University of Iceland 30th April 2007Experimental physics

Characterization of Semiconductors

by Capacitance Methods

Líney Halla KristinsdóttirPétur Gordon HermannssonSigurður Ægir JónssonInstructor:Djelloul Seghier

Contents

Contents 1

1 Introduction 2

2 Theoretical background 22.1 Schottky diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 DLTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Measurments and devices 73.1 Capacitance-voltage proling (CV) . . . . . . . . . . . . . . . . . . . . . . 73.2 DLTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Results 84.1 Capacitance-voltage proling (CV) . . . . . . . . . . . . . . . . . . . . . . 84.2 DLTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5 Conclusion 14

References 15

1

1 Introduction

Semiconductors are one of the greatest discoveries of the 20th century. Diodes andtransistors, which are based on semiconductors, are now be found in almost all modernelectrical devices, such as mobile phones and computers. Still today, semiconductorsare an active eld of research and scientists are always on the lookout for new materialsposessing distinct characteristics which can be useful in modern technology as well asbeing suited for production. In this report we examine the characteristics of Schottkydiodes. We run a CV proling on a mass-produced silicon diode and DLTS (deep leveltransient spectoscopy) on a homemade diode made out of CdO (semiconductor) andgold (metal).

2 Theoretical background

2.1 Schottky diodes

Figure 1. The various importantenergy levels in the metal and thesemiconductor with respect to thevacuum level.

Figure 2. The junction potentialproduced when the metal and semi-conductor are brougt together.

The Schottky diode has characteristics that are similar to those of the classical p − njunction, except that for many applications it has a much faster response which can bedesirable.

When a semiconductor is brougt into a contact with a metal, a barrier is formed in thesemiconductor from which charge carriers are severaly depleted. The barrier layer iscalled the depletion layer and can be seen in gures 1 and 2. There is also depletionlayer in the metal which is so small that in most cases it can be completely ignored[2]. We may consider the bulk region to be electrically neutral and approximate theboundary between the bulk and the depletion region to be sharp.

2

In gures 1 and 2 an n-type semiconductor is brought into contact with a metal to forma Schottky diode. After the transfer of electrons to the conduction band of the metalthe Fermi levels are coincident. Positively charged donor ions are left behind in thisregion which is practically stripped of electrons. Here the Poission equation is

∇D =Nde

ε0

(1)

where D is the electric displacement and Nd is the doping concentration. From 1 theelectric potential is determined by

d2φ

dx2= −Nde

εε0

(2)

where x is the distance from the junction into the semiconductor. If we assume Nd tobe constant we get

φ = −Nde

2εε0

x2 (3)

If we apply a reverse bias Vr to the Schottky diode the total potential becomes Vb + Vr

where Vb is the built-in potential of the junction (see gure 2). From (3) we determinethe thickness of the layer (see gure 2) to be

W =

√2εε0(Vb + Vr)

Nde(4)

Therefore the stored charge in the depletion region can be found by (4)

Q = WNdeA = A√

2εε0Nde(Vb + Vr) (5)

where A is the cross sectional area of the junction.

The capacitance of the junction then can be found by

C =dQ

dVr

=1

2A

√2εε0Nde

Vb + Vr

= εε0A

W⇒ W = εε0

A

C(6)

From equation (6) it can be shown that

1

C2=

2(Vb + Vr)

A2eεNd

(7)

It can also be shown [3] that

Nd(W ) =−C3

eεε0A2

(dC

dVr

)−1

(8)

It turns out that these equations give quite accurate results despite the assumption thatNd is constant.

3

Figure 3. Equvalent circuit forthe Schottky diode. Figure 4. Typical characteristic

of a diode.

In gure 1 we can see en equvalent circuit for the Schottky diode. There one can seethat there are two resistances, Rl connected in parallel (nonlinear leakage resistance)and Rs connected in series. The total impedance is then

Z =Rl + Rs(1 + ω2C2R2

l ) + jωCRl

1 + ω2C2R2l

(9)

In the LCR meter we have to use the approximation that we have a resistance connectedeither serial or parallel, not both. If we dene Cm to be the value measured by the LCRmeter (where we ignore Rs) we can calculate the real capacitance C by

Cm

C=

1

(1 + Rs

Rl)2 + ω2C2R2

s

(10)

if we somehow know the value of Rs. If Rs Rl as expected at reverse voltage we seethat Cm ' C.

A typical I-V diagram is shown in gure 4. By looking at the gure and equation 9 wesee that if Rl Rs the diagram becomes nearly linear at high bias voltages. Then it issafe to assume the resistance in the circuit to only depend on Rs.

