Models for Semiconductor Devices Used in Analog Electronic Systems

97

Note the trade-off between closed-loop gain and bandwidth; the lower the mid-band gain magnitude is, −RF/Rs, the higher the break frequency,

ω =	μRs	(Rs	+ RF )	r s
		τa
b

The choice of transistor for a high-frequency amplifier design is paramount. BJTs must be chosen to have high fT s, and FETs to have high fmax. Although it may never be necessary to design a discrete or an IC multistage amplifier, the factors described earlier that go into the design of broadband amplifiers should appreciated.

2.6Photons, Photodiodes, Photoconductors, LEDs, and Laser Diodes

2.6.1Introduction

This section examines the properties of semiconductor devices that sense photon energy and others that emit photon energy. Photons are an alternate way to describe electromagnetic (EM) radiation generally having wavelengths from 1 ∞ 10−4 m to less than 3 ∞ 10−12 m. These wavelengths include infrared (IR); visible light; ultraviolet (UV); x-rays; and gamma rays. The photon is a quantum EM “particle” used to describe physical interactions of low-power EMR with molecules, atoms, and subatomic particles, such as atomic shell electrons. The electromagnetic wave characterization of EM energy is used throughout the EM spectrum and finds application in describing the operation of antennas, transmission lines, fiber optic cables, and optical elements such as lenses, prisms, and mirrors used in IR, visible, and UV wavelengths. Maxwell’s equations for EM wave propagation are also useful in describing such phenomena as diffraction, refraction, and polarization of light (Balanis, 1989; Hecht, 1987).

A photon has essentially zero mass; it moves at the speed of light in a medium and has an individual energy of ε = hc/λ = hν joules, where h is Planck’s constant (6.6253 ∞ 10−34 joule-second); c is the speed of light in the supporting medium (c = 2.998 ∞ 108 m/sec in vacuo); λ is the wavelength of the EM radiation in meters; and ν is the Hertz frequency of the EMR (ν of visible light is approximately 1014 Hz). Photon energy is also given in electronvolts (eV). To obtain the energy of a photon in eV, divide its energy in joules by 1.602 ∞ 10−19. (1.602 ∞ 10−19 is the magnitude of the charge on an electron.) For example, a photon of blue light with a wavelength of λ = 450 nm has an energy of e = (6.626 ∞ 10−34 J s)(3 ∞ 108 m/s)/(450 ∞ 10−9 m) = 4.41 ∞ 10−19 J or 2.76 eV. Similarly, a photon of λ = 700 nm (red light) has an energy of 2.84 ∞ 10−19 J or 1.77 eV. Figure 2.55 illustrates the EM spectrum.

© 2004 by CRC Press LLC

98	Analysis and Application of Analog Electronic Circuits

ν Hz			Wavelength		Photon
1021			in vacuo	(m)	energy, eV
	Gamma rays
1020			3X10−12		4.13X105
1019	X-rays		3X10−11		4.13X104
1018			3X10−10		4.13X103
1017			3X10−9		4.13X102
1016		UV	3X10−8		4.13X101
1015			3X10−7		4.13
	Visible
1014			3X10−6		4.13X10−1
		NIR
1013	IR	LIR	3X10−5		4.13X10−2
1012		FIR	3X10−4		4.13X10−3
1011			3X10−3		4.13X10−4
1010	Microwave		3X10−2		4.13X10−5
109			3X10−1		4.13X10−6
	Radio frequencies
108			3		4.13X10−7

FIGURE 2.55

The electromagnetic spectrum.

Many sensors are used to measure EM energy in biomedical applications. These include, but are not limited to, photodiodes; phototransistors; photoconductors; pyroelectric IR sensors; photomultiplier tubes (PMTs); scintillation crystals + PMTs; etc. (Northrop, 2002).

