Electron Transport in Amorphous Silicon
`
`A Thesis
`
`Presented to
`
`The Division of Mathematics and Natural Sciences
`
`Reed College
`
`In Partial Fulfillment
`
`of the Requirements for the Degree
`
`Bachelor of Arts
`
`Zachary C. Holman
`
`May 2005
`
`HANWHA 1010
`
`

`

`

`

`Approved for the Division
`
`(Physics)
`
`John Essick
`
`

`

`

`

`A C K N O W L E D G E M E N T S
`
`Special thanks to John Essick for his advising and experimental expertise, David Cohen
`
`for donating the samples, my parents for making a Reed education possible, and Lara
`
`Sands for editing this work, putting up with me when I talked about nothing else, and
`
`giving me a mullet and a mohawk.
`
`

`

`

`

`T A B L E O F C O N T E N T S
`
`1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`
`1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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`1.2 Scope of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
`
`2 Semiconductor Primer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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`2.1 Band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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`2.2 Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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`2.3 Charge generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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`2.3.1 Thermal excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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`1
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`1
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`2
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`5
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`5
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`7
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`9
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`9
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`2.3.2 Photoexcitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
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`2.4 Charge transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
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`2.5 Crystalline vs. amorphous . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
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`2.6 Doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
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`2.7 Junctions
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`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
`
`3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
`
`3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
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`3.2 Assumptions
`
`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
`
`3.3 Recovering ψ(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
`
`3.4 Short times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
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`3.4.1 Electron occupation . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
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`3.4.2 Transient mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
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`3.4.3 Photocurrent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
`
`3.5 Long times
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`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
`
`3.6 High temperatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
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`

`

`Table of Contents
`
`3.7 Transit time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
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`4 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
`
`4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
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`4.2 Equipment
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`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
`
`4.2.1 Laser
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`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
`
`4.2.2 Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
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`4.2.3 Bias voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
`
`4.2.4 Oscilloscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
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`4.2.5 Computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
`
`4.2.6 Temperature control
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`. . . . . . . . . . . . . . . . . . . . . . . . . . . 52
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`4.3 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
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`5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
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`5.1 Qualitative observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
`
`5.2 Transient mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
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`5.2.1 Finding Q0
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`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
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`5.2.2 Normalized current traces . . . . . . . . . . . . . . . . . . . . . . . . 67
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`5.3 Fitting charge curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
`
`6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
`
`A Detailed Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
`
`References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
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`

`

`L I S T O F F I G U R E S
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`1.1 Time-of-flight measurement technique . . . . . . . . . . . . . . . . . . . . .
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`2.1 Band structure of a solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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`2.2 Allowed electron transitions . . . . . . . . . . . . . . . . . . . . . . . . . . .
`
`3
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`6
`
`8
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`2.3 Fermi-Dirac distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
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`2.4 Photoexcitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
`
`2.5 Non-dispersive charge transport . . . . . . . . . . . . . . . . . . . . . . . . . 17
`
`2.6 Dispersive charge transport . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
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`2.7 Two-dimensional crystalline lattice . . . . . . . . . . . . . . . . . . . . . . . 19
`
`2.8 Crystalline silicon lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
`
`2.9 Atomic and band structures of a two-dimensional amorphous solid . . . . . 21
`
`2.10 Atomic structure of hydrogenated amorphous silicon . . . . . . . . . . . . . 21
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`2.11 Atomic and band structures of a doped semiconductor . . . . . . . . . . . . 25
`
`2.12 pn junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
`
`2.13 pin solar cell
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`. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
`
`3.1 Multiple-trapping and hopping transport mechanisms
`
`. . . . . . . . . . . . 31
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`3.2 Electron occupation of band tail states after the initial trapping event . . . 35
`
`3.3 Trapping events experienced by a typical electron . . . . . . . . . . . . . . . 37
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`3.4 Pre-transit electron occupation of band tail states
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`. . . . . . . . . . . . . . 38
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`3.5 Electric fields generated by a drifting sheet of charge . . . . . . . . . . . . . 40
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`3.6 High temperature electron occupation of band tail states . . . . . . . . . . . 43
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`4.1 Experimental apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
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`4.2 Sample mounted on a transistor header
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`. . . . . . . . . . . . . . . . . . . . 55
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`

