Michael Deemer E SC 597c Term Paper

Polaronic Transport of Holes in Silicon Dioxide

A polaron is the result of a charge carrier in a localized state within a solid. A

polaron arises from coulombic interactions between the charge carrier and the

surrounding atoms. While the charge carrier affects many surrounding atoms, these

interactions are usually weak and can be neglected. If considerable distortion is to

occur, a pseudo-particle known as a polaron is formed. This polaron can be

categorized as either small of large, based upon the radius of affected atoms. While

large polarons affect many surrounding atoms to a small extent, small polarons are

the opposite. They affect a lower amount of surrounding atoms to a much higher

extent. This polarons are commonly formed from holes because holes have a larger

effective mass than electrons, and have a much lower mobility. In this paper, I will

focus on small polarons due to the fact that they are type of polaron present in

amorphous silicon dioxide. Figure 1 shows an example of a polaron in an ionic solid.

Figure 1: A Polaron is formed as a result of an electron within an ionic solid.

http://en.wikipedia.org/wiki/Polar

on

A popular and well-known example of a polaron is the Vk center in alkali halides

and solid rare gases such as Argon. In this example, a hole gets trapped between two

adjacent atoms that are attracted toward each other to form a potential well. The

charge carrier and the potential well move as a whole through the material. This

movement through the material can be done in two different ways.

Transport of polarons can occur via excitation or hopping. In excitation, the

charge carrier is excited out of the potential well and later relaxed back into another

potential well. The method of hopping is more likely than this excitation. In the

hopping method, the charge carrier can move from the current potential well, to an

adjacent potential well created from the same displacement of a neighboring pair of

atoms. For hopping to occur, the displacement between adjacent pair must be the

same, resulting in a potential well of the same depth. Although the charge carrier is

the only thing physically moving, it is always being accompanied by a potential well

of the same depth. The charge carrier will remain within a potential well with

constant depth throughout the entire transport process through the silicon dioxide.

This is more easily understood by viewing the charge carrier and the potential well

as one being (a polaron).

As mentioned earlier, the most studied example of a polaron is the Vk center

in alkali halides and solid noble gases. For example, these polarons are formed in the

valence band of solid argon. Two argon atoms will be attracted toward each other to

form a potential well in which a whole can get trapped. This formation of a Vk center

is very similar to the mechanism of the self-trapping of holes in amorphous silicon

dioxide (a-SiO2). Small polarons tend to form in circumstanced involving a narrow

valence band, a large hole-lattice interaction, and the presence of disorder. All three

of these are characteristics of a-SiO2, which shows why holes get self-trapped within

this material.

Another aspect that plays a large role in polaronic transport of holes is the

presence of defects within a-SiO2. These defects can be readily analyzed by using

electron spin resonance (ESR). One if the most common defects within a-SiO2 is the

E’ center. This defect is the result of an electron on a silicon-dangling bond, and can

be visualized by the oxygen-vacancy model represented in figure 2. To analyze the

hole trapping process in particular, ESR must be used in conjunction with

capacitance vs. voltage (CV) measurements.

Figure 2: shows a visual of the oxygen-vacancy model of hole trapping process

within the amorphous silicon dioxide. When a hole comes across an oxygen vacancy

within a-SiO2, it recombines with one of the electrons in the silicon-silicon bond.

This leaves behind one electron between the two silicon atoms, which will localize

on one of them, allowing the oxygen atoms on the opposing silicon to relax and form

sp2 orbitals. This gives rise to the E’ center.

Warren, William, Patrick Lenahan , and Jeffery Brinker. "Experimental evidence for two fundamentally different E'...." Journal of Non-Crystalline Solids 136 (1991): 151-162. Print.

By using ESR and CV measurements together, one can analyze the E’ defect in

a-SiO2 with relatively good precision. An experiment was conducted with the intent

of verifying the oxygen vacancy model, as well as trying to find the density of E’

centers within a sample of a-SiO2. The results are explained below in figure 3.

Figure 3: ESR signature and capacitance vs. voltage curve for a sample of a-SiO2 that

was subjected to various irradiation cycles. Between (a.) and (b.), the sample was

exposed to vacuum ultraviolet radiation that resulted in the oxide being flooded

with holes. Between (b.) and (c.), the sample was subjected to ultraviolet radiation,

which resulted in the oxide being flooded with electrons. This process was repeated

again two more times, which is represented from the figures (d.) through (g.).

Warren, William, Patrick Lenahan , and Jeffery Brinker. "Experimental evidence for two fundamentally different E'...." Journal of Non-Crystalline Solids 136 (1991): 151-162. Print.

From this experiment, one can argue that the E’ center is created by means of

the oxygen-vacancy model. Evidence of this can be seen from both the ESR spectrum

and the CV measurements. First, when comparing the CV curve from parts (a.) and

(b.), you can clearly see the leftward shift, which is indicative of positive charge

buildup within the oxide. Upon investigation of the ESR spectrum, we can clearly see

a signature that is representative of the E’ center. What this means is that after the

oxide was flooded with holes, the E’ centers appeared and were responsible for the

trapping of holes within the oxide. Next, when the results between parts (b.) and (c.)

are examined, it is clear to see the positive shift in the CV curve. This shows that

after the hole was trapped within the E’ center and the oxide was flooded with

electrons for a long period of time (~10 hours), these added electrons react and

combine with the trapped hole and eliminate the E’ center as well as the positive

charge in the oxide (the hole). More evidence of this can be noticed by the lack of the

E’ ESR signature. This cycle of hole flooding and electron flooding was repeated,

yielding the same results. These results verify the oxygen-vacancy model for the E’

center.

