Experimental Methods and Materials
3.5.2 Hall effect.
Hall effect measurements can be use to determine the carrier type and concentration for a semiconductor sample [14]. Figure 3.6 depicts the basic experimental setup of the Hall effect measurement, with an electric field applied along the x-axis and a magnetic field along the z-axis.
Va
Area=A
Figure 3.5. Hall effect measurement setup.
Consider a p-type sample. The magnetic Lorentz force on the hole i.e. q(v x B) =
q Vx Bz, will exert an upward-directed force on holes flowing in the x direction. This causes an accumulation of holes on the top surface of the sample, creating a downward-directed electric field Sy. In the steady state, no current flows along the y axis in the steady state, so Sy exactly balances the Lorentz force [15]:
(3.9) q Ey = q Vx Bz
Once this situation is established, holes travelling in the x direction experience no net force along the y direction and travel un-deflected through the semiconductor sample. The electric field Sy is termed the Hall field and its creation the Hall effect. Measurement of the potential difference between top and bottom surfaces yields the Hall voltage Vh. Substituting equation (3.4) into (3.9) yields: Jp (3.10) £y =
qp
Bz = Rh Jp Bz where; Rh = 1 / q pRh is termed the Hall coefficient and is inversely proportional to the carrier concentration. An analogous relationship for an n-type semiconductor can be developed, with Rhbeing negative: Rh = - 1 / q n
Determination of the Hall coefficient for a semiconductor sample therefore gives the carrier type and concentration. The Hall coefficient can be derived by measurement of the Hall voltage for a known current and magnetic field. From equation (3.4):
(3.11) RH = J p B z / q 8 y = { (I / A) B z }/q (V H /W ) = (I Bz W )/(q Vh A) where w is the sample thickness and A is the cross-sectional area. Again, the Van der Pauw technique can be used for Hall measurements [15]. Here, current is passed through contacts 1 and 3 and voltage measured across 2 and 4 (Figure 3.5). The measured Hall coefficient is averaged over all possible contact combinations and current and magnetic field polarities to minimise errors such as magneto-resistive and thermomagnetic effects [16].
Hall measurements and resistivity measurements can therefore be conducted on the same sample by the same apparatus, provided that the current and voltmeter terminals can be switched between all permutations of the four contacts on the sample.
3 .5 .3 V a ria b le te m p e ra tu re Hall m easurem en ts.
Donor and acceptor impurity states can be represented by energy levels Ed,a
in the semiconductor energy-band diagram [8]. The energy levels are measured with respect to either band-edge and located within the forbidden gap; Ed = Ec - Ed and Ea = Ea - Ey. A donor is neutral if it contains an electron and positively charged if the electron is excited (donated) to the conduction band. Similarly, an acceptor is negatively charged if it captures (accepts) an electron from the valance band and neutral if it remains empty. Both capture and emission processes are ionisation processes, so donors or acceptors can either be ionised or neutral [8]. Donor and acceptor energy levels are shown schematically in Figure 3.6.
Conduction Band
Donors
Energy Gap
Acceptors
Valence Band
The total number of donors (N o ) within an un-compensated n-type semiconductor equals the sum of neutral ( N °d) and ionised donors (N ^o ):
(3.12) Nd= N“d +
An analogous expression can be developed for acceptors within p-type material. Electron occupation probabilities for donors and acceptors for non-degenerate semiconductors are governed by Fermi-Dirac statistics [17]. The concentration of neutral donors within an n-type semiconductor is given by:
(3.13) N°d = Nd fpCEo}
where fF(Eo) is the value of the Fermi-Dirac distribution at donor energy Ed [17]:
(3-14) fF(Eo) = ^ ^
Substitution of equation (3.13) into (3.14) yields an expression for the ionised donor density [17]:
Ed - Ef
(3.15) N^D = Nd [ l - f p ( E D ) ] = Nd
1 + — e x D — k T
where Ef is the Fermi level and g equals the ground state degeneracy of the donor. For GaAs, g equals 2 for donors because the donated electrons can possess up or down spin. Acceptors in p-type material have a value of g equal to four, as electrons can be captured from the light- or heavy-hole band and possess either spin. A extrinsic (doped) semiconductor with a single dominant carrier type e.g. electrons in n-type material, possesses an equilibrium concentration of the other carrier type (holes) given by [8];
(3.16) p = n j ^ / n nj = intrinsic carrier conc. This is a very small figure, as n » ni at room temperature and these carriers are therefore termed minority carriers. The dominant carriers are called majority
carriers.