`

`List of Figures
`
`5.1 Typical photocurrent trace
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`. . . . . . . . . . . . . . . . . . . . . . . . . . . 58
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`5.2 Typical current trace plotted on logarithmic scales . . . . . . . . . . . . . . 59
`
`5.3 Current traces at various temperatures and constant voltage bias . . . . . . 60
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`5.4 Current traces at constant temperature and various voltage biases
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`. . . . . 61
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`5.5 Pre-transit current traces at constant temperature and various voltage biases 62
`
`5.6 Typical photocharge trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
`
`5.7 Photogenerated charge at high bias . . . . . . . . . . . . . . . . . . . . . . . 65
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`5.8 Photogenerated charge at low bias
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`. . . . . . . . . . . . . . . . . . . . . . . 66
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`5.9 Normalized current traces at low temperature . . . . . . . . . . . . . . . . . 68
`
`5.10 Normalized current traces at high temperature . . . . . . . . . . . . . . . . 69
`
`5.11 Normalized charge traces at low temperature . . . . . . . . . . . . . . . . . 70
`
`5.12 Normalized charge traces at high temperature . . . . . . . . . . . . . . . . . 72
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`5.13 Displaceability curves and the mulitple-trapping fit . . . . . . . . . . . . . . 74
`
`

`

`A B S T R A C T
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`Electron transport in an undoped hydrogenated amorphous silicon thin film was studied
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`using the time-of-flight measurement technique. Electrons were injected near the surface
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`of the film with a short laser pulse and pulled through the bulk of the material with an
`
`externally applied electric field. The photocurrent resulting from their transit was moni-
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`tored on an oscilloscope. Current traces were recorded for temperatures between 140 and
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`200 K and voltage biases between 2 and 8 V. The data are observed to exhibit behavior
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`characteristic of dispersive transport, as is expected for low-temperature electron trans-
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`port in amorphous materials. A simple multiple-trapping model based on an exponential
`
`distribution of thermally activated band tail traps is presented. The model is fit to the
`
`collected charge data, obtained by numerical integration of the current traces. The fit is
`found to be quite good and yields 0.11 cm2V−1s−1 for the microscopic mobility, 1.3× 1011
`s−1 for the attempt-to-escape frequency, and 22 meV for the band tail width. These values
`are comparable to those previously reported, and can be used to describe the transient
`
`mobility and displaceability of electrons in the sample.
`
`

`

`

`

`1
`
`I N T R O D U C T I O N
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`1.1 Motivation
`
`I recently traveled to Senegal, West Africa, where clean water is rare and stable electricity
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`is rarer yet. I was pleasantly surprised to discover that one of the most reliable sources of
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`power—and the only source for rural communities—is photovoltaic devices, or solar cells.
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`After an initial investment to purchase a few solar cells and a battery bank, Senegalese
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`families and businesses enjoy free and clean (though judging by their auto emissions this is
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`not their primary concern) electricity for several decades. Sunlight that would otherwise
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`heat an already hot land is instead used to power lightbulbs, simple appliances, and even
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`a few computers.
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`Since Russell Ohl [1] accidentally shone a flashlight on a silicon sample at Bell Labs in
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`1940 and discovered the pn junction, solar cells have improved by leaps and bounds. With
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`reduced cost and increased efficiency, new markets have developed so that photovoltaics
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`are now used to power everying from NASA’s premier satellites to homes to roadside
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`emergency call boxes. The cells, whose intrinsic material properties allow for photons to
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`be passively and automatically converted into electricity, have come into fame because they
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`capitalize on the abundance of free energy striking Earth each day in the form of sunlight.
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`Of ever increasing importance, solar energy is inexhaustible and completely clean, though
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`there are some emissions associated with the production of solar cells. Photovoltaics will
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`

