M any researchers currently working on nitrides have previous experience on GaA s an d rela te d m aterials, and th eir expertise has been readily tran sferred onto th e devel o pm ent of G aN based devices. A lthough this has helped in achieving rap id progress in n itrid e technology, there exist o ther problem s th a t are special to nitrides. O ne such problem is th e presence of another conducting layer in n -ty p e G aN epilayers, m ainly th o u g h t to be an im p u rity band. T here have been m any rep o rts of th is phenom enon, w hich has been observed irrespective of different grow th and processing techniques, and various investigations have been carried o u t to m odel such behaviour. In th is ch ap ter, th e effect of a secondary conducting channel in G aN films has also been s tu d ied, a n d th e experim ental findings and th eir com parisons w ith theo retical analysis will be discussed later in Section 4.2.
4.1
T heoretical considerations
Before presenting analysis and results of th e tra n s p o rt investigations on G aN layers, th e relevant th eory m ust be discussed. T he th eory is broadly divided into th re e p a rts; (a) basic aspects of sem iconductor tra n sp o rt, such as th e H all effect, free carrier s ta tistic s an d sc atterin g m echanism s, providing background m aterial t h a t would be used to characterise non-degenerate G aN bulk sam ples, (b) im p u rity b a n d conduction, a n d (c) th eo retical m odelling, which presents a set of equations employed to fit th e behaviour of b o th th e non-degenerate sam ples and specim ens th a t exhibit im p u rity b an d conduction
4. T ran sp ort in G aN E pilayers 41
(IB C ). Finally, note th a t th e GaN param eters th a t will be used in th e analysis are listed in Section 4.1.5.
4 .1 .1 B a s ic t r a n s p o r t th e o r y
P ro b a b ly th e m ost com m on m easurem ent m ade of th e properties of a sem iconductor is th e electrical conductivity. Consider th e weak-field conductivity arising from tra n s p o rt in an n -ty p e sem iconductor which at equilibrium has n m obile electrons p er u n it vol um e. T h e conductivity is defined as th e pro p o rtio n ality c o n stan t betw een th e applied electric field E and th e induced current density J ,
J = crE . (4.1)
If n electrons all move w ith velocity v, th en th e current density th ey give rise to will be parallel to v. Hence J can also be defined as
J — —n e v , (4.2)
w here e is th e electronic charge. T he m otion of an electron in a sem iconductor crystal can be described by adopting the relaxation tim e approxim ation (A shcroft and M erm in 1976). T his approach assum es th a t the effect of interactions experienced by free carriers is sm all, i.e., elastic collisions. The free electron m ass is replaced by an effective m ass m* to take into account of th e periodic p o ten tia l of th e crystal w ith spherical co n stan t energy surface, and a notion of a drift velocity is introduced, reflecting th e random ly averaged m otion of th e carriers by various scatterin g processes. T hen, th e eq u atio n of m otion for an electron is as follows;
w here (r) denotes th e average relaxation tim e. For low carrier density m aterials, th is average is over th e non-degenerate M axw ell-B oltzm ann d istrib u tio n function, i.e.,
7 4 e x p ( - u ) E
w here is th e B o ltzm an n ’s constant and T is th e tem p e ra tu re . T h e ste a d y s ta te solution of Eq. 4.3 is th en given by
v = - ^ E . (4.5)
4. T ran sp ort in G a N E pilayers 42
Com bining Eqs. 4.2 and 4.5 leads to
. .
,
,.,e|
w here th e positive scalar qu an tity
' - ^
(" I
is th e analytical expression for the electrical conductivity. T h e con d u ctiv ity is other- wisfe known as th e inverse of resistivity p, and can be expressed as
a = = nep , (4.8)
where
^ ^ (4-9)
is th e drift m obility of electrons. T he carrier concentration and drift m obility can n o t be o b tain ed from conductivity m easurem ents alone. T he Hall effect m easurem ent can be used to find n, while a com bination of these m easurem ents enables th e evaluation of p. Hence th e next section presents th e theoretical background on th e electrical c h aracterisation utilising th e Hall effect.
H all ch a ra cterisa tio n
W hen a current is injected in the x-direction in a sam ple and a m agnetic field is applied along th e z-direction, an electric field will build up in th e ^/-direction due to th e Lorentz force exerted on th e carriers (Hall effect). T h is Hall voltage exactly balances th e Lorentz force and can be m easured externally. T h e H all effect has been w idely used by m ost laboratories around the world as an im p o rta n t tool in characterising electrical p ro perties of sem iconductors. Its p o p u larity arises from th e fact th a t th e technique is sim ple to im plem ent, and also because of th e significance of th e p a ra m ete rs it can m easure, nam ely carrier concentration, m obility and resistivity.
E xperim entally, th e Hall m easurem ents em ploy th e van der Pauw (V D P) technique (van der Pauw 1958) where its square an d clover-leaf stru c tu re s are shown in Fig. 4.1. These shapes have m any advantages com pared w ith th e conventional H all b a r configu ration. T he V D P configuration gives a relatively large Hall effect a t th e sam e am ount
4. T ransport in G aN Epilayers 43
of heat dissipation, which is of im portance when m easuring m aterials of relatively low electron m obility (as in this case, GaN). Also, the influence of finite size contacts can be reduced significantly, provided the contacts are w ell-separated and positioned as sym m etrical as possible near the edge of the sam ple w ith th e resistance betw een any pair of contacts being less than approxim ately 10 MQ.
1
12
'32
43
4
F I G . 4 .1 . Hall measurements using the van der Pauw method. Shown are the simple square