Clearly, you cannot change internal states very much at frequencies
much larger than 1/t_{RC}. And note
that space charge regions, or MOS structures always have a capacity C, too. 

Second, if we turn to Lasers for a moment,
we have seen that
we need to feed some of the light produced by stimulated emission back into the
semiconductor by using a suitable mirror assembly. 


Light bounces back and forth between the two
mirrors in the simple system considered  and that means that even after you
turned off the current through the Laser diode, some light will still bounce
back and forth and thus come out until everything eventually calmed down. There
is an obvious time constant 


t_{Q} 
= 
N_{r} · L · n_{r}
c 




With N_{r} = average number
of reflections, L = distance between the mirrors,
n_{r} = refective index of the material, and c = vacuum
velocity of light. 


If, for an order of magnitude guess, we take
L = 100 µm and consider 10 reflections; the
"last" photons to come out would have to travel 10 · 100
µm = 1 mm, which takes them a time t_{Q} = N_{r} · L
· n_{r}/c » 10^{–11}
s = 10 ps. 


In other words, for the example given, it would
not be possible to modulate the light intensity with frequencies in excess of
about 100 GHz. This seems to be a respectable frequency, but keep im
mind that data can now (2001) be transmitted through fibre optics at
frequncies in the THz regime. 

This example, while a bit
farfetched, gives us an important insight: There is a general relation between
a time constant of a system and a typical length of a system mediated by the speed
with which things move. This means that the size of a
device may be important for its frequency response. 


In other words, we can always
ask: How much time does it take to move things over a distance l?
And whenever the output O is some distance away from the input
In, the question of how long it takes to move whatever it takes
from In to O produces a typical time constant of
the system. 


In straightforward simple mechanics
l is linked to its time constant t_{l} by the speed of the moving "things"  for the
photons considered above this was clearly the speed of light (in the medium, to
be correct). 


For our moving statistical ensembles, we have
somewhat more involved relations, e.g. . 




What are the moving things? Well,
besides photons, we essentially are left with electrons and holes; everything
else that might be of interest is usually immobile (dopants, localized
excitons), or so slow that it should not matter for electronic signals (phonons, mechanical movements
(e.g. vibrating parts) in MEMS devices) 


This brings us to a first simple and important
question: How long does it take electrons or holes to move from the source to
the drain in a MOS transistor. Clearly, this will give us another
maximum frequency for operating said transistor. 


The relevant velocity in this case is the
drift velocity v_{D} of the
carriers, usually proportional to the field strength E as driving
force for the movement, and better expressed via the carrier mobility 





With the sourcedrain distance
l_{SD}, and the source drain voltage
U_{SD}, we have E =
U_{SD}/l_{SD} and a "travel time"



t_{l} = 
l_{SD}
v_{D} 
= 
l_{SD}^{2}
m · U_{SD} 




To get a feeling for orders of magnitude, we take
a sourcedrain distance l_{SD} = 1 µm and a
sourcedrain voltage U_{SD} = 5V, obtaining a field
strength of E_{SD} = 5 · 10^{4} V/cm.
Typical
mobilities are m_{Si} = 1000
cm^{2}/Vs for Si. This gives us a drift velocity of 


v_{D} 
= 
1000 
cm^{2}
Vs 
· 5 · 10^{4} 
V
cm 
= 5 · 10^{7} 
cm
s 




Is that a large or small velocity? It might be
good to look up at
an old
exercise at this point 


The "travel time" t_{l} then is 


t_{l} 
= 
l_{SD} · v_{D} = 
10^{–4}
10^{7} 
s = 10^{–11} s 




A "1 µm" Si MOS
transistor thus would not be able to switch frequencies beyound about
10^{11} Hz = 100 GHz if
t_{l} would be the only limiting time
constant of the system. 

Last, there are some ultimate limits
that we should be aware off: 


Nothing moves faster than c, the the speed
of light (in vacuum). The consideration for the Laser from above already gives
an example for this limit. 


The movement of electrons and holes has some
intrinsic constant of its own: The average time
between scattering processes and the
average distance
or mean free path in between. While we are not very aware of the values for
these parameters, the mean free path is in the order of 100 nm. 

This has an important consequence: We
only can use average quantities like drift
velocities, if individual carriers could have many collisions. 


Turning this around implies: If we look at travel
scales around and below 100 nm, everything may change. For transistors
this small, electrons (or holes) might just speed from source to drain without
any collisions in between  much faster than at larger distances. This is the
case of ballistic carrier transport
which must be considered separately. 

