Photonics and Lasers

Chapter 14.3.1 - Carrier Transit Time

 

Carrier Transit Time

Single p-n Junction

When the RC time constant is made sufficiently small, the photodiode response time will be limited by the motion of charge carriers across the device. The fundamental principle needed for this analysis is given by Eq. (13-9), developed in connection with the vacuum photodiode. According to this result, which applies quite generally to any photon-type detector, the current pulse from a single photoexcited electron lasts as long as the electron is moving through a region with high electric field. A similar relation applies to photoexcited holes. Since the E field is high only in the depletion region (see Fig. 10-11), the current pulse will last a time

14_03_Photonics_and_Lasers-11.jpg

known as the transit time, where d is the width of the depletion region and v is the velocity of the charge carrier. At low to moderate electric field strength, the electron velocity is ve = eE, where eis the electron mobility, and the hole velocity is vh = hE.

 

14_03_Photonics_and_Lasers-12.jpg

Figure 14-10 Summary of performance trade-offs in choosing load resistor RLand detector area A.

 

The time response will be limited by the charge carrier with the smallest mobility, which is usually a hole. In this case, the transit time becomes the transit time then becomes

14_03_Photonics_and_Lasers-13.jpg

If the field is sufficiently high (above a value Esat ~2× 106 V/m for holes in Si), then the carrier velocity is no longer proportional to E, saturating at the upper limit vs(~ 105 m/s for holes in Si). Under high-field conditions, the transit time is

14_03_Photonics_and_Lasers-14.jpg

It would appear from Eq. (14-31) that a higher E field gives a shorter transit time. However, the width of the depletion region d increases with increasing E, and this tends to increase the transit time. To see how these two effects offset each other, we express both E and d in terms of the junction potential V0 + VB. Taking E as approximately constant across the depletion region,

14_03_Photonics_and_Lasers-15.jpg

the transit time then becomes

14_03_Photonics_and_Lasers-16.jpg

where Eq. (14-20) has been used with V= -VB.

This result shows that the transit time with low field is actually independent of the applied reverse bias. Apart from the choice of semiconductor material, which determines and h, it depends only on the donor concentration ND. Higher NDwould appear to be best for a fast time response (small ttr). However, we found in Eq. (14-25) that the junction capacitance increases with higher ND, resulting in a slower response. The time response will, therefore, be optimized when the contributions of capacitance and transit time to the response time are approximately equal. The value of NDgiving this optimum response depends on several other parameters, as shown in the following example.

EXAMPLE 14-2

A p+-n silicon photodiode has an junction area of 1 mm2, and a reverse bias of 10 V is applied through a 10 k load resistor. Determine the doping level in the lightly doped n region that minimizes the response time, and determine the junction width for this doping level. Take the hole mobility in Si to be 5 × 10-2 m2/Vs.

Solution: The optimum time response will occur when the capacitance rise time 2.2 RLC is approximately equal to transit time ttr. Using Eqs. (14-25) and (14-33),

14_03_Photonics_and_Lasers-17.jpg

where 0 = 8.85 × 10 -12 F/m is the permitivity of free space, r= 11.9 is the relative dielectric constant for silicon, and V0 + VB VB = 10 V. Solving this for NDgives

14_03_Photonics_and_Lasers-18.jpg

This is a very light doping level, only two orders of magnitude above the "intrinsic" carrier concentration in undoped silicon of ~ 1.4 × 1010 cm-3. The junction width is then obtained from Eq. (14-20) as

14_03_Photonics_and_Lasers-19.jpg

The electric field in the depletion region is E 10 V/10-4m = 105 V/m. This is less than the saturating field value Esat ~2× 106 V/m, which justifies using Eq. (14-33) for the transit time.

 

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