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Admittance spectroscopy in junctions
Solid-State ElectronicsVol. 35, No. 3, pp. 285-297, 1992 Printed in Great Britain.All rights reserved
0038-1101/92 $5.00+ 0.00 Copyright © 1992PergamonPress plc
A D M I T T A N C E SPECTROSCOPY IN JUNCTIONS J. BARBOLLA, S. DUEgrASand L. BAIL6N Departamento de Electricidad y Electr6nica, Facultad de Ciencias, Universidad de Valladolid, 47011-Valladolid, Spain (Received 24 July 1991; in revised form 19 September 1991)
Abstract--The techniques based on the measurement of admittance used for characterization of deep levels are reviewed in this paper. Thermal admittance spectroscopy is a technique which allows the measurement of thermal emission rates and thermal activation energies of deep levels in junctions. Optical admittance spectroscopy allows the measurement of optical capture cross sections and optical threshold energies of deep levels in junctions. Thermal admittance spectroscopy can also be applied to III-V alloy junctions in order to characterize the thermal properties of DX centres, which typically appear in these materials. Finally, we show that optical admittance spectroscopy applied to junctions containing DX centres should yield the optical properties of these centres.
!. INTRODUCTION In this work we review the different techniques based on the measurement of the conductance and capacitance of a junction. From these measurements, the thermal and optical properties of the deep levels which are present in the bandgap of the semiconductor can be obtained. We show that these techniques can also be applied to DX centres in III-V alloys. Thermal admittance spectroscopy[l-7] is a technique which provides thermal emission rates and activation energies of deep levels from the variations of the capacitance and conductance of the junction as a function of temperature and frequency. In the optical admittance spectroscopy technique[8,9] the capacitance and conductance of a p - n junction under illumination are measured as a function of the photon energy hv and of the frequency of the measuring signal co. From these measurements, the optical capture cross sections of deep levels can be determined as a function of the photon energy hv. Also, the optical threshold energies of deep levels can be obtained. These are the minimum photon energies necessary to emit electrons from the deep level to the conduction band or to emit holes from the deep level to the valence band. The DX centres are the levels most commonly studied in III-V semiconductor alloys, such as A1GaAs and GaAsP. This is due to the fact that on many occasions these centres control the electronic properties of these materials. The DX centres exhibit a behaviour which is a mixture of that of the deep levels and of the shallow impurities. They show the emission and capture processes typical of traps while, at the same time, they control the free electron concentration. It has been shown[10,11] that the thermal admittance spectroscopy technique can also be used for the characterization of these kinds of centres. The reason for this is that the DX centres are
partially ionized at the experimental temperatures and the resulting space charge distribution is similar to that of a junction containing a shallow donor and a deep level. In the same way as thermal admittance spectroscopy applied to a junction with DX centres can yield their thermal emission rates, optical admittance spectroscopy should give the optical capture cross-sections of these centres. In this paper we review the main features of these techniques. They can all be treated with a common mathematical approach which stems from the resolution of Poisson's equation in integral form. Thus, the capacitance and conductance of the junction for each of the different above-mentioned situations are obtained. Admittance techniques are also compared to techniques based on transient measurements, such as DLTS, DLOS, CC-DLOS, etc. showing their main advantages and disadvantages, as well as the limitations of each method. 2. THERMAL ADMITTANCE SPECTROSCOPY OF DEEP
LEVELS Thermal admittance spectroscopy[I-7] is a technique which yields thermal emission rates of deep levels from the variations of capacitance and conductance of a junction as a function of temperature and frequency. These variations are due to the change in frequency of the measuring signal with respect to the time constant of charge and discharge processes of the deep levels around the point of the space charge region where both time constants are equal. Under dark conditions, this point coincides with the crossing point of the Fermi level with the deep level. Let us consider, for example, a p +-n junction with a shallow donor concentration ND and a deep level with concentration Nx lower than N D, located in the upper-half of the bandgap at an energy Er below the conduction band: We shall assume that the junction
285
286
J. BARBOLLAet al. P~
N
.................
I-eU
a)
Ec EFn ET
....... i
Ev
i i
P(xJI
bJ
I........ r i [
yI
w
ff
Fig. 1. (a) Energy-band diagram of a reverse-biased p +-n junction containing an acceptor deep level in the upper-half of the bandgap. (b) Space-charge distribution under the darkness condition. The y-point is defined by the condition e t = c,n(y).
is in the dark and is reverse biased with a voltage Ua, which is the sum of a d.c. component Ua and a small a.c. component U~. Figure 1 shows the band diagram and the space charge distribution in the spacecharge region of the junction. The y-point is the point within the space-charge region where the time constants of the thermal emission and capture processes of electrons are equal, e , -_t c , n ( y ) . It coincides with the crossing point of the Fermi level with the deep level. Since the electron profile within the space charge region is exponential, at the right of the y-point the capture processes will prevail over the emission processes, e t ,~ c , n ( x ) , and the deep level will be full of electrons. To the left of the y-point the emission processes will prevail over the capture processes, et>> c ~ n ( x ) , and the deep level will be empty of electrons. The sinusoidal component of the voltage modulates the charge stored in the junction at the y-point and at the edge of the space charge region w. The w-point can follow any voltage variation since this point is determined by free electrons, whose response is given by the dielectric relaxation time constant. At the experimental frequencies this response can be considered instantaneous. Let us now assume that the junction is at a constant temperature and that the frequency of the modulating signal is changed. At frequencies below the thermal emission rate of the deep level, this level follows the voltage variations and there are no energy losses, and therefore the junction conductance is zero. The charge modulated by the alternating signal is the sum of the charge due to free electrons at the w-point and the charge modulated at the y-point due to the deep level. Therefore, the junction responds to the low-frequency signal like a capacitance CLF, equal to the sum
of the capacitances associated to the two charge components mentioned above. As frequency increases, the response of the deep level at the y-point cannot follow the voltage variations. Since the emission and capture processes of the deep level at the y-point are not instantaneous, energy losses associated to non-equilibrium situations arise. In this situation electrons can be trapped at deep levels above the Fermi level. Alternatively, there can be free electrons at the conduction band with energies higher than that of equilibrium. This energy has to be supplied by the a.c. current. Hence, in the response of the junction to the alternating signal a non-zero conductance appears associated to these energy losses. As frequency increases the energy losses will be higher and the conductance, in turn, will also increase. At very high frequency the deep level cannot follow the voltage variations. The energy losses and the conductance reach their maximum. In summary, the G(~o) plot (Fig. 2a) goes from a zero value at low frequency up to a maximum at high frequency, showing an inflexion point which, as will be seen later, is located at frequencies close to the thermal emission rate. As for the C(og) plot, at low frequency the capacitance value is the maximum CLF, since the charge modulated is the sum of that corresponding to the edge of the space-charge region and the charge modulated at the y-point where the deep level can follow the voltage variations. As frequency is increased the deep level cannot follow the voltage variations and the charge modulated at the y-point decreases, which leads to a decrease in the junction capacitance. At high frequency the charge at the y-point cannot be modulated at all and, consequently, the capacitance has only one component, which is due to the modulation at the w-point, and
T =const.
(b)
w=const C
J
ent<<
e >>
Fig. 2. (a) Capacitanceand conductance vs frequency. (b) Capacitance and conductancevs temperatureat a constant
frequency.
Admittance spectroscopy in junctions its value CnF will be minimum. In conclusion, the C(co) curve (Fig. 2a) goes from a high value CLF at low frequency, to a low value CHF at high frequencies, and show an inflexion point located at frequencies close to the thermal emission rate. Let us now consider the case where instead of keeping the temperature constant, the frequency of the measuring signal is fixed and the temperature is scanned. At high temperatures the thermal emission rate has a value which is much higher than the frequency, e~, ,> co, and then the deep level responds instantaneously to the voltage variations. This situation is as that of the low-frequency case: there are no energy losses and the charge modulated is maximum. Therefore, at high temperature the conductance of the junction is zero and the capacitance takes its low-frequency value CLF- As temperature decreases the thermal emission rate of the deep level decreases and the centre cannot respond immediately. In consequence, there are energy losses and the total modulated charge decreases. Therefore, there will be a non-zero conductance value and the capacitance will be lower than CLF. At the same time as the thermal emission rate decreases the electron concentration at the y-point, n ( y ) = e ~ , / c , decreases too. In other words, as temperature decreases the amount of electrons involved in the charge and discharge process of the deep level decreases. At low temperatures the thermal emission rate is zero and, hence, the electron concentration involved in the modulation of the y-point also becomes zero and then there is no charge modulation at that point. In consequence, there are no energetic losses and the junction conductance at low temperature goes to zero again. The modulated charge will have only the component due to the free electrons at the edge of the space charge region and the capacitance will take, therefore, its high frequency value CHFIn summary, the G ( T ) plot (Fig. 2b) vanishes at low and high temperatures and reaches a maximum at a temperature at which the thermal emission rate is equal to the frequency. In the C ( T ) plot (Fig. 2b), the capacitance goes from a maximum value CLF at high temperatures to a minimum, CHF at low temperatures and exhibits an inflexion point at temperatures at which e', is close to the measuring signal frequency co. In the following paragraphs the junction admittance is derived and the exact conditions defining the positions of the maxima and of the inflexion points are determined. The charge stored in the space-charge region of the junction of Fig. 1 can be derived through the Poisson equation, which relates the electric potential and the electrostatic charge:
/'v
Q =
71/~
-2, J0 p(V)dVj ,
the semiconductor. The electrostatic charge is given by: p ( V ) = e ( N D - n - n.r ),
(2)
where e is the magnitude of the electron charge, n is the free-electron density and nr is the density of deep levels filled with electrons. Let Ua and Ub be the absolute values of the applied voltage and of the built-in voltage of the junction, respectively. The voltage drop across the spacecharge region will be: U ~-- --(Ua-{- Ub).
(3)
The voltage applied to the junction is the sum of a d.c. component Ua and a sinusoidal component U~" of frequency co and amplitude 6 Ua: Ua = U~ + R{Va },
(4)
where Ua = 6Ua expjcot and 6 U a ~. U ~ . The voltage in each point of the space-charge region will also be the sum of a d.c. component and an a.c. component: u = u = + R{U-},
(5)
where U = = - ( U ~ + Us) and U ~ = - U ~ ' . Since the amplitude of the a.c. signal is very low, we can develop all the magnitudes up to first order, and write:
V=Vo+R{V-}, n=n
= + R{n ~},
nT = nav + R{n~ }.
(6)
For a Maxwell-Boltzmann free-carrier distribution in the space-charge region: eV
n = nu exp K T "
(7)
Then, we have: eV =
n==nuexp KT' eV ~
n ~ = n = •--
KT
(8)
The density of deep levels filled with electrons follows the Shockley-Read-Hall statistics. In the ease of a deep level located in the upper-half of the bandgap of an n-type semiconductor, that is written a s : dn----S = e , n ( N T -- nv ) -- et, nv. dt
(9)
Developing this equation and neglecting secondorder terms, the expressions of the stationary and sinusoidal components of the density of deep levels filled with electrons are obtained:
(I)
where V is the voltage drop across the space-charge region and E is the dielectric permittivity of SSE 35/3--E
287
dnr dt
O+jom~
=c.n=(Nr-n~)-e~.n~
+e.n~(Nr-n~)-c.n'n~-et.n~
(10)
288
J. BARBOLLAet al.
and then
From eqn (13), a first-order approximation for the current yields:
Cn n =
r / T c.n - - = - +- e t NT'
C = CLFF ( o ) , G = CLFCOH(CO),
Cnetn n ~ =
n-? =(jco + c . n
,
,
+e.)(c.n= +e.)
