8.1.2 Basic Time Consuming Processes

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

t_d	=	ee₀ s

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_rof 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 far-fetched, 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 straight-forward 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. .

L	=	æ è	D · t	ö ø	1/2	for the relation between the diffusion length of the minority carrier and their lifetime

L_Dn	=	æ è	D · t_d	ö ø	1/2	for the relation between the Debye length L_Dn and the dielectric relaxation time

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

m	=	v_D E

With the source-drain 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_SD² m · U_SD

To get a feeling for orders of magnitude, we take a source-drain distance l_SD = 1 µm and a source-drain voltage U_SD = 5V, obtaining a field strength of E_SD = 5 · 10⁴ V/cm. Typical mobilities are m_Si = 1000 cm²/Vs for Si. This gives us a drift velocity of

v_D	=	1000	cm² Vs	· 5 · 10⁴	V cm	= 5 · 10⁷	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^–4 10⁷	s = 10^–11 s

A "1 µm" Si MOS transistor thus would not be able to switch frequencies beyound about 10¹¹ 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.