When we apply the reverse bias voltage a small saturation current Is appears. Thiscurrent is given by

Is = A∗T 2e−eφbkT (11)

where A∗ is the area of the metal-semiconductor interface A multiplied by the so calledRicharson constant R∗ (A∗ = AR∗) [3], T is the temperature and φb is the height of thebarrier as shown in gure 2.

When Rl Rs at low bias the current is given by

I = Is

(e

eVnkT − 1

)(12)

4

where n is the ideality factor of the diode

n = 1− ∂φb

∂V(13)

For ideal diodes φb does not depend on V and thus n = 1. For non-ideal diodes n > 1.

2.2 DLTS

Figure 5. Illustrates the repeti-tive lling and reverse bias pulse se-quence [1].

Figure 6. Shows the diode capac-itance transient as a function of time.[1].

Native crystallographic defects or impurities may create electically active centers withlocalized potentials in the lattice. These defects have rather high thermal ionization en-ergies (deep defects). These defects can often be characterized by the DLTS technique.

The DLTS technique works by observing the capacitance transient associated with thechange in depletion region width as the diode returns to equlibrium from an initial non-equilibrium state. The capacitance transient is measured as a function of temperature.

The bias on the test diode is pulsed between a bias near zero and some reverse biasVr with a repetation time tr as shown in gure 5. The zero bias condition is held fora time tf during which traps are lled with majority carriers. During the reverse biaspulse the trapped carriers are emitted at a rate en producing exponential transient inthe capacitance, which in general form can be written

C(t) = C(∞) + ∆C0 exp (−t/τ) (14)

where the time constant τ is equal to e−1n . This transient is illustrated in gure 6. The

basis of the DLTS method is to feed this transient to a "rate window" which provides amaximum output when the time constant τ is equal to a known preset time constant τref.

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Figure 7. Shows how the tranient is measured using two pulses [1].

This is done by sampling the signal with two gates set at time t1 and t2 from the begin-ning of the transient, and producing an output proportional to the dierence betweenthese two signals. Then it can quite easily be shown that if τ t1 or τ t2 thedierence output is zero, whereas when τ ≈ t1, t2 an output is created. With a smallgate width ∆t τ an exponential capacitance transient gives a steady output signal

S = g∆C0(exp−t1τ− exp−t2

τ) (15)

where g is a calibration factor which also accounts for the gain of the system. The signalvaries as τ changes with temperature, and by dierentiating equation (15) with respectto τ we a get maximum where

τ(T ) = τref =t2 − t1ln t2/t1

(16)

But now is it also known [1] that if we have a diode made of a certain material with atrap energy Ena and capture cross section σna then

en(T ) = γT 2σna exp−Ena

kT(17)

where γ is a constant depending on the semiconductor properties and T is the temper-ature.

This allows us to make measurments and see how the location of the maximum changesfor dierent values of τref. Using the fact that τ = e−1

n and equation (17) we can t thedata and estimate Ena, and σna if know the value of γ beforehand.

We can also calculate the trap concentration Nt by using the equation

Nt = 2Nd∆C

C(18)

where Nd is the donor density.

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3 Measurments and devices

3.1 Capacitance-voltage proling (CV)

First we connected our diode (the mass-produces silicon diode) to a voltage source andto an ammeter and measured current vs voltage (I-V). We checked both polarities of biasby increasing the voltage V , and found both the forward and reverse characteristics. Wewere not able to determine the breakdown-voltage because the voltage-source couldn'treach a high enough voltage. By using a high forward voltage we could estimate thevalue of Rs by observing how the current changed linearly with voltage.

After that we connected our diode to an LCR-meter and measured the capacitance vs.voltage at reverse bias. The LCR-meter was on C||R mode (we ignored the resistanceRs in gure 3 since the I-V measurements showed it could be safely ignored).

3.2 DLTS

Figure 8. Diagram of a DLTS system [1].

In gure 8 we see how the DLTS system works. The sample (now the CdO/GoldSchottky diode) was put in a cryostat in which the temperature was controlled withliquid nitrogen and a built-in heating element The polarity of the bias was checked tomake sure it was reverse. The voltage values of the pulse and bias were xed using theoscilloscope. In our setup the ratio t1/t2, was xed at 13/3. We cooled down the sampleand measured for dierent values of ∆t = t2 − t1 using the DLTS program.

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4 Results

4.1 Capacitance-voltage proling (CV)

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

−30 −25 −20 −15 −10 −5 0

I [nA

]

V [V]

Figure 9. I-V for reverse bias.