N T.
where
Replacing eqns (2), (8) and (11) in eqn (1), the following expression for the charge stored in the space-charge region of the junction is obtained:
Q =(2eQ,/2[R{_foVNDdV+f U nN exp ~e V
fo
K T nN
rl ~ d g
cnen
The first two integrals correspond to the charge contribution of both ionized shallow donors and the free electrons. The third integral is the stationary term contribution of filled traps. The fourth integral corresponds to the deep level charge-state variation due to the sinusoidal component of the voltage. Solving these integrals, the following expression for the charge is obtained: KT n N
e ND
NT EF ND e - -
e ( U : + Ub)J"
ET
ND ~] e ( U : + Ub) --
co \
°
arctan -27 - arctan en
°)]
cnnN
and H(co)=~X /
(17)
EF--ET et. e(U7 + Ub)2co
x[In(l+e~f)-In(l+~) ]. (14) O92
Taking the derivative of the expression of the charge with respect to time, we can obtain the following expressions for the capacitance and conductance of the junction: C = 1 (dQ/dt)q co 6 Ua ' G = (dQ/dt)ph 6Ua '
(18)
Finally, taking into account that at any temperature in the neutral region it is verified that e t < c.nu and that around the maximum or inflexion point co - e~., we can write, around these points, the expressions of the capacitance and conductance of the junctions: et C ~-- Cav -~- (CLF -- CHF )
G-
(CLF
-m
co
co arctan
_-7, en
1+
--
.
(19)
2 The inflexion point of the capacitance as a function of the frequency is given by: e~(T)=
where
-
]
e
ET
+ F(co)6UaCOScot + H(co)6Uasincot) 1/2, (13)
EF-
o
}]= (12)
F(co) = 1 _NT /
NT EF -- ET'~ 1/2,
e N D N
is the low-frequency capacitance, while for a highfrequency (co--*co), we obtain:
NT(jco+c.n=+et)(c.n=+et)
Q(2eeND)I/2(U a + Ub
/o N \m/
dV
+tUNT c. n= jo c.n = + e', d V
+
(16)
(11)
(15)
where (dQ/dt)q and (dQ/dt)ph are, respectively, the current components in-quadrature and in-phase with the applied signal.
col . 0.825
(20)
The inflexion point of the conductance as a function of the frequency verifies: et.(T)=coi
(21)
.
Therefore, the C(co) and G(co) plots at a fixed temperature allow us to obtain the thermal emission rate of the deep level at that temperature. On the other hand, from eqn (19) we deduce that the maximum of the conductance as a function of temperature is located at: etn(Tm) --
co
(22)
1.980
and, if we assumed for the thermal emission rate the expression: e~(T) = A T ~ exp
ET KT'
(23)
which, apart from the physical significance of the A and E parameters, is known to be always followed on the experimental range of temperature, it follows from eqn (19) that the inflexion point of the C ( T ) plot, at a constant frequency, is defined as: co
et~(T~) = 1.825"
(24)
Admittance spectroscopy in junctions So, the C(T) and G(T) measurements allow us to determine the variation of the thermal emission rate with temperature and, from that, we can obtain the thermal activation energy ET of the deep level. In summary, we can see that thermal admittance spectroscopy yields the thermal emission rates of the deep levels from the capacitance and conductance curves as a function of frequency, at a constant temperature, or from the plots obtained varying the temperature at a fixed frequency. However, the measurements made scanning the temperature are most convenient, since they present two advantages from the experimental point of view: Firstly, the conductance curves as a function of temperature show maxima. The measurement of these maxima are much easier than that of the inflexion points obtained when the frequency is scanned. On the other hand, the curves obtained varying the temperature are made at only one frequency, and it is only necessary to adjust the phase shift introduced by the experimental circuit at the beginning of the experiment. Therefore, the G(T) plots give a higher accuracy, continuous curves and are less time-consuming than the G(co) plots. For these reasons, admittance measurements of deep levels in junctions are usually performed by measuring the conductance as a function of temperature at a given frequency. The heights of the peaks of conductance are obtained substituting eqn (22) into eqn (19): G(Tm) = 0.402~(CLF -- Car )
NT / EF-- ET
= 0"402°9 ~D ~/e (-~ -~ Ub)"
(25)
From this expression it is possible to obtain the deep-level concentration once the Fermi level and the thermal activation energy E T are known. In summary, thermal admittance spectroscopy is a technique which gives the thermal emission rates, the thermal activation energy and the concentration of a deep level located in the upper-half of the bandgap for an n-type semiconductor or in the lower-half for a p-type semiconductor. There are several interesting points that should be mentioned regarding the physical nature of the processes involved in admittance measurements: In the first instance, it should be noticed that the variations of the capacitance and conductance of the junction are due to the variation of the time constant of the charge and discharge processes of the deep level relative to the measuring signal frequency around the crossing point of the Fermi level with the deep level. At this point the electric field is low and the values obtained for the thermal emission rates will not be affected by possible dependencies with electric field.
289
Another important feature of thermal admittance spectroscopy is that it is a spectroscopic technique. This means that, for example, in the case of an n-type semiconductor with several deep levels located in the upper-half of the bandgap, the junction conductance will be the sum of the contributions of each of those centres whose thermal emission rates are close to the measuring signal frequency will contribute to the conductance. Deep levels with emission rates much higher or much lower than frequency will be in the low- or high-frequency cases, respectively, and will, therefore, have no contributions to the junction conductance. Hence, a plot of the conductance as a function of temperature will exhibit a maximum for each deep level located in the upper-half of the semiconductor bandgap. Experimental results reported by different authors using this technique show that the ability of thermal admittance spectroscopy to resolve two close levels is similar to that of DLTS. As for its sensitivity, thermal admittance spectroscopy has been able to detect deep levels in concentrations 106 times lower than the shallow impurities, which again is comparable to the DLTS sensitivity. Finally, with this technique it has been possible to measure emission rates of deep levels relatively close (about 100 meV) to the conduction band in n-type semiconductors or to the valence band in p-type semiconductors. However, thermal admittance spectroscopy cannot detect deep levels located in the lower half of the bandgap in n-type semiconductors or in the upper half of the bandgap in p-type semiconductors. Examples of experimental results obtained by thermal admittance spectroscopy for p +-n junctions containing deep levels related to Ti in Si[6] are shown in Fig. 3. Several curves of G(T) and C(T) at different frequencies are plotted. The two peaks appearing in
I
I
1 Hz -
I
-
~
10 Hz
100 Hz 1000 Hz
.4 B Q
G
I 100
I 200 T(K)
I 300
Fig. 3. Plots of C and G/to vs T for a p+-n junction containing Ti at several frequencies[6].
290
J. BARBOLLAet al. P÷
N
t
I I
......i-!!
y' W X" Fig. 4. (a) Energy-band diagram of a reverse-biasedp +-n junction containing an acceptor deep ]eve] in the upper-half of the bandgap. (b) Space-charge distribution under the
illumination condition. The y' point is defined by the condition et~+ e~ = c,,n(y'). the conductance curves and the inflexion points of the capacitance curves reveal the presence of two energy levels associated to Ti in n-type Si. The Arrhenius plots obtained from the positions of the maxima and of the inflexion points indicate that Ti in Si creates two deep levels located at 238 and 512 meV below the conduction band. 3. OPTICAL
ADMITTANCE
SPECTROSCOPY
OF
DEEP
LEVELS
Optical admittance spectroscopy[8] is a technique which allows the measurement of the optical emission rates of a deep level from the variations of the capacitance and conductance of a junction under monochromatic light when the photon energy hv and measuring signal frequency co are changed. Once the optical emission rates of the deep levels have been measured the optical capture cross-sections can be determined from the following expressions: e ° = a~,(hv), dp(hv), ep = a~ (hv ) . dp(hv ).
(26)
where e ° and e~ are the optical emission rates of electrons and holes, respectively, a~° and a~ are the optical capture cross-sections of electrons and holes, respectively and dp(hv) is the photon flux of the monochromatic light. In addition to this, the spectra of the optical emission rates of a deep level yield the optical threshold energies, E ° and E~, which are the minimum photon energies necessary to the emission of an electron to the conduction band or of a hole to the valence band, respectively. Let us deal, as in the preceding section, with a p +-n junction with a deep level located in the upper-half of the bandgap and in a concentration lower than that
of the shallow impurities. We shall assume that the junction is illuminated by a monochromatic photon flux q~(hv) and that the temperature is low enough that the thermal emission rate of electrons is much lower than the optical emission rates, et, <~e ° + e~. The band diagram and the space-charge distribution of the reverse-biased junction are shown in Fig. 4. The y'-point is the point within the space-charge region where the time constants of the emission and capture processes of the deep level are equal, c , , n ( y ' ) - e , - o +e~,. Since the electron density at y , n(y') = (e ° + e~)/c,,, is much higher than that at the crossing point of the Fermi level with the deep level, n ( y ) = et,,/c,,, the y'-point will be located at the right of y and, therefore, the y'-point will be closer to the neutral region. To the right of y ' the capture processes are predominant and the deep levels are full of electrons. To its left emission processes are predominant and the ionization factor of the centre is determined by the balance between electron and hole emissions, f . r -_e p /o( e , , o +e~). In the case where the optical emission rate of electrons is much higher than that of holes the deep level will be empty of electrons to the left of y ' . In any other case f r will have some value between 0 and 1. The a.c. voltage component will modulate the space-charge region at y ' and w. The response to this voltage at w is given by the dielectric relaxation time constant and can be considered instantaneous for all experimental frequencies. The response at y ' is determined by the optical emission rates of the deep level and will be faster the larger these rates are with respect to frequency co. Optical admittance spectroscopy measurements are carried out changing the photon energy hv at a given temperature and frequency. Let us assume that the photon energy hv is low enough that the optical emission rates are much lower than the frequency e ° + ep ,~ co. Under such conditions the deep level cannot respond to the modulating signal, the electron concentration at y ' is very low and there are no energy losses. Therefore, the conductance is zero and the capacitance value is the high-frequency value CHF, since only the charge modulated at w will contribute to it. At a photon energy high enough, the optical emission rates will be much higher than the frequency e ° +e~ >>co. In consequence, the deep level will follow the voltage variations, there will be no energy losses and the charge modulated will be the sum of the contribution at y ' and w. Therefore, the conductance will be zero again and the capacitance value will be the low-frequency value, CLF. At intermediate photon energies the deep level cannot fully follow the voltage variations, there will be energy losses and the charge modulated at y ' will be less than in the low-frequency case. In consequence, the conductance curves as a function of hv take a zero value at low and high energies and show a maximum at a photon energy at which the total optical emission rate is close to the frequency. The
291
Admittance spectroscopy in junctions uJ=const
I
where the low- and high-frequency capacitance are given by:
values of the
(30)
h-4
Fig. 5. Capacitance and conductance vs photon energy at a constant temperature and frequency. capacitance curves go from a low value, Cur at low photon energy, to a high value CL, at high energies and exhibit an inflexion point at a photon energy at which the total optical emission rate ez + ei, is close to the frequency w. These curves are depicted in Fig. 5. The calculation of the expressions of the capacitance and the conductance of the junction as a function of photon energy hv is similar to that developed in the thermal case. Under illumination conditions and at temperatures low enough that thermal emission rates can be neglected, the Shockley-Read-Hall equation can be written as:
dnr
ht- -
-e&
Solving this equation terms we find: n; =
c,n’+e’
+ (c,n + 67;)(NT - nT).
and neglecting
(27)
second-order
-NT,
c,n’+ez+e,”
c,ein-
n;=(iw
+
NT.
c,nE+e~+epO)(cnn=+e~+epO)
(28) The electrostatic charge density and the free electron distribution are again given by eqns (2) and (8), respectively. Substituting all these expressions into Poisson’s equation, the charge stored in the space-charge region can be obtained. Taking the derivative of this expression with respect to time, the following expressions for the capacitance and conductance of the junction are derived:
e,O+eO -112 ez ----NT KT In P 3 ) ND e ez+e,O wN l---
C,F=CLF
[
eZ NT ei+e,oN, KT 1 --1nP e Ua=+Ub
e,”+ e” wN
(2%
From eqn (29) we can deduce that the curves of conductance as a function of photon energy at constant temperature and frequency show a maximum defined by the condition:
Mwd
w
+ e,“@v,)= 1.980.