`

`2
`
`Chapter 1
`
`never be able to escape the role of “supporting power supply” because they depend on
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`weather and location. Nevertheless, they are extremely useful in the right places. With
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`fossil fuel costs permanently on the rise and solar cell technology continually developing, I
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`expect the day is not far off when it will be more economical—not to mention healthier—to
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`look to the sun instead of the Middle East for power.
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`Historically, photovoltaic research has proceeded by a sort of guess-and-check explo-
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`ration of semiconducting materials. New materials are developed and then characterized,
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`and changes are made according to the results. In this search the name of the game is
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`maximizing the efficiency-to-cost ratio. Many mechanisms reduce the output of a solar
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`cell including imperfect incident light absorption, recombination of charge carriers, and
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`poor charge transport through the cell [2–6]. In characterizing new materials, these loss
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`mechanisms are studied so that the underlying processes can be understood and the ma-
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`terials improved. This thesis is intended to be a small step in the materials development
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`marathon.
`
`It should be noted that, while I have couched this discussion in the context of photo-
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`voltaics, the applications of this thesis are actually farther reaching. The material inves-
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`tigated may also be used for other electronic devices including transistors and flat-panel
`
`displays.
`
`1.2 Scope of this thesis
`
`Electron transport is studied in undoped hydrogenated amorphous silicon (a-Si:H) using
`
`the time-of-flight (TOF) measurement technique. Amorphous silicon, the younger sibling
`
`of the crystalline silicon used in computer chips, is currently a popular photovoltaic mate-
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`rial. While not as efficient as its crystalline brethren, it is significantly cheaper and more
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`convenient to manufacture. Undoped a-Si:H comprises the so-called intrinsic middle layer
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`of solar cells. It is the thickest layer in the cells and the majority of the electrons’ time
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`during transport is spent traversing it. As a result, the electrical properties of the intrinsic
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`layer are a critical factor in determining how well a solar cell will work.
`
`The TOF technique, which was developed in the early 1950’s to analyze crystalline
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`

`

`Introduction
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`3
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`−e
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`−e
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`−e
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`−e
`−e
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`−e
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`−e
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`−e
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`I(t)
`
`t
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`− +
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`Figure 1.1: Time-of-flight measurement technique. A short laser pulse is incident on the
`sample surface from the left. A voltage bias is simultaneously applied across the width
`of the sample, causing the photoexcited electrons to traverse the thickness of the sample.
`The resulting current is recorded on an oscilloscope.
`
`semiconductors during the transistor boom [7–11], investigates charge transport in a ma-
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`terial by measuring the current that carriers produce in response to an electric field. The
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`basic idea is described below.
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`A semiconductor (such as silicon) is made of atoms bonded together in some ordered
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`fashion. In their lowest energy configuration, electrons participate in bonding and their
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`movement is confined. In the TOF experiment, a laser is flashed briefly on a thin semi-
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`conducting sample. Some of the electrons near the surface absorb the light’s energy, and
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`subsequently they become free to move. A uniform voltage bias is applied across the width
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`of the sample so that the front contact is negatively charged. The photoexcited electrons
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`are repelled from the front contact and begin to traverse the bulk of the sample. In doing
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`so they create a current, which is measured on an oscilloscope (Figure 1.1).
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`Information about the mobility, or velocity per unit electric field, of the charges can be
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`gleaned from the current they produce. If the electrons travel as a packet with constant
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`

`

`4
`
`Chapter 1
`
`velocity, a steady current will be observed which disappears suddenly when the electrons
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`have traversed the sample.
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`If the electrons slow down or speed up as they cross the
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`sample but continue to move as a coherent group, the current will decrease or increase,
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`respectively, until the electrons complete their transit at which time it will vanish. If the
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`electrons do not move as a packet but instead travel with different velocities and traverse
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`the sample in different times, this also will be reflected in the current, and it will fall off
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`slowly as the charges reach the back contact.
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`In addition to experimentally measuring how charges move, the TOF technique can
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`provide insight as to why they move as they do. Theories exist which describe possible
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`mechanisms by which electrons interact with the structure of an amorphous material. Each
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`mechanism predicts a particular behavior for electrons subject to an electric field. TOF
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`experiments, which give direct information about electron movement, can thus support or
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`undermine theories about the structure of semiconductors. In this thesis it is found that
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`the multiple-trapping model explains the a-Si:H TOF data well.
`
`