Another thing to think about when discussing the E’ center is the fact that

these trapped holes (polarons) have a larger radius, and larger effective mass than

the hole alone. This means that the capture cross-section of the positively charged E’

center can more readily capture an electron. This is due to the coulombic interaction

between the positively charged hole within the E’ center and the electron outside

the E’ center, promoting recombination within the E’ center. By knowing the value

of this capture cross-section as well as the number of injected electrons, you can

calculate the density of E’ centers within a-SiO2. These equations are relatively

simple, but for the sake of time I will not get much into the equations.

Another study that I found interesting was the field-dependent hole

transport in a-SiO2. This experiment was carried out between temperature of 79K

and room temperature, on samples of a-SiO2 that were wet-grown on silicon

substrates at temperatures of 950 degrees Celsius, to a thickness of almost 1000

angstroms. The goal of this experiment was to explain the method of polaronic

transport of holes within a-SiO2.

In this study, the samples were irradiated to form electron-hole pairs. There

was an applied negative voltage to sweep the electrons out of the sample, and leave

the holes within the oxide. After the negative voltage was removed, a positive

increasing voltage was applied to encourage the transport of hole through the oxide.

Results of this portion of the experiment can be explained in figure 4.

Figure 4: This figure described the relationship between the flat-band voltage

recovery (normalized to account for the removal of the initial negative bias) and the

time after the sample was irradiated.

Mott, N. F., and E. A. Davis. Electronic processes in non-crystalline materials,. Oxford: Clarendon Press, 1971. Print.

From these results, we can analyze the difference in recovery (or hole

transport) between various applied electric fields with respect to time. There are

many important aspects of hole transport that stick out from this graph. Firstly, it is

clear that in less than 10-4s, little or no recovery has taken place at all. By looking at

the curve that correlates with 3MV/cm, you can determine that throughout the

whole time frame of this experiment (~800s), there is still little to no hole transport

through the oxide. Another key feature to note on this figure, it that all the curves

have the same general shape, by increasing the electric field at the gate only the time

frame in which these curves are ‘stretched’ is effected by the externally applied field.

And Lastly, from the upper curve corresponding with an oxide field of 6MV/cm,

almost full recovery has taken place within the oxide. That is, almost all the holes

are pushed through the oxide into the semiconductor region of the device.

In this same experiment, the temperature dependence of hole transport was

studied. A curve was made that plotted various field slopes vs. inverse temperature

(1000/T). From these results, figure 5 was created. The purpose of this curve is to

use the relationship between the field slopes to the temperature at which the device

will recover at approximately 50%. By looking at the graph of this plot, we can

determine the means by which the holes are being transported. From the semi-

classical hopping method of polaronic transport, the graph of this curve should

represent (ea/2k), where a is the hopping distance between adjacent potential

wells. This slope should show a value of 9+/- 2 angstroms. From this, we can

determine that hopping is the most common method of transport in the oxide.

Figure 5: Temperature dependence of hole transport in a-SiO2. The solid line on the

plot fits the curve for 50% recovery. This recovery of the flat-band voltage refers to

the total percentage of holes are the pushed from the oxide, or the holes being

transported through the oxide.

Mott, N. F., and E. A. Davis. Electronic processes in non-crystalline materials,. Oxford: Clarendon Press, 1971. Print.

In conclusion, we have discussed the basis upon what polarons are and how

they can move through a-SiO2. Polarons are pseudo-particles formed when a hole

encounters a potential well and gets trapped. This can be easily represented by

means of the Vk center in alkali halides and solid noble gases, which is also very

similar to the mechanism behind a self-trapped hole within a-SiO2. Next, I talked

about defects within a-SiO2, with a focus on the E’ center, and verified the previous

model of the E’ center arising due to an oxygen vacancy within the oxide. This

oxygen vacancy results in a silicon-silicon bond that is relatively weak. A hole can

get trapped by this oxygen vacancy and result in the ESR signature referring to the

E’ center. Then, I talked about how the applied electric field and the temperature can

affect polaronic transport within amorphous silicon dioxide. After the removal of

the initial negative voltage that is used to force the electrons out of the oxide, as the

positive voltage is increased, the speed at which the holes are transported through

the oxide also increases. Lastly, we concluded that the mode of polaronic transport

that is most commonly seen within a-SiO2 is hopping of the hole between adjacent

potential wells of equal depth, which was proven due to the last figure showing the

slope of the 50% recovery samples at a value of 9+/- 2 angstroms which

corresponds almost perfectly with the expected semi-classical hopping model for

polaronic transport of holes.

References:

1. "Polaron." Wikipedia. Wikimedia Foundation, 4 Aug. 2014. Web. 1 May 2014.

<http://en.wikipedia.org/wiki/Polaron>.

2. Warren, William, Patrick Lenahan , and Jeffery Brinker. "Experimental evidence for

two fundamentally different E'...." Journal of Non-Crystalline Solids 136 (1991): 151-

162. Print.

3. Mott, N. F., and E. A. Davis. Electronic processes in non-crystalline materials,. Oxford:

Clarendon Press, 1971. Print.

4. Mott, N F, and A M Stoneham. "The lifetime of electrons, holes and excitons before

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Print.

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