(31)
The height of the maxima is given by: i/2 G(hv,) = 0.4020 2
w l.980c,nN >
y&In a
b eZ(hv,)
’ eX(hv,) + e,“(hv,) ’
(32)
Thus, the curves of the conductance as a function of hv at constant T and w enables us to determine the spectral shape of the optical emission rates of electrons ei(hv) and holes ei(hv). To determine the exact condition satisfied by the inflexion points of the capacitance it would be necessary to have a prior knowledge of the dependence of the optical emission rates of electrons and holes on the photon flux. However, unlike the thermal case, this dependence is unknown before measurements are made. In order to estimate the sensitivity of optical emittance spectroscopy, we can evaluate the ratio between the peak height obtained by this technique and that obtained by thermal admittance spectroscopy for the same measuring frequency. From eqns (25) and (32):
WV,)
CW,)
G(T,)=e~(hv,)+e,“(hv,)
T
0 if
112
’
(33)
where To is the temperature at which the optical admittance spectroscopy measurements are performed. Obviously, To < T,,, to meet the condition that the thermal emission rate can be neglected. From eqn (33) we can see that the sensitivity of optical admittance spectroscopy is approximately equal to that of thermal admittance spectroscopy except when the optical emission rate of electrons is much lower than the optical emission rate of holes. When in the semiconductor bandgap there is more than one optically active deep level, optical admittance spectroscopy reveals itself as a useful tool as a spectroscopic technique. It is so because each of those deep levels will contribute to the conductance with a non-zero value only when the photon energy is such that the optical emission rate of a deep level is close to the measuring signal frequency w. Deep levels with
292
J. BARBOLLAet al.
T • 1,50 K
1 Hz
T = 200 K
"2.
1.5 Hz
//~
T • 270 K
0.7
0.9
1.1
1.3
1.5
hv (eV)
Fig. 6. Plots of G/to vs photon energy for a OaAs Schottky barrier containing the EL2 centre at T = 150, 200 and 250 K[9].
optical emission rates much higher or much lower than ~ will be in the low- or high-frequency condition and will, therefore, have no contribution to the conductance. The plot of the conductance as a function of the photon energy hv will show a maximum for each optically active deep level located in the upper-half of the bandgap. Among the different techniques for optical characterization of deep levels based on transient measurements, the most commonly used are DLOS (deep level optical spectroscopy)[12] and CC-DLOS (constant-capacitance deep level optical spectroscopy) [13]. These techniques determine the optical emission rates of the deep levels for the slope at the origin of the photocapacitance or photovoltage transients. When there is more than one deep level in the semiconductor bandgap, the initial slope of those transients cannot separately yield the contributions of each level, since it is the sum of all of them. For this reason, DLOS and CC-DLOS, unlike optical admittance spectroscopy, are not spectroscopic techniques.
Another advantage of optical admittance spectroscopy is that the conduction is measured in a small region, around the y'-point, where the electric field is low. Therefore, the optical emission rates thus measured will not be affected by the electric field present in the space-charge region. This electric field, however, does affect the measurements carried out by DLOS and CC-DLOS, since the optical emission rates are values averaged across the whole spacecharge region. It can be seen from eqns (31) and (32) that, when the electron and hole optical emission rates of a deep level are of the same order of magnitude, optical admittance spectroscopy can measure both of them. When the optical emission rate of majority carriers is much higher than that of minority carriers, only the former can be measured. Finally, if the optical emission rate of minority carriers is much higher than that of majority carriers the height of the conductance peaks is practically zero and none of them can be measured. The optical emission rates of a deep level are proportional to the photon flux and to the optical capture cross section. Therefore, in order to measure capture cross sections as low as possible, it is necessary to have an experimental set-up capable of measuring at very low frequencies while supplying a photon flux as high as possible. The need for using low frequencies makes optical admittance spectroscopy a slow technique. This is probably the main disadvantage of this technique from the practical point of view. In addition, since the a.c. current generated in the sample is proportional to the frequency, the measured values can be extremely small, and for this reason high-sensitivity electrometers are required. As an example, Fig. 6 shows some experimental curves[9] for the conductance as a function of photon energy, obtained by optical admittance spectroscopy applied to GaAs samples containing the EL2 level, at three different temperatues (150, 200 and 270 K). The measurements were performed within a temperature range where optical admittance spectroscopy can be applied to this level. This range is limited at both low and high temperatures. The upper limit is determined by the temperature at which thermal emission begins to be relevant, which occurs at temperatures above 300 K. The lower limit is imposed by photoquenching, a phenomenon characteristic of the EL2 level, by which this level becomes insensitive to any change in photon energy at temperatures below 140 K. From the positions of the conductance peak, the spectra of the photoionization cross-section of EL2 at the three experimental temperatures are obtained (Fig. 7). In these spectra the contributions corresponding to transitions from the EL2 level to each of the three conduction band minima of GaAs (F, L and X) can be seen. These spectra suggested that EL2 is formed by a family of levels located near the midgap of GaAs. This family can be explained by a microscopic
Admittance spectroscopy in junctions
T=150 K
•
°/° ~ar'-"
.oJ
0/° I
"°I
I
I
I
w T = 200 K
•
,/.-'o\
o/ o/
o~
o/O ~ ° / .e /
I
I
I
T=270 K
• o/'O-'~ON
/c "/
•
"\..
,d ° I
0.7
0.9
I
1.1 hv (eV)
I
1.3
293
Let us consider a p +-n junction of III-V alloy with a DX centre located at an energy Eo below the conduction band and in concentration Nd. In the dark, at the experimental temperatures (ED ~>KT) the ionization factor I of the DX centre in the neutral region coincides with the number of ionized centres (about 1-2%). The charge distribution within the spacecharge region is shown in Fig. 8. The edge of the space charge region w0 is defined as the point where the electron concentration is half of the concentration in the neutral region. To the left of w0, the electron concentration decreases rapidly and there will be a t At this point the time point wI where c,n(w])= e,. constant of the emission and capture processes of the DX centres are equal and the ionization factor of the centre is 1/2. This point coincides with the crossing point of the Fermi level with the energy level of the DX centre and is separated from the edge of the space-charge region by a distance which is of the order of the extrinsic Debye length LD(T). To the left of wl the emission processes prevail and therefore the DX centre is completely ionized. To the right of w1the DX centre has the same ionization factor as in the neutral region. In the neutral region the number of electrons emitted by the DX centre per unit time must equal that of electrons captured. Taking into account that in the neutral region the electron concentration equals the number of ionized DX centres, we have:
I)Nd,
C,,(INd) 2 = e'.(1 -
1.,5
Fig. 7. Electron optical capture cross-section a ° vs photon energy for the EL2 centre in G a A s at T = 150, 200 and 270 K. The solid lines are the curves obtained using the semi-empirical model of Chantre and co-workers[12]. The points (o) are our experimental results[9].
from where the following expression, relating the emission and capture rates and the DX ionization factor, is derived: et
12
(35)
N. ,
c.
model in terms of Ar clusters, as has been proposed by Mochizuki and Ikoma[14].
P+
1-I
-°"
N
_................
4. ADMrVrANCE SPECTROSCOPY OF DX CENTRES
The electronic properties of III-V compounds are often controlled by centres exhibiting a behaviour which is a mixture of that of shallow impurities and that of deep levels. These centres are called DX centres or deep donors[15]. As shallow impurities they control the free electron concentration, while as deep levels they exhibit the emission and capture processes characteristic of a trap. In AlxGa~_xAs alloys these donors are relatively deep for x > 0.2 and control the material conductivity within the range 0.2 < x < 0.6. In recent works[10, 11], thermal admittance spectroscopy has been used to study DX centres in A1GaAs junctions. This is possible due to the fact that since at the experimental temperatures the ionization factor of DX centres is very low, the charge distribution within the space-charge region of the reversebiased junction is very similar to that of a junction with a deep level and a fully-ionized shallow impurity.
(34)
a]
l_eU
] .......
ii n Ev
eNd
1
I bl
w1
w0
x
Fig. 8. (a) Energy-band diagram of a reverse-biased p ÷-n junction containing an D X centre. (b) Space-charge distribution under the darkness condition. The wI point is defined
by the condition e~ = c.n(wO.
294
J. BARBOLLAet al.
A small a.c. component superimposed to the reverse bias of the junction will modulate the charge stored in the space-charge region at points wI and w0. At w0 the response is instantaneous since it is determined by the dielectric relaxation time constant. At wl, however, the response will depend upon the relaxation between the time constant of the charge and discharge processes of the centre and the frequency of the modulating signal. At low temperatures the thermal emission rate of the centre is zero. Therefore, the electron density at wi is zero and there is no charge modulation at this point. In consequence, the junction conductance is zero and the capacitance value is the high-frequency value CnF since only the charge modulated at the edge of the space charge region will contribute to the junction capacitance. At very high temperatures the thermal emission rate of the DX centre is much higher than the measuring signal frequency. Therefore, the charge modulated at wI can follow the voltage variations, there are no energy losses and, in consequence, the conductance is zero again and the capacitance returns to its low-frequency value CLF. At intermediate temperatures the DX centre cannot respond instantaneously to the voltage variations and hence energy losses arise which lead to a non-zero conductance while the capacitance values are between CLF and CHF* In summary, at a given frequency the conductance vs temperature plot will exhibit a maximum and the capacitance vs temperature will display an inflexion point. The temperatures at which the maxima and inflexion points appear are those at which the thermal emission rate of the DX centre has a value close to the measuring signal frequency. In order to determine the exact position of these points we will follow the same procedure already sketched in preceding sections for the derivation of the expressions of capacitance and conductance of a junction with a DX centre. In this case the expression of the charge stored in the space-charge region is given by: p = e(IN,~ - n),
(36)
where I is the ionization factor of the DX centre. The density of ionized DX centres will have a constant component I = N d and a sinusoidal component I ~Nd. These components can be obtained from the Shockely-Read-Hall statistics, which in this case are written as: dn d d[(1 - / ) N d ] -- = dt dt
etn(1 -- I ) N d + c, n l N d .