Next, pn junction photon sensors, photoconductors and certain solid-state photon sources will be considered. It will be demonstrated that the interaction of photons in a certain energy range with a semiconductor pn junction can cause the generation of a photovoltaic EMF; this EMF is the basis for

© 2004 by CRC Press LLC

Models for Semiconductor Devices Used in Analog Electronic Systems

99

photodiode behavior and the solar cell as an energy transducer (photons to electrical current flow) capable of doing work. Photodiodes and solar cells can be used to measure incident EM radiation intensity. Also treated in the following sections are the generation of photon energy by special semiconductor diode structures, the light-emitting diode (LED) and the laser diode. Both types of photon-emissive devices have found wide application in biomedical instrumentation (Northrop, 2002).

2.6.2PIN Photodiodes

Photodiodes are used as sensors for EMR ranging from near infrared (NIR) to near ultraviolet (UVA). Even a standard small-signal silicon pn junction diode with a transparent glass envelope will respond to incident EMR of appropriate wavelength, as will a reverse-biased LED. However, photodiodes used for photonic measurements have specialized junction structures that maximize the area over which incident photons are absorbed. Photodiodes are used in a broad range of biomedical instruments, including blood pulse oximeters; finger-tip heart-rate sensors; single-drop blood glucose meters; fiber-optic-based spectrophotometers (used to sense analytes in blood, urine, etc.); spectrophotometric detection of tumors using endoscopes; etc.

Photodiodes fall into two broad categories: (1) three-layer, PIN diodes (“I” stands for intrinsic semiconductor) and (2) avalanche photodiodes (APDs), which are basically four-layer structures (P+IPN). Figure 2.56 illustrates schematic cross sections through two types of PIN devices. Note that to improve the efficiency of photon capture, a thin, λ/4 layer of antireflective (AR) coating is used on the surface of the PD, similar to the AR coatings commonly used on binocular and camera lenses. Assume an incident photon with the appropriate energy passes through the AR coating, enters the P+ diffusion layer, and interacts (collides) with a valence-band electron. If the electron gains energy greater than the band-gap energy, Eg, it is pulled up into the conduction band, leaving a hole in the valence band.

These electron–hole pairs are formed throughout the P+ layer, the depletion layer, and the N-layer materials. In the depletion layer, the E-field accelerates these photoelectrons toward the N-layer and the holes toward the P-layer. Electrons from the electron–hole pairs generated by photons in the N-layer, along with electrons that have arrived from the P-layer, are left in the N-layer conduction band. Meanwhile the holes diffuse through the N-layer up to the depletion layer while being accelerated and are collected in the P-layer valence band. By these mechanisms, electron–hole pairs, which are generated in proportion to the amount of incident light, are collected in the N- and P-layers of the PD. This results in a positive charge in the P-layer and a negative charge in the N-layer.

When an external circuit is connected between the P- and N-layers, photocurrent electrons will flow away from the N-layer and holes will flow away from the P-layer, toward the opposite electrode. The PIN PD and the external

© 2004 by CRC Press LLC

100	Analysis and Application of Analog Electronic Circuits

	hν
AR coating
		Anode
SiO2
					−
p+ diffusion					VB
Depletion region
					+
N- silicon bulk
				IP
AR coating	Anode
N+ diffusion
					+
				R	Vo
Al electrode	A	Cathode		−	−
	−	Vo	+	IP
		R
AR coating	Ring cathode
			Guard ring
n+
Depletion region
P-type Si bulk					VB
p+ diffusion
Al electrode	B	Anode

FIGURE 2.56

(A) Layer cake schematic of a three-layer PIN Si photodiode. (B) Layer cake schematic of a three layer, NIP Si photodiode. The AR coating minimizes reflection (and thus maximizes photon absorption) in the range of wavelengths in which the PD is designed to work. The guard ring minimizes dark current.

circuit are shown in Figure 2.56(A). Note that in normal operation of the PD, it is reverse-biased, so the photocurrent is a reverse current flowing in the same direction as the thermally generated leakage current that flows in a reverse-biased pn diode.