0

1

2

3

4

5

6

7

8

9

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

I [m

A]

V [V]

Figure 10. I-V for forward bias.

In gure 10 we can see the I-V relation for a forward bias. By looking at the slope atthe endpoints (high forward voltage) we applied Ohms law and determined

Rs = (19± 1) Ω (19)

−1

0

1

2

3

4

5

6

7

8

9

10

−30 −25 −20 −15 −10 −5 0 5

I [m

A]

V [V]

Figure 11. I-V for both reverse and forward voltage.

8

−22

−20

−18

−16

−14

−12

−10

−8

−6

−4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

ln(I

)

V [V]

Figure 12. Data from the I-V for forward bias tted to determine theideality factor.

By using the I-V for a forward bias and equation (12) we tted the data and determinedthe ideality factor (see gure 12) to be

n = 1.55± 0.05 (20)

which is a little more than expected.

Looking at equation (7), we see that we should get straight line if we plot 1/C2 vs V ,but from gure 13 we see that clearly isn't the case, and therefore Nd isn't constant. Bytting the data for the low values of V (gure 14), where Nd appears to be a constant,we determined Vb to be

Vb = −0.3V (21)

And by using equations (8) and (6) we can get gure 15 from the data, where we seethat the donor concentration is varying a little but is close too

Nd ≈ 1013 cm−3 (22)

9

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0 5 10 15 20 25 30 35

1/C

2 [1/p

F2 ]

V [V]

Figure 13. Data from the C-V measurement.

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0 0.5 1 1.5 2 2.5 3

1/C

2 [1/p

F2 ]

V [V]

Figure 14. The low voltage data from C-V measurements.

10

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

x 10−5

1012

1013

1014

W [m]

Nd [c

m−

3 ]

Figure 15. Donor concentration.

11

4.2 DLTS

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

320 330 340 350 360 370 380 390 400

∆C [p

F]

T [K]

∆t=20.7 ms∆t=41.4 ms∆t=82.8 ms

∆t=165.6 ms∆t=414.0 ms

Figure 16. The amplitude ∆C0 plotted as function of temperature T fordierent values of τ .

In gure 17 we see how the amplitude ∆C0 changes with temperature T for few dierentvalues of ∆t = t2 − t1. The peak values were used to plot gure 17, from which we candetermine the values of the trap energy Ea and their capture cross section σa. By usingequations 16 and 17 we see that gure 17 where ln en/T

2 is plotted vs 1000/T shouldgive us straight line with slope −1000×Ea/k and should intersect the y-axis at ln γσa.We know that γ = 2.28× 1020 cm−2 s−1 K−2 and from gure 17 we nd

Ea = (0.70± 0.05) eV σa = 3.5× 10−15 cm2 (23)

and we see that Ea is rather deep compared to the CdO band gap of 2.5 eV. By usingequation 18 (where Nd ≈ 1015 cm−3) and ∆C = S/0.43 where S is the height of theDLTS peak we can nd the trap concentration Nt in each case

τ [ms] Nt[cm−3]14.4 1.08×1014

28.2 1.08×1014

56.4 9.83×1013

112.9 9.74×1013

282.3 9.05×1013

12

−10.5

−10

−9.5

−9

−8.5

−8

−7.5

2.55 2.6 2.65 2.7 2.75 2.8 2.85 2.9 2.95

ln(e

n/T

2 )

1000/T [K−1]

Figure 17. The peak values from gure 16 used to plot straight line anddetermine Ea and σa.

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5 Conclusion

For our mass-produced silicon diode we got an ideality factor of n = (1.55 ± 0.05).However, we expected value closer to 1. We estimated that Nd ≈ 1013 cm−3 although itwas varying a bit.

For the CdO/Gold diode we found a defect at Ea ≈ (0.7±0.5) eV having a capture crosssection σa = 3.5× 10−15 cm2 and a trap concentration of Nt ≈ 1014 cm−3.

Líney Halla Kristinsdóttir

Pétur Gordon Hermannsson

Sigurður Ægir Jónsson

14

References

[1] Blood, P., and J.W. Orton. 1992. The Electrical Characterization Of Semicon-

ductors: Majority Carriers And Electon States. Academic Press, USA.

[2] Kittel, C. 2005. Solid State Physics. John Wileys & Sons, USA.

[3] Singh, J. 1994. Semiconductor Devices. McGraw-Hill, USA.

[4] Wikipedia 2007. Internet: http://en.wikipedia.org/wiki/Deep_Level_Transient_Spectroscopy.

[5] Wikipedia 2007. Internet: http://en.wikipedia.org/wiki/Schottky_diode.

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