(37)
From this expression we obtain the two components of the ionization factor mentioned above: et 1 = = _ _ Cn n = + etn ' Cnn - e t
I~ =
(c,,n = + e t) (c~ + et,, + j o g ) "
(38)
Substituting eqn (38) into eqn (36) and assuming a Maxwell-Boltzmann free-electron distribution in the space-charge region, the charge stored in the spacecharge region is obtained from Poisson's equation. Evaluating the derivative of the charge with respect to time, the current generated in the junction is obtained. The in-phase and in-quadrature components of the current yield the following expressions for the capacitance and conductance of the junction: C : CHF "~-(CLF
-
2
CHF)
O9 arctan ~ , e'. O9 en
e~ln 1 +
,
(39)
where the low- and the high-frequency values of the capacitance are given by: =
- -
(u2
+
uu
e
\e(-6-~--~~i) j
'
(40)
Equations (39) are equal to that obtained in the case of junctions with a deep level and fully-ionized shallow donor. Therefore, the conductance and capacitance curves of a junction with a DX centre as a function of frequency, at a constant temperature, will display inflexion points defined by e t ( T ) = ~oi/0.825 and e~(T)=og~, respectively. When the measurements are made varying the temperature, at a constant frequency, the conductance curves will present maxima defined by the condition e t , ( T ~ ) = 0~/1.980, and the capacitance will show inflexion points which verify: et,(Ti) = o9/1.825. In summary, thermal admittance spectroscopy is a technique used for the characterization of deep levels in semiconductors which can also be applied to junctions of III-V alloys with DX centres. The advantages of thermal admittance spectroscopy as compared to those techniques based on transients are of particular relevance in the case of DX centres. Many works[16-19] have shown that the isothermal transients associated to DX centres are strongly non-exponential even in the case of constantcapacitance transients. This non-exponentiality is attributed to effects related to the electric field present in the space-charge region, or to the capture processes taking place in the Debye tail region, or to the fact that the ionization factor changes with temperature[18]. However, none of these effects will affect the results obtained by thermal admittance spectroscopy since the relevant processes involved in this technique occur at a point within the space-charge region, near the neutral region, where the electric field is low. Figure 9 shows several experimental curves[l l] corresponding to thermal admittance spectroscopy applied to A10.26Ga0.74Asjunctions doped with Si to a concentration of 3.5 x 1017crn -3. These curves show two maxima related to two DX centres associated to Si in AIGaAs. From the location of these
Admittance spectroscopy in junctions
100 Hz 500 Hz 1 KHz 5KH
100
200
300
T(K]
Fig. 9. Plots of G/co vs T for a p +-n junction of Si-doped A10.26Ga074Asat several frequencies[11]. maxima the activation energies of both centres were obtained. The thermal activation energy of the first level, called DX-I, was 370 meV whereas that of the second one, called DX-II, was 415 meV.
ation factor of the centre is 1/2• To the left of this point the emission processes are predominant and then the ionization factor is determined by the balance between the optical emission rates of electrons and holes• To the right of Wl the ionization factor of the DX centre will be equal to that of the neutral region• In consequence, we can see that the charge distribution in the space-charge region of a junction with a DX centre under illumination conditions is similar to that of a junction with a deep level and fullyionized shallow impurity. Therefore, optical admittance spectroscopy will be able to measure the optical emission rates of the DX centres. The charge density in the space-charge region will be given by eqn (36)• In this equation the ionization factor will be determined by the Shockley-Read-Hall statistics, which under illumination conditions are written as: d(1 - I)Nd
e°(1 -- I)N d + (c,n + e~)INd.
dt 5. O P T I C A L A D M I T T A N C E S P E C T R O S C O P Y CENTRES
O F DX
In the same way as thermal admittance spectroscopy yields the thermal emission rates of DX centres, the application of optical admittance spectroscopy to junctions with DX centres should yield their optical emission rates. Let us consider a junction with a DX centre under illumination and at a temperature low enough that the thermal emission rate is much lower than the optical emission rates. In the neutral region it must verify that the number of electrons emitted by the DX centre per unit time must equal that of electrons captured. The electrons captured can be free electrons captured from the conduction band or holes emitted to the valence band from the DX centre. Taking into account that in the neutral region the electron concentration equals the number of ionized DX centres, we have:
(c,I°Nd + ep)l°N~ = e°(1 - I°)Nd,
(41)
where I ° is the ionization factor of the DX centre in the neutral region under illumination conditions. From eqn (41) the following expression relating the emission and capture rates and the ionization factor of the DX centre can be obtained: e°
c, I°No + ep
Io - -
1 -- I °
Nd.
(42)
In the cases in which the optical emission rate of electrons is much higher than c, Nd and than ep, the DX centre is fully ionized and the situation is equal to that of a junction with only a shallow impurity. In any other case the DX centre is partially ionized in the neutral region, and the space-charge distribution in the space-charge region will be similar to that of Fig. 8. The w~ point will be defined now by the condition c,n(wj)= e ° + e$. In this point the ioniz-
295
(43)
From this expression the stationary and sinusoidal components of the ionization factor of the DX centre can be obtained: I=~
o en
c,n = + e ° + e ~ ' c n n ~ e n°
I ~
(c,n = + e ° + ep) (c~ + e ° + ep +jo9) (44)
Replacing these values in Poisson's equation, and taking the derivative of the charge with respect to time, the following expressions for the capacitance and conductance of the junction as a function of photon energy can be derived:
C(hv) = CuF + (CL~ -- Car) e° + ep arctan o9
e ° + ep'
G(hv)=(CLF--~CaF) (e° +e~)ln f 1 -~ (e°-+-e~) ] (45) where the low- and high-frequencies of the capacitance are given by: le£M
\1/2 /
KT e,o e e°+ep 0 "Jr- O'X -- 1/2
erl c,I°Nd],
xlne"
e° (KT 1 CHe=CLF l - - e ° + e ~ , \ e U ~ - U b e ° + e°\l/2-1 ×ln~/ /.
c,I Nd,] _]
(46)
Eqns (45) are equal to that obtained in the case of junctions with a deep level and a fully-ionized shallow donor. Therefore, the conductance of a junction
J. BARBOLLAet al.
296
with a DX centre as a function of photon energy hv, at constant frequency and temperature will present a maximum defined by the condition e~(hvm)+ e~(hvm)=Og/1.980. The height of the maximum verifies the following relation of proportionality:
e°(hv~) G(hvm)~C° e°(hvm) + e~(hv~)"
(47)
We see, therefore, that the application of optical admittance spectroscopy to junctions with DX centres should yield the electron optical emission rates of the DX centres provided that e ° is not much less than e~, whereas if both were of the same order of magnitude both the electron and hole optical emission rates would be measured. All the advantages of optical admittance spectroscopy applied to deep levels in junctions would be, then, translated automatically to the case of DX centres. The optical emission rates measured in this way will not be affected by possible dependencies on the electric fields, nor by the other effects responsible for the non-exponentiality of transients which are specially remarkable in the case of DX centres. In conclusion, admittance spectroscopy is a characterization technique of deep levels in junctions which can also be applied to junctions with DX centres, in order to measure the electron thermal emission rates and the electron and hole optical emission rates of these centres. 6. CONCLUSIONS
This work has reviewed the admittance spectroscopy techniques used in the characterization of deep levels and DX centres in junctions. Thermal admittance spectroscopy can be used to measure the thermal emission rates and thermal activation energies of deep levels from the variations of capacitance and conductance of the junction as a function of temperature and of measuring signal frequency. The measurement is carried out in a small region around the crossing point of the Fermi level with the deep level. At this point the electric field is low and the resulting values, therefore, are not affected by the electric field present across the spacecharge region. In the case where there is more than one deep level this technique separately yields the thermal emission rates of each level. Thermal admittance spectroscopy is, therefore, a truly spectroscopic technique. Its sensitivity and resolution powers are comparable to those of DLTS, a technique widely used in the characterization of deep levels. Using thermal admittance spectroscopy it has been possible to measure relatively shallow energy levels of about 100 meV. The range of possible energies of deep levels detectable with this technique is restricted to the upper-half of the bandgap in n-type semiconductors and to the lower-half in p-type semiconductors.
Optical admittance spectroscopy yields the optical emission rates of deep levels from the variations of capacitance and conductance of a junction as a function of the photon energy of a monochromatic light applied to the junction, at a given temperature and frequency. The measurement is performed in a small region around the point within the spacecharge region where the electric field is low and the resulting values are, therefore, not affected by such field. The sensitivity of optical admittance spectroscopy is similar to that of thermal admittance spectroscopy except when the optical emission rate of majority carriers is much less than that of minority carriers since, in this case, the conductance vanishes. When the optical emission rates of electrons and holes are of the same order of magnitude, this technique can measure both optical emission rates. When the optical emission rate of majority carriers is much higher than that of minority carriers, only the former can be measured. This technique is again a spectroscopic technique which can determine separately the optical properties of each optically active deep level present in the semiconductor bandgap. On the contrary DLOS and CC-DLOS, techniques commonly used in the optical characterization of deep levels, cannot separate the contribution of each level and give only the sum of all of them. The main drawback of optical admittance spectroscopy is the fact that it is a slow technique. This is so because the optical thermal emission rates are proportional to the photon flux and to the optical capture cross sections and, in order to measure optical cross-sections as low as possible, very low frequencies have to be used. Thermal admittance spectroscopy can also be applied to the characterization of DX centres in III-V compounds. These centres exhibit a behaviour which is a mixture of that of deep level and of shallow impurity. Accordingly, the charge distribution in the space-charge region of the junction is very similar to that obtained in a junction with a deep level and a fully-ionized shallow impurity. The use of thermal admittance spectroscopy is particularly advantageous in the case of DX centres since these centres exhibit strong non-exponentiality, which is commonly attributed to effects relating to the electric field present in the space-charge region, or to the capture processes occurring in the Debye tail region or to the variation of the ionization factor of the DX centre with temperature. None of these effects affect the results of thermal admittance spectroscopy. Finally, we have shown in this work how the application of optical admittance spectroscopy to junctions with DX centres should give their optical emission rates. The results obtained in this way would not be affected by any possible dependence on electric fields or by any other effects leading to the non-exponential isothermal transients particularly relevant in the case of DX centres.
Admittance spectroscopy in junctions REFERENCES
I. D. L. Losc¢, Appl. Phys. Lett. 21, 54 (1972). 2. G. Vincent, D. Bois and P. Pinard, J. appl. Phys. 46, 5173 (1975). 3. G. Vincent, Solid-St. Commun. 19, 559 (1976). 4. J. L. Pautrat, B. Katircioglu, N. Magnea, D. Bensahel, J. C. Pfister and L. Revoil, Solid-St. Electron. 23, 1159 (1980). 5. J. E. Carceller, P. Cartujo, J. R. Morante, J. Barbolla, J. C. Brabant and M. Brousseau, Rev. Phys. Appl. 15, 843 (1980). 6. J. R. Morante, J. E. Carceller, P. Cartujo and J. Barbolla, Solid-St. Electron. 26, 1 (1983). 7. M. Jaraiz, S. Duefias, J. Vicente, L. Bai16n and J. Barbolla, Solid-St. Electron. 29, 883 (1986). 8. S. Duefias, M. Jaraiz, J. Vicente, E. Rubio, L. Bail6n and J. Barbolla, J. appl. Phys. 61, 2541 (1987). 9. S. Duefias, E. Cast,in, A. de Dios, L. Bail6n and J. • Barbolla, J. appl. Phys. 67, 6309 (1990).