`

`2
`
`S E M I C O N D U C T O R P R I M E R
`
`Unfortunately, there are so many variables to consider when discussing solids that gener-
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`alizations are difficult and exceptions seem to be the rule. What governs the properties of
`
`one material may play only a minimal role in a seemingly similar material. I am primarily
`
`concerned with introducing important concepts that will be used in later chapters and
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`sometimes I will ignore special cases. Consult Keer [12] for a brief introduction to solid
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`state physics, Ashcroft and Mermin [13] for a more comprehensive and accurate treatment
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`of the subject, Zallen [14] for information about amorphous solids, Street [15] for details
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`about a-Si:H, B¨oer [16] or Shur [17] for semiconductor device information, and M¨oller [6]
`
`for details regarding photovoltaics.
`
`2.1 Band structure
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`In a single atom, electrons are allowed to take on only discrete energy, angular momen-
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`tum, z-component angular momentum, and spin values. These quantities (called quantum
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`numbers) uniquely define a quantum state, and we say that electrons may occupy only
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`discrete states. According to molecular-orbital theory, bonding between two atoms creates
`a new set of available states. In a solid with ∼ 1023 bonded atoms per cubic centimeter,
`these quantum states are so numerous that they blend together to form quasi-continuous
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`bands of available states separated by inaccessible gaps. This band structure is unique to
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`

`

`6
`
`Chapter 2
`
`Conduction band
`
`Band gap
`
`Valence band
`
`Energy
`
`Figure 2.1: Band structure of a solid. Note that no information is given on the horizontal
`axis. The allowed electron energies form bands separated by energy gaps. The uppermost
`occupied band (occupation is indicated by shading) at absolute zero temperature is iden-
`tified as the valence band; the band above it is the conduction band. The solid depicted
`here is either insulating or semiconducting since the valence band is completely filled.
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`the solid and determines its electronic, magnetic, thermal, and optical properties.
`
`Consider just the energy quantum number. When the available energy states in a solid
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`are plotted, the bands and gaps are seen, as in Figure 2.1. If we were to place electrons
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`one-by-one into a solid, they would naturally move into the lowest available energy states.
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`Since there are a finite number of states available at each energy and the Pauli exclusion
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`priniciple allows only one electron to occupy a given quantum state, the energy bands
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`would fill up as we continued to add electrons. In a solid, the highest energy band that is
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`occupied by electrons at absolute zero temperature is called the valence band. The next
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`highest energy band is called the conduction band. The energy gap εg between these bands
`
`is the band gap. All the interesting physics happens in the valence and conduction bands,
`
`and higher and lower energy bands can be ignored.
`
`

`

`Semiconductor Primer
`
`7
`
`2.2 Conductivity
`
`The current density (cid:4)J produced in an Ohmic material1 by an applied electric field (cid:4)E is
`
`proportional to the field:
`
`(cid:4)J = σ (cid:4)E.
`
`(2.1)
`
`The constant of proportionality σ is known as the electrical conductivity. Solids are
`grouped in three broad categories based on their conductivities: insulators (σ < 10−8 Ω−1m−1),
`semiconductors (10−7 Ω−1m−1 < σ < 104 Ω−1m−1), and conductors (σ > 105 Ω−1m−1).
`What does the conductivity tell us about a solid? Recalling that
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`(cid:4)J = nq(cid:4)v,
`
`(2.2)
`
`where n is the number density of conducting charges, q is their charge, and (cid:4)v is their
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`velocity, Eqn (2.1) informs us that in a material with a large conductivity there will be
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`more (or quicker) moving charges for a given electric field than in a small-conductivity
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`material. Apparently then, each charge in a conductor is more mobile than each charge in
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`an insulator, there are more charges moving in a conductor than in an insulator, or some
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`combination of the two.
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`For a group of charges with zero initial net velocity to conduct a current, they must
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`change states in such a way that they gain a net momentum. In a solid, electrons need
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`nearby unoccupied states (in energy space) for this to occur. Electrons that find themselves
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`surrounded by filled energy states cannot leave their present states; they have nowhere to
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`go. This is shown schematically in Figure 2.2.
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`Charge transport, or the movement of electrons between free states, happens very
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`differently in conductors and insulators. In metals, electrons can travel within the valence
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`band with little difficulty. As we shall soon see, electron transport in insulators and
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`semiconductors can happen only in the conduction band.
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`In insulators, a formidable
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`energy gap (εg > 3 V) keeps electrons from accessing the conduction band, so that charges
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`do not move at all (on average). Semiconductors are simply insulators with band gaps
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`1An Ohmic material is any material that obeys Eqn (2.1).
`
`