Charge neutrality must be observed in a doped semiconductor in equilibrium and is given by [8];
Negative = Positive (3.17) n + N'A = P + N^d
ionised donor and acceptor densities. For an uncompensated n-type semiconductor, the net electron concentration equals the ionised donor concentration, as the density of minority holes can be neglected:
(3.18) n = N^d
and by use of equation (3.13) a quadratic equation for the carrier concentration in non-degenerate material is obtained [7]:
(3.19) n ^ - Nn N e e g 1 + — n Nc 6
g
-Ed / kT =0
where Nc equals the conduction band density of states and the other symbols have their usual meaning. At low temperatures most electrons occupy donor states because the thermal energy required to excite electrons into the conduction band is not available and n is much smaller than Nq. Equation (3.19) reduces to: 1 / 2 (3.20) — N D N c 2 e x p -Ed 2 k T
As the temperature of the semiconductor is raised, more of the donors become ionised until full ionisation is achieved and the carrier concentration then saturates at a value equal to No. Increasing the temperature causes the
intrinsic carrier concentration n\ to rise and it finally surpasses the doping concentration, with valance band electrons being thermally excited directly into the conduction band [8]. These carriers then dominate the overall carrier concentration, which is seen to further increase. Three separate carrier concentration temperature regimes can be identified; a).Thermal ionisation; b). Saturation; c). Intrinsic and these are shown in Figure 3.8.
Carrier Conc. Ln (n) Intrinsic n = p a exp (-Eg / kT) Saturation n = No Ionisation n a exp (-Ed / 2kT) Reciprocal Temperature 1 / T
Rearranging equation (3.20) allows the donor ionisation energy Ed to be evaluated from the slope of graph:
where k is a constant. Measurement of the carrier temperature dependence allows determination of Ed. A similar treatment can be applied to p-type semiconductors, with Ea derived from the p vs 1 /T relationship.
Compensated semiconductors contain both donors and acceptors. This behaviour is seen in Si-doped GaAs, as silicon atoms can occupy either Ga (donor) or As (acceptor) sites. Both carrier types are therefore present in non trivial quantities, although one normally dominates. Carrier concentrations for compensated semiconductors are described by:
net n-type; n = - N'a
net p-type; p = N 'a -
The treatment used for a single carrier type semiconductor can be applied to compensated semiconductors, but neither carrier type can be neglected for this case. In a lightly-compensated n-type semiconductor it can be shown that for low temperatures [8]; 1 / 2 (3.22)
n
=
— Nc
2
N d-N Aexp
-Ed
kT
N
ACarrier density temperature dependence for compensated n-type material is half that for an un-compensated one. However, as the measurement temperature is increased, Nd » % Nc exp(-Ed / kT) » N'a and equation (3.22) reduces to equation (3.20). Therefore, donor or acceptor energy levels (Ed,Ea) for compensated semiconductors can only be determined to within a factor of two by this technique. A mass action law approach can be utilised to derive the same results [8].
3 .5 .4 Hall sa m p le prep aratio n.
Hall samples consist of approximately 2 to 5mm square portions of the sem iconductor wafer, that have been cleaved to shape. Cleaving was achieved by scribing a deep groove into the epitaxial layer using a diamond scriber. The groove was then aligned along the edge of two sandwiching glass slides and a
third slide used to apply sufficient pressure to break off the unwanted section of semiconductor. This procedure was repeated until the desired shape was achieved.
The epilayer surface was cleaned by consecutive boiling for 2 minute intervals, in acetone and isopropanol baths heated on a hotplate set at 150-C. Clean indium wire segments (0.5mm x 0.5mm) were placed at the extreme corners of the sample and the assembly furnaced at 400-0 for a 3 minute period, within a forming gas atmosphere. Molten In diffused into the semiconductor surface and created an ohmic contact with the doped layer. For p-type samples In/Zn wire was used, as zinc is a p-type dopant in GaAs and similarly In/Sn wire was used for n-type material, tin being an n-type dopant.
The generated contacts were tested for their ohmic nature by an IV curve tracer and if satisfactory, the sample was then measured on a Polaron Van der
Pauw machine.