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10. S. Chakravarty, S. Subramanian, D. K. Sharma and B. M. Arora, J. appl. Phys. 66, 3955 (1989). 11. S. Duefias, I. Izpura, J. Arias, L. Enriquez and J. Barbolla, J. appl. Phys. 69, 4300 (1991). 12. A. Chantre, G. Vincent and D. Bois, Phys. Rev. B23, 5335 (1981). 13. J. Vilanova, S. Duefias, E. Rubio, L. Bail6n, J. Barbolla and E. Lora-Tamayo, Solid-St. Electron. 32, 287 (1989). 14. Y. Mochizuki and T. Ikoma, Jpn. J. appl. Phys. 24, 895 (1985). 15. D. V. Lang and R. A. Logan, Phys. Rev. Lett. 39, 635 (1977). 16. P. M. Mooney, E. Callcja, S. L. Wright and M. Heiblum, Defects in Semiconductors, Vols 10-12, p. 417. Material Science Forum (1986). 17. E. Calleja, P. M. Mooney, S. L. Wright and M. Heiblum, Appl. Phys. Lett. 48, 657 (1986). 18. J. C. Bourgoin, S. L. Feng and H. J. Von Bardelebcn, Appl. Phys. Lett 53, 1841 (1988). 19. P. J. Landsberg and E. H. Shaben, J. appl. Phys. 61, 5055 (1987).
0038-1101/92 $5.00+ 0.00 Copyright © 1992PergamonPress plc
A D M I T T A N C E SPECTROSCOPY IN JUNCTIONS J. BARBOLLA, S. DUEgrASand L. BAIL6N Departamento de Electricidad y Electr6nica, Facultad de Ciencias, Universidad de Valladolid, 47011-Valladolid, Spain (Received 24 July 1991; in revised form 19 September 1991)
Abstract--The techniques based on the measurement of admittance used for characterization of deep levels are reviewed in this paper. Thermal admittance spectroscopy is a technique which allows the measurement of thermal emission rates and thermal activation energies of deep levels in junctions. Optical admittance spectroscopy allows the measurement of optical capture cross sections and optical threshold energies of deep levels in junctions. Thermal admittance spectroscopy can also be applied to III-V alloy junctions in order to characterize the thermal properties of DX centres, which typically appear in these materials. Finally, we show that optical admittance spectroscopy applied to junctions containing DX centres should yield the optical properties of these centres.
!. INTRODUCTION In this work we review the different techniques based on the measurement of the conductance and capacitance of a junction. From these measurements, the thermal and optical properties of the deep levels which are present in the bandgap of the semiconductor can be obtained. We show that these techniques can also be applied to DX centres in III-V alloys. Thermal admittance spectroscopy[l-7] is a technique which provides thermal emission rates and activation energies of deep levels from the variations of the capacitance and conductance of the junction as a function of temperature and frequency. In the optical admittance spectroscopy technique[8,9] the capacitance and conductance of a p - n junction under illumination are measured as a function of the photon energy hv and of the frequency of the measuring signal co. From these measurements, the optical capture cross sections of deep levels can be determined as a function of the photon energy hv. Also, the optical threshold energies of deep levels can be obtained. These are the minimum photon energies necessary to emit electrons from the deep level to the conduction band or to emit holes from the deep level to the valence band. The DX centres are the levels most commonly studied in III-V semiconductor alloys, such as A1GaAs and GaAsP. This is due to the fact that on many occasions these centres control the electronic properties of these materials. The DX centres exhibit a behaviour which is a mixture of that of the deep levels and of the shallow impurities. They show the emission and capture processes typical of traps while, at the same time, they control the free electron concentration. It has been shown[10,11] that the thermal admittance spectroscopy technique can also be used for the characterization of these kinds of centres. The reason for this is that the DX centres are
partially ionized at the experimental temperatures and the resulting space charge distribution is similar to that of a junction containing a shallow donor and a deep level. In the same way as thermal admittance spectroscopy applied to a junction with DX centres can yield their thermal emission rates, optical admittance spectroscopy should give the optical capture cross-sections of these centres. In this paper we review the main features of these techniques. They can all be treated with a common mathematical approach which stems from the resolution of Poisson's equation in integral form. Thus, the capacitance and conductance of the junction for each of the different above-mentioned situations are obtained. Admittance techniques are also compared to techniques based on transient measurements, such as DLTS, DLOS, CC-DLOS, etc. showing their main advantages and disadvantages, as well as the limitations of each method. 2. THERMAL ADMITTANCE SPECTROSCOPY OF DEEP
LEVELS Thermal admittance spectroscopy[I-7] is a technique which yields thermal emission rates of deep levels from the variations of capacitance and conductance of a junction as a function of temperature and frequency. These variations are due to the change in frequency of the measuring signal with respect to the time constant of charge and discharge processes of the deep levels around the point of the space charge region where both time constants are equal. Under dark conditions, this point coincides with the crossing point of the Fermi level with the deep level. Let us consider, for example, a p +-n junction with a shallow donor concentration ND and a deep level with concentration Nx lower than N D, located in the upper-half of the bandgap at an energy Er below the conduction band: We shall assume that the junction
285
286
J. BARBOLLAet al. P~
N
.................
I-eU
a)
Ec EFn ET
....... i
Ev
i i
P(xJI
bJ
I........ r i [
yI
w
ff
Fig. 1. (a) Energy-band diagram of a reverse-biased p +-n junction containing an acceptor deep level in the upper-half of the bandgap. (b) Space-charge distribution under the darkness condition. The y-point is defined by the condition e t = c,n(y).
is in the dark and is reverse biased with a voltage Ua, which is the sum of a d.c. component Ua and a small a.c. component U~. Figure 1 shows the band diagram and the space charge distribution in the spacecharge region of the junction. The y-point is the point within the space-charge region where the time constants of the thermal emission and capture processes of electrons are equal, e , -_t c , n ( y ) . It coincides with the crossing point of the Fermi level with the deep level. Since the electron profile within the space charge region is exponential, at the right of the y-point the capture processes will prevail over the emission processes, e t ,~ c , n ( x ) , and the deep level will be full of electrons. To the left of the y-point the emission processes will prevail over the capture processes, et>> c ~ n ( x ) , and the deep level will be empty of electrons. The sinusoidal component of the voltage modulates the charge stored in the junction at the y-point and at the edge of the space charge region w. The w-point can follow any voltage variation since this point is determined by free electrons, whose response is given by the dielectric relaxation time constant. At the experimental frequencies this response can be considered instantaneous. Let us now assume that the junction is at a constant temperature and that the frequency of the modulating signal is changed. At frequencies below the thermal emission rate of the deep level, this level follows the voltage variations and there are no energy losses, and therefore the junction conductance is zero. The charge modulated by the alternating signal is the sum of the charge due to free electrons at the w-point and the charge modulated at the y-point due to the deep level. Therefore, the junction responds to the low-frequency signal like a capacitance CLF, equal to the sum
of the capacitances associated to the two charge components mentioned above. As frequency increases, the response of the deep level at the y-point cannot follow the voltage variations. Since the emission and capture processes of the deep level at the y-point are not instantaneous, energy losses associated to non-equilibrium situations arise. In this situation electrons can be trapped at deep levels above the Fermi level. Alternatively, there can be free electrons at the conduction band with energies higher than that of equilibrium. This energy has to be supplied by the a.c. current. Hence, in the response of the junction to the alternating signal a non-zero conductance appears associated to these energy losses. As frequency increases the energy losses will be higher and the conductance, in turn, will also increase. At very high frequency the deep level cannot follow the voltage variations. The energy losses and the conductance reach their maximum. In summary, the G(~o) plot (Fig. 2a) goes from a zero value at low frequency up to a maximum at high frequency, showing an inflexion point which, as will be seen later, is located at frequencies close to the thermal emission rate. As for the C(og) plot, at low frequency the capacitance value is the maximum CLF, since the charge modulated is the sum of that corresponding to the edge of the space-charge region and the charge modulated at the y-point where the deep level can follow the voltage variations. As frequency is increased the deep level cannot follow the voltage variations and the charge modulated at the y-point decreases, which leads to a decrease in the junction capacitance. At high frequency the charge at the y-point cannot be modulated at all and, consequently, the capacitance has only one component, which is due to the modulation at the w-point, and
T =const.
(b)
w=const C
J
ent<<
e >>
Fig. 2. (a) Capacitanceand conductance vs frequency. (b) Capacitance and conductancevs temperatureat a constant
frequency.
Admittance spectroscopy in junctions its value CnF will be minimum. In conclusion, the C(co) curve (Fig. 2a) goes from a high value CLF at low frequency, to a low value CHF at high frequencies, and show an inflexion point located at frequencies close to the thermal emission rate. Let us now consider the case where instead of keeping the temperature constant, the frequency of the measuring signal is fixed and the temperature is scanned. At high temperatures the thermal emission rate has a value which is much higher than the frequency, e~, ,> co, and then the deep level responds instantaneously to the voltage variations. This situation is as that of the low-frequency case: there are no energy losses and the charge modulated is maximum. Therefore, at high temperature the conductance of the junction is zero and the capacitance takes its low-frequency value CLF- As temperature decreases the thermal emission rate of the deep level decreases and the centre cannot respond immediately. In consequence, there are energy losses and the total modulated charge decreases. Therefore, there will be a non-zero conductance value and the capacitance will be lower than CLF. At the same time as the thermal emission rate decreases the electron concentration at the y-point, n ( y ) = e ~ , / c , decreases too. In other words, as temperature decreases the amount of electrons involved in the charge and discharge process of the deep level decreases. At low temperatures the thermal emission rate is zero and, hence, the electron concentration involved in the modulation of the y-point also becomes zero and then there is no charge modulation at that point. In consequence, there are no energetic losses and the junction conductance at low temperature goes to zero again. The modulated charge will have only the component due to the free electrons at the edge of the space charge region and the capacitance will take, therefore, its high frequency value CHFIn summary, the G ( T ) plot (Fig. 2b) vanishes at low and high temperatures and reaches a maximum at a temperature at which the thermal emission rate is equal to the frequency. In the C ( T ) plot (Fig. 2b), the capacitance goes from a maximum value CLF at high temperatures to a minimum, CHF at low temperatures and exhibits an inflexion point at temperatures at which e', is close to the measuring signal frequency co. In the following paragraphs the junction admittance is derived and the exact conditions defining the positions of the maxima and of the inflexion points are determined. The charge stored in the space-charge region of the junction of Fig. 1 can be derived through the Poisson equation, which relates the electric potential and the electrostatic charge:
/'v
Q =
71/~
-2, J0 p(V)dVj ,
the semiconductor. The electrostatic charge is given by: p ( V ) = e ( N D - n - n.r ),
(2)
where e is the magnitude of the electron charge, n is the free-electron density and nr is the density of deep levels filled with electrons. Let Ua and Ub be the absolute values of the applied voltage and of the built-in voltage of the junction, respectively. The voltage drop across the spacecharge region will be: U ~-- --(Ua-{- Ub).