`

`8
`
`Chapter 2
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`−e
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`Energy
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`Figure 2.2: Allowed electron transitions. Only electrons with empty nearby states can
`acquire a net velocity. In a semiconductor this occurs in the conduction band.
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`small enough (εg < 3 V) that electrons can sometimes cross the gap and conduct.
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`For semiconductors, there is an alternative formulation for electrical conductivity that
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`is written in terms of the density of mobile electrons n and the ease with which they move
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`in response to an applied electric field, or mobility, μn:
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`σ = n|e|μn + p|e|μp.
`
`(2.3)
`
`Here, e is the electronic charge. What, you may ask, is the p term in Eqn (2.3)? I have
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`been slightly misleading thus far: Electron motion is only half the story in conduction.
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`Each electron that is excited into the conduction band leaves behind a hole of missing
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`charge where it left the valence band. Another electron in the valence band may move
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`into this hole, creating a new hole in its former location. In this way, holes “move” around
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`the valence band much like the open parking space changes location in the Reed lot on
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`a busy afternoon.2 As a result it is useful (and customary) to think of holes as positive
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`charge carriers. Thus, p in Eqn (2.3) is the density of free holes and μp is the hole mobility.
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`2Do not take this analogy too seriously—to be completely accurate the cars would need to rearrange
`themselves without leaving the lot.
`
`

`

`Semiconductor Primer
`
`9
`
`For now, we will concern ourselves only with pure (or intrinsic) crystalline semiconductors
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`for which one mobile hole is generated in the valence band during each electron excitation
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`and hence, n = p. With this simplification, Eqn (2.3) reduces to
`
`σ = n|e|(μn + μp).
`
`(2.4)
`
`2.3 Charge generation
`
`2.3.1 Thermal excitation
`
`How exactly are electron-hole pairs generated, and how do they affect semiconductor
`
`electronic properties? That is, what determines n, and what can it tell us? In the dark,
`
`with no externally applied fields, only thermal excitation is possible. In any solid, the
`
`probability that a state with energy ε will be occupied with an electron at temperature T
`
`is given by the Fermi-Dirac distribution3
`
`fn(ε) =
`
`1
`e (ε−μ)/kT + 1
`
`,
`
`(2.5)
`
`where μ is known as the chemical potential (not to be confused with mobility) and k is
`
`Boltzmann’s constant. The Fermi distribution is shown at several temperature values in
`
`Figure 2.3.
`
`The density of occupied states in the energy range ε to ε + dε is given by the product
`
`of the number of states in that range and the probability that each state is occupied. That
`
`is,
`
`dn(ε) = g(ε)fn(ε) dε,
`
`(2.6)
`
`where g(ε) is the density of energy states (number per unit energy and volume). When
`
`Eqn (2.6) is integrated over all energies we must get back the total density of electrons
`
`3The Fermi-Dirac distribution gives back the classical Maxwell-Boltzmann distribution at high temper-
`atures.
`
`

`

`10
`
`Chapter 2
`
`T
`10 T
`20 T
`
`μ
`
`Energy
`
`0
`
`Occupation probability
`
`1
`
`Figure 2.3: The Fermi-Dirac distribution. The probability that a state above the chem-
`ical potential μ is occupied with an electron is nearly zero at low temperatures, and
`increases with increasing temperature. One minus the value of the Fermi distribution is
`the probability that a state is occupied with a hole.
`
`