© 2004 by CRC Press LLC

Models for Semiconductor Devices Used in Analog Electronic Systems

101

		iD		iD
	h		−	VB
			vD
	+		+
	Vo = IPR	R	IP	Ip q η (Pi λ/hc)
				(Pi in watts)
	−VB			Pi = 0
				(dark)
			Vo	vD
				IP
	Increasing light
	intensity
				Load-line
				−VB /R

FIGURE 2.57

Top: simple series circuit for PIN PD. Bottom: iD vs. vD curves as a function of absorbed photon power, Pi. The load line is determined by the Thevenin equivalent circuit that the PD “sees.” Note that the PD’s photocurrent, IP, flows in the reverse direction. Vo across the load resistor can be determined graphically by the intersection of the iD = f(Pi) line with the load line.

Figure 2.57 illustrates typical PIN PD volt–ampere curves. The load line represents a graphical solution of the PD’s (nonlinear) volt–ampere curves with the (linear) Thevenin circuit “seen” by the PD. The zero-current intercept of the load line on the VD axis is at −Thevenin open-circuit voltage; the zero-voltage intercept is −Thevenin short-circuit current. The load line permits a graphical solution of:

	iD (vD )	=	VB − vD		(2.185A)
			R
	or
	vD = VB − iD R, (VB < 0 in reverse-biased diode.)				(2.185B)

The slope of the load line is easily seen to be −1/R and depends only on the Thevenin model parameters. Most PDs are operated in the third quadrant, either with a reverse-bias OCV or under short-circuit conditions (vD = 0). PD signal-conditioning circuits will be described later.

© 2004 by CRC Press LLC

102	Analysis and Application of Analog Electronic Circuits

The total PD current can be modeled by:

iD = Irs [exp(vD VT )− 1]− IP

(2.186)

where the photocurrent, IP, flows in the reverse direction and is given by:

IP	=	ηq Pi	λ	amperes	(2.187)
		hc

where Pi is the total photon power incident on the PD active surface in watts; VT = kT/q, η is the capture efficiency (approximately 0.8); and Irs is the reverse saturation current. Irs is very temperature dependent; it can be approximated by (Navon, 1975; Millman, 1979):

Irs (T) = Irs (To )2(T−To )10

or

rs	rs		[		(		]
I	I = 1	2		3 T + φ		kT2	)	0.08	∞K	(2.188B)

where T is the Kelvin temperature; To is the Kelvin reference temperature; φ is the silicon energy gap (1.15 eV); and k is Boltzmann’s constant (1.380 ∞ 10−23 J/K). The simple PD model given by Equation 2.186 does not include the ohmic leakage of the reverse-biased PD. Such leakage can be an appreciable portion of the reverse dark iD for large reverse vD.

If the PIN PD is operated at zero iD, photon power produces an opencircuit voltage given by:

			q ηλ P ˘
vDoc	= (kT q)ln 1	+	i	˙ open-circuit volts	(2.189)
			hc Irs ˚

Silicon PIN PDs are useful over a wavelength band covering approximately 200 to 1100 nm; their spectral sensitivity rises slowly to a peak at approximately 800 nm, then falls off rapidly. Sensitivity is given by the PD’s responsivity in amps per watt; R(λ) = iD(0)/Pi. R(λ) peaks at approximately 0.55 A/W at approximately 850 nm. Figure 2.58 shows a “typical” Si PIN PD responsivity plot. PDs can also be fabricated using germanium (Ge) and InGaAs. The former material responds from approximately 300 to 1600 nm and the latter composition has a spectral responsivity range from approximately 800 to 2600 nm. Depending on operating conditions, PIN PDs are useful over a range of picowatts to milliwatts of optical power.