(3)
The voltage applied to the junction is the sum of a d.c. component Ua and a sinusoidal component U~" of frequency co and amplitude 6 Ua: Ua = U~ + R{Va },
(4)
where Ua = 6Ua expjcot and 6 U a ~. U ~ . The voltage in each point of the space-charge region will also be the sum of a d.c. component and an a.c. component: u = u = + R{U-},
(5)
where U = = - ( U ~ + Us) and U ~ = - U ~ ' . Since the amplitude of the a.c. signal is very low, we can develop all the magnitudes up to first order, and write:
V=Vo+R{V-}, n=n
= + R{n ~},
nT = nav + R{n~ }.
(6)
For a Maxwell-Boltzmann free-carrier distribution in the space-charge region: eV
n = nu exp K T "
(7)
Then, we have: eV =
n==nuexp KT' eV ~
n ~ = n = •--
KT
(8)
The density of deep levels filled with electrons follows the Shockley-Read-Hall statistics. In the ease of a deep level located in the upper-half of the bandgap of an n-type semiconductor, that is written a s : dn----S = e , n ( N T -- nv ) -- et, nv. dt
(9)
Developing this equation and neglecting secondorder terms, the expressions of the stationary and sinusoidal components of the density of deep levels filled with electrons are obtained:
(I)
where V is the voltage drop across the space-charge region and E is the dielectric permittivity of SSE 35/3--E
287
dnr dt
O+jom~
=c.n=(Nr-n~)-e~.n~
+e.n~(Nr-n~)-c.n'n~-et.n~
(10)
288
J. BARBOLLAet al.
and then
From eqn (13), a first-order approximation for the current yields:
Cn n =
r / T c.n - - = - +- e t NT'
C = CLFF ( o ) , G = CLFCOH(CO),
Cnetn n ~ =
n-? =(jco + c . n
,
,
+e.)(c.n= +e.)
N T.
where
Replacing eqns (2), (8) and (11) in eqn (1), the following expression for the charge stored in the space-charge region of the junction is obtained:
Q =(2eQ,/2[R{_foVNDdV+f U nN exp ~e V
fo
K T nN
rl ~ d g
cnen
The first two integrals correspond to the charge contribution of both ionized shallow donors and the free electrons. The third integral is the stationary term contribution of filled traps. The fourth integral corresponds to the deep level charge-state variation due to the sinusoidal component of the voltage. Solving these integrals, the following expression for the charge is obtained: KT n N
e ND
NT EF ND e - -
e ( U : + Ub)J"
ET
ND ~] e ( U : + Ub) --
co \
°
arctan -27 - arctan en
°)]
cnnN
and H(co)=~X /
(17)
EF--ET et. e(U7 + Ub)2co
x[In(l+e~f)-In(l+~) ]. (14) O92
Taking the derivative of the expression of the charge with respect to time, we can obtain the following expressions for the capacitance and conductance of the junction: C = 1 (dQ/dt)q co 6 Ua ' G = (dQ/dt)ph 6Ua '
(18)
Finally, taking into account that at any temperature in the neutral region it is verified that e t < c.nu and that around the maximum or inflexion point co - e~., we can write, around these points, the expressions of the capacitance and conductance of the junctions: et C ~-- Cav -~- (CLF -- CHF )
G-
(CLF
-m
co
co arctan
_-7, en
1+
--
.
(19)
2 The inflexion point of the capacitance as a function of the frequency is given by: e~(T)=
where
-
]
e
ET
+ F(co)6UaCOScot + H(co)6Uasincot) 1/2, (13)
EF-
o
}]= (12)
F(co) = 1 _NT /
NT EF -- ET'~ 1/2,
e N D N
is the low-frequency capacitance, while for a highfrequency (co--*co), we obtain:
NT(jco+c.n=+et)(c.n=+et)
Q(2eeND)I/2(U a + Ub
/o N \m/
dV
+tUNT c. n= jo c.n = + e', d V
+
(16)
(11)
(15)
where (dQ/dt)q and (dQ/dt)ph are, respectively, the current components in-quadrature and in-phase with the applied signal.
col . 0.825
(20)
The inflexion point of the conductance as a function of the frequency verifies: et.(T)=coi
(21)
.
Therefore, the C(co) and G(co) plots at a fixed temperature allow us to obtain the thermal emission rate of the deep level at that temperature. On the other hand, from eqn (19) we deduce that the maximum of the conductance as a function of temperature is located at: etn(Tm) --
co
(22)
1.980
and, if we assumed for the thermal emission rate the expression: e~(T) = A T ~ exp
ET KT'
(23)
which, apart from the physical significance of the A and E parameters, is known to be always followed on the experimental range of temperature, it follows from eqn (19) that the inflexion point of the C ( T ) plot, at a constant frequency, is defined as: co
et~(T~) = 1.825"
(24)
Admittance spectroscopy in junctions So, the C(T) and G(T) measurements allow us to determine the variation of the thermal emission rate with temperature and, from that, we can obtain the thermal activation energy ET of the deep level. In summary, we can see that thermal admittance spectroscopy yields the thermal emission rates of the deep levels from the capacitance and conductance curves as a function of frequency, at a constant temperature, or from the plots obtained varying the temperature at a fixed frequency. However, the measurements made scanning the temperature are most convenient, since they present two advantages from the experimental point of view: Firstly, the conductance curves as a function of temperature show maxima. The measurement of these maxima are much easier than that of the inflexion points obtained when the frequency is scanned. On the other hand, the curves obtained varying the temperature are made at only one frequency, and it is only necessary to adjust the phase shift introduced by the experimental circuit at the beginning of the experiment. Therefore, the G(T) plots give a higher accuracy, continuous curves and are less time-consuming than the G(co) plots. For these reasons, admittance measurements of deep levels in junctions are usually performed by measuring the conductance as a function of temperature at a given frequency. The heights of the peaks of conductance are obtained substituting eqn (22) into eqn (19): G(Tm) = 0.402~(CLF -- Car )
NT / EF-- ET
= 0"402°9 ~D ~/e (-~ -~ Ub)"
(25)
From this expression it is possible to obtain the deep-level concentration once the Fermi level and the thermal activation energy E T are known. In summary, thermal admittance spectroscopy is a technique which gives the thermal emission rates, the thermal activation energy and the concentration of a deep level located in the upper-half of the bandgap for an n-type semiconductor or in the lower-half for a p-type semiconductor. There are several interesting points that should be mentioned regarding the physical nature of the processes involved in admittance measurements: In the first instance, it should be noticed that the variations of the capacitance and conductance of the junction are due to the variation of the time constant of the charge and discharge processes of the deep level relative to the measuring signal frequency around the crossing point of the Fermi level with the deep level. At this point the electric field is low and the values obtained for the thermal emission rates will not be affected by possible dependencies with electric field.
289
Another important feature of thermal admittance spectroscopy is that it is a spectroscopic technique. This means that, for example, in the case of an n-type semiconductor with several deep levels located in the upper-half of the bandgap, the junction conductance will be the sum of the contributions of each of those centres whose thermal emission rates are close to the measuring signal frequency will contribute to the conductance. Deep levels with emission rates much higher or much lower than frequency will be in the low- or high-frequency cases, respectively, and will, therefore, have no contributions to the junction conductance. Hence, a plot of the conductance as a function of temperature will exhibit a maximum for each deep level located in the upper-half of the semiconductor bandgap. Experimental results reported by different authors using this technique show that the ability of thermal admittance spectroscopy to resolve two close levels is similar to that of DLTS. As for its sensitivity, thermal admittance spectroscopy has been able to detect deep levels in concentrations 106 times lower than the shallow impurities, which again is comparable to the DLTS sensitivity. Finally, with this technique it has been possible to measure emission rates of deep levels relatively close (about 100 meV) to the conduction band in n-type semiconductors or to the valence band in p-type semiconductors. However, thermal admittance spectroscopy cannot detect deep levels located in the lower half of the bandgap in n-type semiconductors or in the upper half of the bandgap in p-type semiconductors. Examples of experimental results obtained by thermal admittance spectroscopy for p +-n junctions containing deep levels related to Ti in Si[6] are shown in Fig. 3. Several curves of G(T) and C(T) at different frequencies are plotted. The two peaks appearing in
I
I
1 Hz -
I
-
~
10 Hz
100 Hz 1000 Hz
.4 B Q
G
I 100
I 200 T(K)
I 300
Fig. 3. Plots of C and G/to vs T for a p+-n junction containing Ti at several frequencies[6].
290
J. BARBOLLAet al. P÷
N
t
I I
......i-!!
y' W X" Fig. 4. (a) Energy-band diagram of a reverse-biasedp +-n junction containing an acceptor deep ]eve] in the upper-half of the bandgap. (b) Space-charge distribution under the
illumination condition. The y' point is defined by the condition et~+ e~ = c,,n(y'). the conductance curves and the inflexion points of the capacitance curves reveal the presence of two energy levels associated to Ti in n-type Si. The Arrhenius plots obtained from the positions of the maxima and of the inflexion points indicate that Ti in Si creates two deep levels located at 238 and 512 meV below the conduction band. 3. OPTICAL
ADMITTANCE
SPECTROSCOPY
OF
DEEP
LEVELS
Optical admittance spectroscopy[8] is a technique which allows the measurement of the optical emission rates of a deep level from the variations of the capacitance and conductance of a junction under monochromatic light when the photon energy hv and measuring signal frequency co are changed. Once the optical emission rates of the deep levels have been measured the optical capture cross-sections can be determined from the following expressions: e ° = a~,(hv), dp(hv), ep = a~ (hv ) . dp(hv ).
(26)
where e ° and e~ are the optical emission rates of electrons and holes, respectively, a~° and a~ are the optical capture cross-sections of electrons and holes, respectively and dp(hv) is the photon flux of the monochromatic light. In addition to this, the spectra of the optical emission rates of a deep level yield the optical threshold energies, E ° and E~, which are the minimum photon energies necessary to the emission of an electron to the conduction band or of a hole to the valence band, respectively. Let us deal, as in the preceding section, with a p +-n junction with a deep level located in the upper-half of the bandgap and in a concentration lower than that
of the shallow impurities. We shall assume that the junction is illuminated by a monochromatic photon flux q~(hv) and that the temperature is low enough that the thermal emission rate of electrons is much lower than the optical emission rates, et, <~e ° + e~. The band diagram and the space-charge distribution of the reverse-biased junction are shown in Fig. 4. The y'-point is the point within the space-charge region where the time constants of the emission and capture processes of the deep level are equal, c , , n ( y ' ) - e , - o +e~,. Since the electron density at y , n(y') = (e ° + e~)/c,,, is much higher than that at the crossing point of the Fermi level with the deep level, n ( y ) = et,,/c,,, the y'-point will be located at the right of y and, therefore, the y'-point will be closer to the neutral region. To the right of y ' the capture processes are predominant and the deep levels are full of electrons. To its left emission processes are predominant and the ionization factor of the centre is determined by the balance between electron and hole emissions, f . r -_e p /o( e , , o +e~). In the case where the optical emission rate of electrons is much higher than that of holes the deep level will be empty of electrons to the left of y ' . In any other case f r will have some value between 0 and 1. The a.c. voltage component will modulate the space-charge region at y ' and w. The response to this voltage at w is given by the dielectric relaxation time constant and can be considered instantaneous for all experimental frequencies. The response at y ' is determined by the optical emission rates of the deep level and will be faster the larger these rates are with respect to frequency co. Optical admittance spectroscopy measurements are carried out changing the photon energy hv at a given temperature and frequency. Let us assume that the photon energy hv is low enough that the optical emission rates are much lower than the frequency e ° + ep ,~ co. Under such conditions the deep level cannot respond to the modulating signal, the electron concentration at y ' is very low and there are no energy losses. Therefore, the conductance is zero and the capacitance value is the high-frequency value CHF, since only the charge modulated at w will contribute to it. At a photon energy high enough, the optical emission rates will be much higher than the frequency e ° +e~ >>co. In consequence, the deep level will follow the voltage variations, there will be no energy losses and the charge modulated will be the sum of the contribution at y ' and w. Therefore, the conductance will be zero again and the capacitance value will be the low-frequency value, CLF. At intermediate photon energies the deep level cannot fully follow the voltage variations, there will be energy losses and the charge modulated at y ' will be less than in the low-frequency case. In consequence, the conductance curves as a function of hv take a zero value at low and high energies and show a maximum at a photon energy at which the total optical emission rate is close to the frequency. The
291
Admittance spectroscopy in junctions uJ=const
I
where the low- and high-frequency capacitance are given by:
values of the
(30)
h-4
Fig. 5. Capacitance and conductance vs photon energy at a constant temperature and frequency. capacitance curves go from a low value, Cur at low photon energy, to a high value CL, at high energies and exhibit an inflexion point at a photon energy at which the total optical emission rate ez + ei, is close to the frequency w. These curves are depicted in Fig. 5. The calculation of the expressions of the capacitance and the conductance of the junction as a function of photon energy hv is similar to that developed in the thermal case. Under illumination conditions and at temperatures low enough that thermal emission rates can be neglected, the Shockley-Read-Hall equation can be written as:
dnr
ht- -
-e&
Solving this equation terms we find: n; =
c,n’+e’
+ (c,n + 67;)(NT - nT).
