Sensing of Biomolecules - Bionanoelectronics: Bioinquiring and Bioinspired Devices - page 65

a

b

E V G

E C G

E F1
E F1 - eV D

E F1 C D

height of the
potential at the
top of the barrier
U top

k C S

V D

top of the
barrier

E F1 - eV D

V S

x

Fig. 2.7 ( a ) Energy map for the ballistic HEMT in the Datta-Lundstrom model and ( b ) the Datta-
Lundstrom equivalent circuit

In this model, the maximum potential energy value between source and drain is

U top D eŒ.C G V G =C tot / C .C D V D =C tot / C .C S V S =C tot / C e 2 n mob =C tot ; (2.16)

where C tot D C G C C S C C D and C G C S ;C D . The density of mobile charges
in the channel, n mob , is calculated by filling with electrons the C k and k states in
conformity with the source and drain Fermi energies, respectively, starting from the
bottom of the conduction band at U top , assumed known. U top is computed iteratively
by increasing the density of mobile charges until convergence is reached, and I D
is estimated from the known electron populations in the two halves of the energy
dispersion curve above U top .
Nevertheless, contrary to common expectations, the mobility of ballistic carriers
in short-channel HEMTs, is much lower than in long-channel HEMTs ( Shur 2002 ).
The effective mobility can be written as

1= eff D 1= ball C 1= 0 ; (2.17)

where 0 denotes the mobility in the long-channel regime, in which collisions dom-
inate, and ball is the mobility associated to ballistic carriers. In the nondegenerate
case, the ballistic mobility has the expression ball D 2eL=m v th , whereas in
the degenerate case, the thermal velocity v th D .8E th =m/ 1=2 in this relation, with
E th D k B T , must be replaced with the Fermi velocity v F .From( 2.17 ), it follows that
the mobility decreases significantly when the gate length decreases, in agreement
with experiments; for instance, in GaAs, the mobility is 10;000 cm 2 =Vs for a gate
length of 10m and only 3;000 cm 2 =Vs for a gate length of 150 nm.
For low drain voltage values, the expression of the drain current can be
simplified as

I D D Wq i .0/ ball V D =L; (2.18)

where q i .0/ denotes the electron distribution at the source. On the contrary, for high
drain voltages, we obtain