© 2004 by CRC Press LLC

Models for Semiconductor Devices Used in Analog Electronic Systems

103

R(λ) (A/ W) 0.7

0.6

0.5

0.4

0.3

0.2

0.1

λ, nm

0.0

200 400 600 800 1000 1200

FIGURE 2.58

A typical responsivity plot for a Si PIN PD. See text for discussion.

Figures of merit for PDs include the responsivity R(λ), described previously; the noise-equivalent (optical) power (NEP); and the detectivity (D*). The NEP is the incident Pi at λ required to generate a short-circuit response current, IP, equal to the RMS noise current of the detector system (unity output signal-to-noise ratio.) NEP is a measure of the minimum detectable noise power at a given wavelength and bandwidth. In other words,

	NEP(λ) =	rms noise current	Watts	(2.190)
		responsivity @ λ

The noise generated by a PD operating under reverse bias is due to shot noise generated in the dark leakage current and Johnson (thermal) noise generated in the equivalent shunt resistance of the PD. Shot and thermal noises are broadband and considered to have flat white power density spectra (See Chapter 9). The mean-squared shot noise can be shown to be given by:

i 2	= 2 q I	DL	B msA	(2.191)
sn

where q is the magnitude of the electron charge; IDL is the dark leakage current in amperes (IDL is zero for a zero-biased PD); and B is the equivalent

© 2004 by CRC Press LLC

104	Analysis and Application of Analog Electronic Circuits

Hertz noise bandwidth over which the noise current is measured. The meansquared thermal noise can be shown to be given by:

i 2	= 4 k T GB msA	(2.192)
tn

where k = Boltzmann’s constant (1.38 ∞ 10−38 J/K); T = Kelvin temperature of PD; B = equivalent Hertz noise bandwidth; and G = net thermal noise producing resistance. The total MS diode noise current is found by adding the two MS current noises:

i 2	= i 2		+ i 2		msA	(2.193)
n		sn		tn

Thus the total RMS diode noise is:

in = in2 = (2 q IDL + 4 k T G) B rmsA

(2.194)

Johnson noise dominates as the dark current 0. Note that NEP depends on λ, IDL (thus the PD operating circuit), the noise bandwidth B, T, and the net (Norton) conductance, G, in parallel with the photocurrent and noise current sources. The NEP for Si PIN PDs ranges from approximately 10–14 W/ Hz for small-area (A = 1 mm2) low-noise PDs, to over 2 ∞ 10–13 W/ Hz for very large area cells (A = 100 mm2). Obviously, the NEP is desired to be as small as possible. Note that manufacturers give NEP independent of the noise Hertz bandwidth, B. NEP must be multiplied by the

B to get the actual NEP in watts.

Often the input light power, Pi, is chopped; that is, the beam is periodically interrupted by a chopper wheel, effectively modulating the beam by multiplying it by a 0,1 square wave. The chopping rate is generally at audio frequencies (e.g., 1 kHz) and the bandwidth of the associated band-pass filter used to condition the PD output determines B. Chopping is used to avoid the excess 1/f diode noise present at DC and very low frequencies.

Figure 2.59 illustrates the equivalent model for a reverse-biased Si PIN PD, showing signal, noise, and dark current sources, the diode small-signal capacitance, Cd, which is a depletion capacitance that depends on −vDQ, and diode semiconductor doping and geometry. Note that the DC reverse leakage current has two components: a constant small Irs and a voltage-dependent dark current, IDL. (IDL + Irs) are used to calculate isn. Normally, RL R, so the thermal noise current is found using the external load resistor, R (1/R = G). The junction depletion capacitance, Cd, decreases as vDQ goes more negative. Large Cd is deleterious to PD high-frequency response because it shunts IP(jω) to ground. Note that Cd increases with illumination and is smallest in the dark

(using the circuit of Figure 2.57). Cd is on the order of picofarads. For example, one Si PIN PD with 1 mm2 active area and a vDQ = −10 V has a Cd 4 pF.

© 2004 by CRC Press LLC