and neglecting
(27)
second-order
-NT,
c,n’+ez+e,”
c,ein-
n;=(iw
+
NT.
c,nE+e~+epO)(cnn=+e~+epO)
(28) The electrostatic charge density and the free electron distribution are again given by eqns (2) and (8), respectively. Substituting all these expressions into Poisson’s equation, the charge stored in the space-charge region can be obtained. Taking the derivative of this expression with respect to time, the following expressions for the capacitance and conductance of the junction are derived:
e,O+eO -112 ez ----NT KT In P 3 ) ND e ez+e,O wN l---
C,F=CLF
[
eZ NT ei+e,oN, KT 1 --1nP e Ua=+Ub
e,”+ e” wN
(2%
From eqn (29) we can deduce that the curves of conductance as a function of photon energy at constant temperature and frequency show a maximum defined by the condition:
Mwd
w
+ e,“@v,)= 1.980.
(31)
The height of the maxima is given by: i/2 G(hv,) = 0.4020 2
w l.980c,nN >
y&In a
b eZ(hv,)
’ eX(hv,) + e,“(hv,) ’
(32)
Thus, the curves of the conductance as a function of hv at constant T and w enables us to determine the spectral shape of the optical emission rates of electrons ei(hv) and holes ei(hv). To determine the exact condition satisfied by the inflexion points of the capacitance it would be necessary to have a prior knowledge of the dependence of the optical emission rates of electrons and holes on the photon flux. However, unlike the thermal case, this dependence is unknown before measurements are made. In order to estimate the sensitivity of optical emittance spectroscopy, we can evaluate the ratio between the peak height obtained by this technique and that obtained by thermal admittance spectroscopy for the same measuring frequency. From eqns (25) and (32):
WV,)
CW,)
G(T,)=e~(hv,)+e,“(hv,)
T
0 if
112
’
(33)
where To is the temperature at which the optical admittance spectroscopy measurements are performed. Obviously, To < T,,, to meet the condition that the thermal emission rate can be neglected. From eqn (33) we can see that the sensitivity of optical admittance spectroscopy is approximately equal to that of thermal admittance spectroscopy except when the optical emission rate of electrons is much lower than the optical emission rate of holes. When in the semiconductor bandgap there is more than one optically active deep level, optical admittance spectroscopy reveals itself as a useful tool as a spectroscopic technique. It is so because each of those deep levels will contribute to the conductance with a non-zero value only when the photon energy is such that the optical emission rate of a deep level is close to the measuring signal frequency w. Deep levels with
292
J. BARBOLLAet al.
T • 1,50 K
1 Hz
T = 200 K
"2.
1.5 Hz
//~
T • 270 K
0.7
0.9
1.1
1.3
1.5
hv (eV)
Fig. 6. Plots of G/to vs photon energy for a OaAs Schottky barrier containing the EL2 centre at T = 150, 200 and 250 K[9].
optical emission rates much higher or much lower than ~ will be in the low- or high-frequency condition and will, therefore, have no contribution to the conductance. The plot of the conductance as a function of the photon energy hv will show a maximum for each optically active deep level located in the upper-half of the bandgap. Among the different techniques for optical characterization of deep levels based on transient measurements, the most commonly used are DLOS (deep level optical spectroscopy)[12] and CC-DLOS (constant-capacitance deep level optical spectroscopy) [13]. These techniques determine the optical emission rates of the deep levels for the slope at the origin of the photocapacitance or photovoltage transients. When there is more than one deep level in the semiconductor bandgap, the initial slope of those transients cannot separately yield the contributions of each level, since it is the sum of all of them. For this reason, DLOS and CC-DLOS, unlike optical admittance spectroscopy, are not spectroscopic techniques.
Another advantage of optical admittance spectroscopy is that the conduction is measured in a small region, around the y'-point, where the electric field is low. Therefore, the optical emission rates thus measured will not be affected by the electric field present in the space-charge region. This electric field, however, does affect the measurements carried out by DLOS and CC-DLOS, since the optical emission rates are values averaged across the whole spacecharge region. It can be seen from eqns (31) and (32) that, when the electron and hole optical emission rates of a deep level are of the same order of magnitude, optical admittance spectroscopy can measure both of them. When the optical emission rate of majority carriers is much higher than that of minority carriers, only the former can be measured. Finally, if the optical emission rate of minority carriers is much higher than that of majority carriers the height of the conductance peaks is practically zero and none of them can be measured. The optical emission rates of a deep level are proportional to the photon flux and to the optical capture cross section. Therefore, in order to measure capture cross sections as low as possible, it is necessary to have an experimental set-up capable of measuring at very low frequencies while supplying a photon flux as high as possible. The need for using low frequencies makes optical admittance spectroscopy a slow technique. This is probably the main disadvantage of this technique from the practical point of view. In addition, since the a.c. current generated in the sample is proportional to the frequency, the measured values can be extremely small, and for this reason high-sensitivity electrometers are required. As an example, Fig. 6 shows some experimental curves[9] for the conductance as a function of photon energy, obtained by optical admittance spectroscopy applied to GaAs samples containing the EL2 level, at three different temperatues (150, 200 and 270 K). The measurements were performed within a temperature range where optical admittance spectroscopy can be applied to this level. This range is limited at both low and high temperatures. The upper limit is determined by the temperature at which thermal emission begins to be relevant, which occurs at temperatures above 300 K. The lower limit is imposed by photoquenching, a phenomenon characteristic of the EL2 level, by which this level becomes insensitive to any change in photon energy at temperatures below 140 K. From the positions of the conductance peak, the spectra of the photoionization cross-section of EL2 at the three experimental temperatures are obtained (Fig. 7). In these spectra the contributions corresponding to transitions from the EL2 level to each of the three conduction band minima of GaAs (F, L and X) can be seen. These spectra suggested that EL2 is formed by a family of levels located near the midgap of GaAs. This family can be explained by a microscopic
Admittance spectroscopy in junctions
T=150 K
•
°/° ~ar'-"
.oJ
0/° I
"°I
I
I
I
w T = 200 K
•
,/.-'o\
o/ o/
o~
o/O ~ ° / .e /
I
I
I
T=270 K
• o/'O-'~ON
/c "/
•
"\..
,d ° I
0.7
0.9
I
1.1 hv (eV)
I
1.3
293
Let us consider a p +-n junction of III-V alloy with a DX centre located at an energy Eo below the conduction band and in concentration Nd. In the dark, at the experimental temperatures (ED ~>KT) the ionization factor I of the DX centre in the neutral region coincides with the number of ionized centres (about 1-2%). The charge distribution within the spacecharge region is shown in Fig. 8. The edge of the space charge region w0 is defined as the point where the electron concentration is half of the concentration in the neutral region. To the left of w0, the electron concentration decreases rapidly and there will be a t At this point the time point wI where c,n(w])= e,. constant of the emission and capture processes of the DX centres are equal and the ionization factor of the centre is 1/2. This point coincides with the crossing point of the Fermi level with the energy level of the DX centre and is separated from the edge of the space-charge region by a distance which is of the order of the extrinsic Debye length LD(T). To the left of wl the emission processes prevail and therefore the DX centre is completely ionized. To the right of w1the DX centre has the same ionization factor as in the neutral region. In the neutral region the number of electrons emitted by the DX centre per unit time must equal that of electrons captured. Taking into account that in the neutral region the electron concentration equals the number of ionized DX centres, we have:
I)Nd,
C,,(INd) 2 = e'.(1 -
1.,5
Fig. 7. Electron optical capture cross-section a ° vs photon energy for the EL2 centre in G a A s at T = 150, 200 and 270 K. The solid lines are the curves obtained using the semi-empirical model of Chantre and co-workers[12]. The points (o) are our experimental results[9].
from where the following expression, relating the emission and capture rates and the DX ionization factor, is derived: et
12
(35)
N. ,
c.
model in terms of Ar clusters, as has been proposed by Mochizuki and Ikoma[14].
P+
1-I
-°"
N
_................
4. ADMrVrANCE SPECTROSCOPY OF DX CENTRES
The electronic properties of III-V compounds are often controlled by centres exhibiting a behaviour which is a mixture of that of shallow impurities and that of deep levels. These centres are called DX centres or deep donors[15]. As shallow impurities they control the free electron concentration, while as deep levels they exhibit the emission and capture processes characteristic of a trap. In AlxGa~_xAs alloys these donors are relatively deep for x > 0.2 and control the material conductivity within the range 0.2 < x < 0.6. In recent works[10, 11], thermal admittance spectroscopy has been used to study DX centres in A1GaAs junctions. This is possible due to the fact that since at the experimental temperatures the ionization factor of DX centres is very low, the charge distribution within the space-charge region of the reversebiased junction is very similar to that of a junction with a deep level and a fully-ionized shallow impurity.
(34)
a]
l_eU
] .......
ii n Ev
eNd
1
I bl
w1
w0
x
Fig. 8. (a) Energy-band diagram of a reverse-biased p ÷-n junction containing an D X centre. (b) Space-charge distribution under the darkness condition. The wI point is defined
by the condition e~ = c.n(wO.