`

`Semiconductor Primer
`
`11
`
`(number per unit volume) in the solid:4
`
`dn(ε)
`
`g(ε)fn(ε) dε
`
`g(ε)
`e (ε−μ)/kT + 1
`
`dε.
`
`(2.7)
`
`(cid:2) ∞
`(cid:2) ∞
`(cid:2) ∞
`
`0
`
`0
`
`0
`
`=
`
`=
`
`=
`
`N V
`
`Here, and in the rest of the calculations in this chapter, we take the ground state energy
`
`to be zero. Equation (2.7) uniquely defines the chemical potential for a given temperature
`
`and density of states.
`Referring back to Eqn (2.5), notice that as T → 0,
`
`⎧⎪⎪⎨
`⎪⎪⎩0 ε > μ
`
`1 ε < μ
`
`fn(ε) →
`
`and hence, all states below the chemical potential are occupied at absolute zero tempera-
`
`ture, while all states above it are empty.5 If some of the electrons can manage (at non-zero
`
`temperatures) to reach states above the chemical potential, they will have a whole sea of
`
`empty states to swim in. Thus, μ separates conducting states (for electrons) from non-
`
`conducting states, and is an important material parameter.
`
`In a metal, the chemical
`
`potential lies in the middle of the valence band and electrons immediately below it need
`
`very little energy to hop into the conducting states above it. This energy is provided
`
`thermally even at low temperatures, causing metals to conduct well.
`
`Where is the chemical potential in an insulator or semiconductor? For intrinsic, crys-
`
`talline semiconductors it lies in the energy gap. This means that at T = 0 every state in
`
`the valence band is occupied and every state in the conduction band is empty. As a result,
`
`in the valence band the electrons cannot obtain a net velocity (Figure 2.2), and in the
`
`conduction band where they are free to move, there are none to move. Unlike in a metal,
`
`4Note that I am using lowercase letters for number densities and uppercase letters for numbers.
`5Many authors are sloppy and use the term Fermi energy interchangeably with the chemical potential.
`The Fermi energy is the energy of the highest occupied state at T = 0 and is constant, while the chemical
`potential is temperature dependent. At T = 0 the two quantities take on the same value, and this has
`given rise to incorrect use of the terms.
`
`

`

`12
`
`Chapter 2
`
`a few lattice vibrations will not remedy the situation. In this case, promoting an electron
`
`above the chemical potential requires exciting it all the way across the band gap into the
`
`empty conduction band. Doing so requires more energy than is thermally available at low
`
`temperatures and thus, no conduction occurs.
`
`However, the Fermi distribution tells us that with increasing temperature, it is in-
`
`creasingly likely that states with energy greater than μ (including those in the conduction
`
`band) will be occupied.6 The density of electrons in the conduction band can be calculated
`
`from Eqn (2.6) if we know the density of available states:
`
`n =
`
`εc max
`
`g(ε)fn(ε) dε.
`
`(2.8)
`
`εc
`
`The limits of the integral are the upper and lower energy bounds of the conduction band.
`
`The upper bound may be extended to infinity since at finite temperatures fn(ε) drops
`
`off sharply above μ, and g(ε) has no such radical behavior that might compensate. This
`
`is important: qualitatively it means that the conducting electrons are concentrated near
`
`the conduction band edge, and it justifies our neglect of higher energy bands. Now, for
`
`energies several kT above μ the Fermi distribution approximately reduces to the simpler
`
`Boltzmann distribution,
`
`(cid:2)
`
`−(ε−μ)/kT .
`
`fn(ε) ≈ e
`Thus, if εc − μ (cid:5) kT , Eqn (2.8) can be written7
`(cid:2) ∞
`
`n ≈
`
`g(ε) e
`
`εc
`
`−(ε−μ)/kT dε.
`
`(2.9)
`
`(2.10)
`
`Finally, it can be shown that the density of states near the conduction band edge8 has the
`form g(ε) ∝ (ε − εc)1/2 [18]. With this result, Eqn (2.10) can be easily solved and yields
`
`6Of course, even in the high temperature limit it will never be more likely for a high energy state to be
`occupied than a low energy state.
`7This is a reasonable assumption for temperatures at or below room temperature. At 300 K, kT = 0.026
`eV, which is significantly smaller than half the band gap energy in most semiconductors.
`8The conduction band edge is assumed to be parabolic.
`
`

`

`a conduction electron density
`
`Semiconductor Primer
`
`n ∝ (kT )3/2 e
`
`−(εc−μ)/kT .
`
`13
`
`(2.11)
`
`A similar procedure can be used to determine the density of mobile holes in the valence
`
`band. If each excited electron leaves a hole in its wake, the probability that a state in the
`valence band will be occupied by a hole is given by fp(ε) = 1 − fn(ε) (Figure 2.3). The
`valence hole density is then
`(cid:2)
`
`g(ε) [1 − e
`
`−(ε−μ)/kT ] dε,
`
`(2.12)
`
`p ≈
`
`εv
`
`0
`
`where g(ε) is the density of states near the valence band edge εv, and the lower limit of
`
`the integral has been extended to zero since fp(ε) is appreciably large only for states close
`to the top of the valence band.9 It has also been assumed that μ − εv (cid:5) kT so that the
`Fermi distribution can be replaced by the Boltzmann distribution. As in the conduction
`band, the density of states near the valence band edge goes as g(ε) ∝ (εv − ε)1/2, and
`hence,
`
`p ∝ (kT )3/2 e
`
`−(μ−εv)/kT .
`
`(2.13)
`
`Now, recalling that each mobile electron in the conduction band has a corresponding
`
`hole in the valence band,
`
`n = p
`√
`
`=
`np
`∝ (kT )3/2 e
`= (kT )3/2 e
`
`−(εc−εv)/2kT
`−εg/2kT .
`
`(2.14)
`
`There are several interesting things to notice here. First, this result is independent of the
`
`chemical potential. Second, (kT )3/2 varies slowly and the carrier density is dominated by
`
`9This implies that hole conduction is concentrated near the valence band edge and lower energy bands
`can be ignored.
`
`

