(Ng et al. 1998). The circles that lie on near the solid line correspond to samples whose transport is dominated by ionised impurity scattering.
B ased on th e dislocation m odel using Eq. 4.28, Ng et al. (1998) have investi g ated tra n s p o rt in n-G aN films grown by E C R -M B E w ith concentrations in th e range 10^® — 10^*^ cm "^. T hey found th a t th e 300 K m obility versus carrier co n centration followed a fam ily of bell-shaped curves as shown in Fig. 4.2, w here for th e sam ples w ith lower N^ish tra n s p o rt is m ainly controlled by ionised im p u rity scatterin g . For a given carrier concentration, m uch lower m obilities are observed for th e sam ples w ith higher dislocation densities. It should be noted th a t m ost of th e sam ples stu d ied in this thesis have m obilities th a t correspond to tra n s p o rt d o m in ated by ionised im pu rity scattering, and th u s dislocation scatterin g is not expected to play a m ajo r role (this issue will be explained more fully in Section 4.2.1). T h ere are a few sam ples w ith very low m obilities, th u s pointing tow ards to dom inant dislocation scatterin g . U nfortunately, transm ission electron m icroscopy (TEM ) studies have n o t been carried out on th e sam ples th a t have been characterised in this thesis, and th e ex act level of dislocation density of th e sam ples is unknow n, th u s m aking th e in co rp o ratio n of
4. T ran sp ort in G a N E pilayers 51
dislocation scatterin g in th e analysis difficult. Hence, th e m obility lim ited by disloca tio n sc atterin g is excluded in the calculations, and is theoretically presented here for com pleteness.
D efo rm a tio n p o te n tia l sca tterin g
Free carriers can be subjected to scatterin g by th erm al v ib rations of th e lattic e atom s, represented by quasi-particles as acoustic phonons, which can p ro p ag ate th ro u g h th e c ry sta l by longitudinal and transverse waves. D uring propagation, an ad d itio n al peri odic p o ten tia l is superim posed on th e periodic field of th e in ternal cry stal field. T his a lters th e energy stru c tu re of the crystal, and th e equivalent a lte rn a tin g p o ten tia l en ergy of th e carriers is known as th e deform ation p o ten tial. T h e m obility lim ited by acoustic-m ode deform ation potential scatterin g is (A nderson and A spley 1989):
„ = 2 (2 7 r)V y ;fi^ e
3S2p(m*)V2(fcBT)V2 ^ • ’
w here g is th e density of mass, Vg is th e crystal sound velocity, and is th e defor m atio n potential.
P ie z o e le c tr ic sca tter in g
T h is is an o th er form of th e acoustic phonon scattering, which arises in ionic sem icon d u ctors, w here th e acoustic vibrations produce a piezoelectric field th a t p e rtu rb s th e crystal. A nderson and Aspley (1989) have also given an analytical expression for th e m obility lim ited by piezoelectric scattering, which is given by:
^ 16(27r)V^gs^^^e
w here hpz is th e piezoelectric constant.
P o la r o p tica l sca tterin g
P o lar sem iconductors such as GaN possess a high degree of ionicity, th u s th e in teractio n betw een carriers and th e optical vibrations of th e lattic e atom s is likely to be strong. T h is ty p e of scatterin g is different from acoustic phonon scattering, w here th e energy exchange during th e absorption and th e emission of acoustic phonons is sm all, and
4. T ran sp ort in G a N E pilayers 52
th e sc atterin g is regarded as elastic. However, judging from th e high polar optical phonon energy (91.2 m eV ), th e sc atterin g by polar optical phonon is grossly inelastic, and becom es effective m ainly a t high tem p e ra tu re s. As a result, it is not suitable to tre a t optical scatterin g w ith th e relaxation tim e approxim ation. In stead , G elm ont et al. (1995) have developed an analytical th eo ry to describe optical phonon scatterin g , w here th e m obility lim ited by polar optical phonons is given by:
1 ~ Eg
(4.31) \/2 m * e $ o p (l + ^ap/Eg)
where n = (l//^oo — 1/ko)~^ is th e coupling constant, Kqo = Eoo/^o an d kq = Sg/so are th e high frequency and sta tic dielectric co nstant ratios, respectively, Eq is th e free p erm ittivity,
Fp = [exp{^op/Vih) - 1]- 1
is th e P lanck function, Vth = k B T / e is th e th erm al voltage, is th e p olar optical phonon energy, and Eg is th e energy gap.
4 .1 .3 I m p u r it y b a n d c o n d u c t io n
Before discussing th e origin of an im p u rity b an d and its im plications on G aN tra n s p o rt, th e Mott transition needs to be considered (M ott and Tw ose 1961). It refers to an in su lato r-to -m etal tra n sitio n occurring in sem iconductors a t high doping concentra tions. T h e tra n sitio n occurs when th e distance betw een im purities becom es com parable to th e B ohr radius, i.e., when th e im p u rity concentration exceeds a critical value Ncrit, where
Oq is defined as th e effective Bohr radius and is given by (w ith th e donor ground s ta te of j = 1)
=
w
■In GaN, this corresponds to ab o u t 24 Â, which consequently yields th e M o tt (critical) co n centration of approxim ately 1 x 10^® cm “ ^.
4. T ran sp ort in G a N E pilayers 53
T h e im p u rity ban d form ation is graphically illu strate d in Fig. 4.3. A t im p u rity concentrations well below th e critical M o tt concentration, im purities can be consid ered as isolated, non-interacting entities. As th e concentration increases b u t still well below th e M o tt concentration, im purities begin to in teract. C harge tra n s p o rt a t low tem p e ra tu re s occurs via th erm ally assisted hopping conduction. A t still higher im pu rity concentrations, overlapping of th e im p u rity w avefunction results in th e form ation of an im p u rity band. A t low tem p eratu res, carriers can p ro p ag ate w ithin th e im pu rity b an d w ithout entering th e conduction band. At even higher doping densities, th e im p u rity b an d widens and merges w ith th e conduction band, which occurs at Nc b = l/4 7 ra ^ ~ 6 x 10^® cm~^ (M a tsu b ara and Toyozawa 1961). I t should be po in ted o u t th a t there is always a degree of com pensation in G aN layers. Hence, th e existence of acceptors m eans th a t although th e free electron density falls way below th e critical concentration, th ere still could be num erous ionised donors com pensated by acceptors, and th e to ta l donor density could still be around or above th e 1 x 10^® cm “ ^ level.
(a )
( b )
■ ■■ AE,
F I G . 4 .3 . (a) Donor impurity level/band at N o Ncrit, Nd 4 Ncrit and Nd ^ Ncrit for