294
J. BARBOLLAet al.
A small a.c. component superimposed to the reverse bias of the junction will modulate the charge stored in the space-charge region at points wI and w0. At w0 the response is instantaneous since it is determined by the dielectric relaxation time constant. At wl, however, the response will depend upon the relaxation between the time constant of the charge and discharge processes of the centre and the frequency of the modulating signal. At low temperatures the thermal emission rate of the centre is zero. Therefore, the electron density at wi is zero and there is no charge modulation at this point. In consequence, the junction conductance is zero and the capacitance value is the high-frequency value CnF since only the charge modulated at the edge of the space charge region will contribute to the junction capacitance. At very high temperatures the thermal emission rate of the DX centre is much higher than the measuring signal frequency. Therefore, the charge modulated at wI can follow the voltage variations, there are no energy losses and, in consequence, the conductance is zero again and the capacitance returns to its low-frequency value CLF. At intermediate temperatures the DX centre cannot respond instantaneously to the voltage variations and hence energy losses arise which lead to a non-zero conductance while the capacitance values are between CLF and CHF* In summary, at a given frequency the conductance vs temperature plot will exhibit a maximum and the capacitance vs temperature will display an inflexion point. The temperatures at which the maxima and inflexion points appear are those at which the thermal emission rate of the DX centre has a value close to the measuring signal frequency. In order to determine the exact position of these points we will follow the same procedure already sketched in preceding sections for the derivation of the expressions of capacitance and conductance of a junction with a DX centre. In this case the expression of the charge stored in the space-charge region is given by: p = e(IN,~ - n),
(36)
where I is the ionization factor of the DX centre. The density of ionized DX centres will have a constant component I = N d and a sinusoidal component I ~Nd. These components can be obtained from the Shockely-Read-Hall statistics, which in this case are written as: dn d d[(1 - / ) N d ] -- = dt dt
etn(1 -- I ) N d + c, n l N d .
(37)
From this expression we obtain the two components of the ionization factor mentioned above: et 1 = = _ _ Cn n = + etn ' Cnn - e t
I~ =
(c,,n = + e t) (c~ + et,, + j o g ) "
(38)
Substituting eqn (38) into eqn (36) and assuming a Maxwell-Boltzmann free-electron distribution in the space-charge region, the charge stored in the spacecharge region is obtained from Poisson's equation. Evaluating the derivative of the charge with respect to time, the current generated in the junction is obtained. The in-phase and in-quadrature components of the current yield the following expressions for the capacitance and conductance of the junction: C : CHF "~-(CLF
-
2
CHF)
O9 arctan ~ , e'. O9 en
e~ln 1 +
,
(39)
where the low- and the high-frequency values of the capacitance are given by: =
- -
(u2
+
uu
e
\e(-6-~--~~i) j
'
(40)
Equations (39) are equal to that obtained in the case of junctions with a deep level and fully-ionized shallow donor. Therefore, the conductance and capacitance curves of a junction with a DX centre as a function of frequency, at a constant temperature, will display inflexion points defined by e t ( T ) = ~oi/0.825 and e~(T)=og~, respectively. When the measurements are made varying the temperature, at a constant frequency, the conductance curves will present maxima defined by the condition e t , ( T ~ ) = 0~/1.980, and the capacitance will show inflexion points which verify: et,(Ti) = o9/1.825. In summary, thermal admittance spectroscopy is a technique used for the characterization of deep levels in semiconductors which can also be applied to junctions of III-V alloys with DX centres. The advantages of thermal admittance spectroscopy as compared to those techniques based on transients are of particular relevance in the case of DX centres. Many works[16-19] have shown that the isothermal transients associated to DX centres are strongly non-exponential even in the case of constantcapacitance transients. This non-exponentiality is attributed to effects related to the electric field present in the space-charge region, or to the capture processes taking place in the Debye tail region, or to the fact that the ionization factor changes with temperature[18]. However, none of these effects will affect the results obtained by thermal admittance spectroscopy since the relevant processes involved in this technique occur at a point within the space-charge region, near the neutral region, where the electric field is low. Figure 9 shows several experimental curves[l l] corresponding to thermal admittance spectroscopy applied to A10.26Ga0.74Asjunctions doped with Si to a concentration of 3.5 x 1017crn -3. These curves show two maxima related to two DX centres associated to Si in AIGaAs. From the location of these
Admittance spectroscopy in junctions
100 Hz 500 Hz 1 KHz 5KH
100
200
300
T(K]
Fig. 9. Plots of G/co vs T for a p +-n junction of Si-doped A10.26Ga074Asat several frequencies[11]. maxima the activation energies of both centres were obtained. The thermal activation energy of the first level, called DX-I, was 370 meV whereas that of the second one, called DX-II, was 415 meV.
ation factor of the centre is 1/2• To the left of this point the emission processes are predominant and then the ionization factor is determined by the balance between the optical emission rates of electrons and holes• To the right of Wl the ionization factor of the DX centre will be equal to that of the neutral region• In consequence, we can see that the charge distribution in the space-charge region of a junction with a DX centre under illumination conditions is similar to that of a junction with a deep level and fullyionized shallow impurity. Therefore, optical admittance spectroscopy will be able to measure the optical emission rates of the DX centres. The charge density in the space-charge region will be given by eqn (36)• In this equation the ionization factor will be determined by the Shockley-Read-Hall statistics, which under illumination conditions are written as: d(1 - I)Nd
e°(1 -- I)N d + (c,n + e~)INd.
dt 5. O P T I C A L A D M I T T A N C E S P E C T R O S C O P Y CENTRES
O F DX
In the same way as thermal admittance spectroscopy yields the thermal emission rates of DX centres, the application of optical admittance spectroscopy to junctions with DX centres should yield their optical emission rates. Let us consider a junction with a DX centre under illumination and at a temperature low enough that the thermal emission rate is much lower than the optical emission rates. In the neutral region it must verify that the number of electrons emitted by the DX centre per unit time must equal that of electrons captured. The electrons captured can be free electrons captured from the conduction band or holes emitted to the valence band from the DX centre. Taking into account that in the neutral region the electron concentration equals the number of ionized DX centres, we have:
(c,I°Nd + ep)l°N~ = e°(1 - I°)Nd,
(41)
where I ° is the ionization factor of the DX centre in the neutral region under illumination conditions. From eqn (41) the following expression relating the emission and capture rates and the ionization factor of the DX centre can be obtained: e°
c, I°No + ep
Io - -
1 -- I °
Nd.
(42)
In the cases in which the optical emission rate of electrons is much higher than c, Nd and than ep, the DX centre is fully ionized and the situation is equal to that of a junction with only a shallow impurity. In any other case the DX centre is partially ionized in the neutral region, and the space-charge distribution in the space-charge region will be similar to that of Fig. 8. The w~ point will be defined now by the condition c,n(wj)= e ° + e$. In this point the ioniz-
295
(43)
From this expression the stationary and sinusoidal components of the ionization factor of the DX centre can be obtained: I=~
o en
c,n = + e ° + e ~ ' c n n ~ e n°
I ~
(c,n = + e ° + ep) (c~ + e ° + ep +jo9) (44)
Replacing these values in Poisson's equation, and taking the derivative of the charge with respect to time, the following expressions for the capacitance and conductance of the junction as a function of photon energy can be derived:
C(hv) = CuF + (CL~ -- Car) e° + ep arctan o9
e ° + ep'
G(hv)=(CLF--~CaF) (e° +e~)ln f 1 -~ (e°-+-e~) ] (45) where the low- and high-frequencies of the capacitance are given by: le£M
\1/2 /
KT e,o e e°+ep 0 "Jr- O'X -- 1/2
erl c,I°Nd],
xlne"
e° (KT 1 CHe=CLF l - - e ° + e ~ , \ e U ~ - U b e ° + e°\l/2-1 ×ln~/ /.
c,I Nd,] _]
(46)
Eqns (45) are equal to that obtained in the case of junctions with a deep level and a fully-ionized shallow donor. Therefore, the conductance of a junction
J. BARBOLLAet al.
296
with a DX centre as a function of photon energy hv, at constant frequency and temperature will present a maximum defined by the condition e~(hvm)+ e~(hvm)=Og/1.980. The height of the maximum verifies the following relation of proportionality:
e°(hv~) G(hvm)~C° e°(hvm) + e~(hv~)"
(47)
We see, therefore, that the application of optical admittance spectroscopy to junctions with DX centres should yield the electron optical emission rates of the DX centres provided that e ° is not much less than e~, whereas if both were of the same order of magnitude both the electron and hole optical emission rates would be measured. All the advantages of optical admittance spectroscopy applied to deep levels in junctions would be, then, translated automatically to the case of DX centres. The optical emission rates measured in this way will not be affected by possible dependencies on the electric fields, nor by the other effects responsible for the non-exponentiality of transients which are specially remarkable in the case of DX centres. In conclusion, admittance spectroscopy is a characterization technique of deep levels in junctions which can also be applied to junctions with DX centres, in order to measure the electron thermal emission rates and the electron and hole optical emission rates of these centres. 6. CONCLUSIONS
This work has reviewed the admittance spectroscopy techniques used in the characterization of deep levels and DX centres in junctions. Thermal admittance spectroscopy can be used to measure the thermal emission rates and thermal activation energies of deep levels from the variations of capacitance and conductance of the junction as a function of temperature and of measuring signal frequency. The measurement is carried out in a small region around the crossing point of the Fermi level with the deep level. At this point the electric field is low and the resulting values, therefore, are not affected by the electric field present across the spacecharge region. In the case where there is more than one deep level this technique separately yields the thermal emission rates of each level. Thermal admittance spectroscopy is, therefore, a truly spectroscopic technique. Its sensitivity and resolution powers are comparable to those of DLTS, a technique widely used in the characterization of deep levels. Using thermal admittance spectroscopy it has been possible to measure relatively shallow energy levels of about 100 meV. The range of possible energies of deep levels detectable with this technique is restricted to the upper-half of the bandgap in n-type semiconductors and to the lower-half in p-type semiconductors.
Optical admittance spectroscopy yields the optical emission rates of deep levels from the variations of capacitance and conductance of a junction as a function of the photon energy of a monochromatic light applied to the junction, at a given temperature and frequency. The measurement is performed in a small region around the point within the spacecharge region where the electric field is low and the resulting values are, therefore, not affected by such field. The sensitivity of optical admittance spectroscopy is similar to that of thermal admittance spectroscopy except when the optical emission rate of majority carriers is much less than that of minority carriers since, in this case, the conductance vanishes. When the optical emission rates of electrons and holes are of the same order of magnitude, this technique can measure both optical emission rates. When the optical emission rate of majority carriers is much higher than that of minority carriers, only the former can be measured. This technique is again a spectroscopic technique which can determine separately the optical properties of each optically active deep level present in the semiconductor bandgap. On the contrary DLOS and CC-DLOS, techniques commonly used in the optical characterization of deep levels, cannot separate the contribution of each level and give only the sum of all of them. The main drawback of optical admittance spectroscopy is the fact that it is a slow technique. This is so because the optical thermal emission rates are proportional to the photon flux and to the optical capture cross sections and, in order to measure optical cross-sections as low as possible, very low frequencies have to be used. Thermal admittance spectroscopy can also be applied to the characterization of DX centres in III-V compounds. These centres exhibit a behaviour which is a mixture of that of deep level and of shallow impurity. Accordingly, the charge distribution in the space-charge region of the junction is very similar to that obtained in a junction with a deep level and a fully-ionized shallow impurity. The use of thermal admittance spectroscopy is particularly advantageous in the case of DX centres since these centres exhibit strong non-exponentiality, which is commonly attributed to effects relating to the electric field present in the space-charge region, or to the capture processes occurring in the Debye tail region or to the variation of the ionization factor of the DX centre with temperature. None of these effects affect the results of thermal admittance spectroscopy. Finally, we have shown in this work how the application of optical admittance spectroscopy to junctions with DX centres should give their optical emission rates. The results obtained in this way would not be affected by any possible dependence on electric fields or by any other effects leading to the non-exponential isothermal transients particularly relevant in the case of DX centres.
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