`

`14
`
`Chapter 2
`
`the exponential dependence on temperature. As a result, conductivity in semiconductors
`
`increases drastically with increasing temperature. This is contrasted with conductivity
`
`in metals, which decreases with heating because mobility is reduced by temperature-
`
`dependent scattering and carrier density undergoes no marked increase. Finally, Eqn
`
`(2.14) states quantitatively what was previously only conceptually clear: at a given tem-
`
`perature, larger band gaps εg correspond to smaller densities of thermally generated car-
`
`riers (electrons and holes) available for charge transport. This quality is what divides
`
`semiconductors and insulators. Semiconductors have small enough band gaps that sub-
`
`stantial conductivities are achieved at practical temperatures, and insulators do not.
`
`2.3.2 Photoexcitation
`
`When exposed to light, electrons in semiconductors experience photoexcitation, and this
`
`is the foundation of photovoltaics. Suppose monochromatic photons of energy ε = hν are
`
`incident on a semiconducting material. If hν < εg then the light will not be absorbed by the
`sample. If, however, hν ≥ εg then each photon may promote exactly one electron from the
`valence band to the conduction band (Figure 2.4).10 In this way, mobile electron-hole pairs
`
`are created via absorbed light, and the number of generated carriers is proportional to the
`
`light intensity. Electrons that are excited high in the conduction band by very energetic
`
`photons quickly fall back down to the conduction edge by interacting with the lattice
`
`and releasing the excess energy through lattice vibrations. This thermalization process is
`
`usually very short, on the order of a picosecond. This means that, as far as conduction
`
`is concerned, the carriers effectively gain only energy εg. Blasting a semiconductor with
`light of energy hν (cid:5) εg is no more useful than using light of energy hν = εg when trying
`to create mobile charge carriers. In fact, it is actually detrimental in the case of a-Si:H
`
`due to the creation of additional light-induced defects through a phenomenon known as
`
`the Staebler-Wronski effect [19].
`
`It is useful to introduce an optical absorption coefficient α(λ) defined as the inverse of
`
`10Some materials require a simultaneous lattice vibration at the same location that the photon “strikes,”
`even if the photon is sufficiently energetic. See §2.5 for details.
`
`

`

`Semiconductor Primer
`
`15
`
`−e
`
`−e
`
`−e
`
`e−
`
`−e
`
`−e
`
`−e
`
`−e
`
`−e
`
`−e
`
`−e
`
`−e
`
`−e
`
`−e
`
`−e
`
`Energy
`
`Figure 2.4: A sufficiently energetic photon may excite a valence electron into the con-
`duction band, leaving a hole in its wake.
`
`the distance required for the incident (wavelength λ) photon intensity to fall off by a factor
`
`of e. This parameter describes how well a material absorbs each wavelength of light and
`
`can be used to find the density of electron-hole pairs created per second at each depth from
`
`the incident surface. For an illuminated thin-film sample of specified thickness, the larger
`
`its absorption coefficient, the better it will conduct. Put another way, to photogenerate
`
`a given density of free carriers, more material is required if the absorption coefficient is
`
`small.
`
`2.4 Charge transport
`
`In an attempt to understand conductivity in semiconductors we have examined n and p
`
`in Eqn (2.3), which has amounted to investigating charge carrier generation. That μn
`
`and μp enter into Eqn (2.3) indicates that we must also be concerned about how free
`
`charges move. Suppose a packet of electron-hole pairs is created by thermal excitation
`
`or photoexcitation.
`