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ASTR 3130, Majewski [SPRING 2015]. Lecture Notes

ASTR 3130 (Majewski) Lecture Notes


DETECTORS:
CONDUCTION IN A SOLID AND THE ORIGIN OF BANDS

As we have seen, individual atoms have discrete, allowable energy levels. However, something interesting happens when we discuss an ensemble of atoms in proximity, as in a crystal lattice.

The Pauli Exclusion Principle states that no two Fermions (particles of 1/2 spin, like an electron) can have identical energy quantum states. Thus, quantum mechanics leads to an interesting consequence for solids:

  • Energy level splitting occurs as atoms are brought into proximity so that individual electrons can exist in different states.
  • In a solid, so many atoms are close together that the many multiply-split levels become a pseudo-continuous band.
  • To conduct electricity, electrons must move freely. However, in a solid:
    • Bound electrons remain bound.
    • If the valence band is full, no electrons may move in valence energy levels (semiconductor, insulator) (all bonds in the solid are "satisfied").
    • The conduction band is normally unfilled (at least with the system in the ground state), so some electron in the valence band has to absorb energy to lift it into the unpopulated energy levels in the conduction band. Only then is there an (excited) electron that is then free to move and conduct electricity.

  • Two other ways to have a solid be electrically conductive:
    • If the valence band is not full, electrons can easily move in valence band (e.g., solids made of Group Ia element "metals" like Li, K, Na).
    • If the conduction bands broaden sufficiently to overlap the valence band, the bands merge and electrons easily find new states in which to move --> Conductor (like more familiar metals Fe, Sn, Pb).
    • Recall:


    • Insulator: Solid for which the energy gap is so large that electrons can't get to conduction band (without huge energy absorption). Thus, these solids do not conduct electricity except under extraordinary conditions of a huge energy influx.

  • The band of excited states is called the Conduction Band because, while valence electrons are rather tightly held to atoms, excited electrons are not very localized to specific atoms and can move rather freely --> Produces a current (similar to electrons in valence 1 metals Li, K, Na).
    • Example: Valence electron density in germanium (Ge) crystal = |Ψ|2 = Square of quantum mechanical wave function.

SEMICONDUCTORS AS TUNED PHOTON DETECTORS

  • The key to usefulness of semiconductors for visible and infrared photon detectors is that their natural bandgap energies match those of single visible / IR photons.
    • E.g. 5500 Å (green/visible) photon E = hf = 2.26 eV can be detected (i.e., absorbed) by the first four listed semiconductor materials below:

      Semiconductor Egap (eV) Photon
      InSb 0.18 IR
      Ge 0.67 IR
      Si 1.11 NIR, Optical
      GaAs 1.43 Optical
      AgBr 2.81 Optical
      SiC 2.86 Optical
      Insulator (e.g. NaCl) > 4  

  • Thus, detectors made out of the above elements or compounds are able to detect photons with higher energies than the bandgap energies listed. Thus, these energies define the red side of the spectral response of the detector, and bluer photons than this are detectable.

  • Photons with the appropriate energy are absorbed by electrons in the valence band which then become excited into the conduction band.

  • Many "traditional" astronomy detectors have used the above semiconductors.

    • Standard photographic plates, the earliest permanent type detector, depend on the blue/UV sensitivity of AgBr and AgCl crystals. The bandgap energy above corresponds to <4400 Å.

      • The first astronomical photograph was taken of the moon by Henry Draper in 1840.

        This is not the first astronomical photograph, but a similar one taken by W.C. Bond at Harvard College Observatory in 1850 that obtained much publicity at the time. Click here to learn more about the history of astronomical photography.

    • Photomultiplier tubes used in astronomy for about the last half century have often made use of GaAs photocathodes.

      One of the first phototubes used widely to do astronomical photoelectric photometry was the RCA 1P21 tube.
    • Many of the earliest infrared semiconductor detectors made use of germanium cells.

      Aside:

      • But the first infrared detectors made use of thermography -- measuring the heat in IR radiation. William Herschel first did this with thermometers, indeed used this to discover infrared radiation in 1800. Later, infrared radiometry (measuring infrared flux) was done by putting wet black paper in the focal plane and measuring the rate at which the paper dries.

DARK CURRENT

Unfortunately, photon absorption is not the only way to excite electrons. A particularly troublesome alternative energy source is thermal excitation.

  • From statistical mechanics, the probability of an electron near the top of the valence band to be thermally excited into the conduction band follows quantum mechanical Fermi-Dirac statistics (which applies to the highly degenerate state of concentrated electrons in a solid).

    Probability of a state being filled according to Fermi-Dirac statistics for four different temperatures. From Wikipedia.
  • However, in the limit where the temperature is much larger than absolute zero (the typical situation of interest here), the probability of an electron transition to conduction band tends towards classical Maxwell-Boltzmann statistics (which may be more familiar to you and looks like the yellow line in the plot above), and therefore goes approximately as:

    Probability ∝ e-(Egap/2kT)

    where:

    k = Boltzmann constant 1.38 x 10-23 J/degree

    T = Temperature in degrees K

  • For reference, at room temperature, kT ~ 0.025 eV; so for silicon and the data in the table above, (Egap/2kT) = 22

  • Thus the key to understanding the thermal production of electrons is that it follows an exponential dependence on temperature, as shown in this plot of the relative conductivity of silicon as a function of temperature.

  • From the plot of e-(Egap/2kT) above, it can be seen how sensitive the production of thermally excited electrons is to temperature over 250 < T < 310 K.
    • Note: Due to the relative availability of thermally excited electrons in the conduction band, silicon is a much better conductor of electricity (from thermally excited electrons) at room temperature than at very cold temperatures, where it's an insulator.

  • The electrical current produced from thermally excited electrons is known as Dark Current.
    • Obviously, one can not discriminate thermally excited electrons in the conduction band from those electrons that have been photo-excited into the conduction band -- the kind of excited electrons we are generating by exposing the detector to an astronomical source!

    • Thus it follows that if you do not want your current signal dominated by thermally excited electrons, you must cool your detector.
    • Even if you cannot completely eliminate thermally excited electrons, for example because you cannot cool your detector sufficiently, if the rate of production of these dark current electrons is constant (i.e. if the temperature of the detector can be kept constant), the dark current can be measured, calibrated, and corrected for.

      (But note, it is always better for the dark current to be a lesser contributor to the total excited electron production.)


ELECTRON / HOLE MIGRATION AND ELECTRICAL CURRENT

  • Note: When electrons elevate to conduction band, they leave behind empty positions, or HOLES, in the valence state with an effective positive charge in the crystal lattice.
  • Other ways to think about this is to consider the occupancy of distinct energy levels in the bands of the solid. We can think of these energy levels being occupied either by electrons or by "holes":

  • If a hole is created, it can be filled by a valence electron from a neighboring atom, but this leaves a new hole. The hole is said to migrate through the valence band.
  • One can think of the bonds in the lattice being filled and unfilled as the electrons jump from bond to bond:

  • Although the holes are not real subatomic particles, they behave in many situations as if they were and it is convenient to discuss them as positive counterparts to electrons with attributes of mass, charge, and velocity. For example:

  • The total electric current in a semiconductor has contribution from motions of both conduction electrons and "e+" or holes.


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Bands image from mrsec.wisc.edu/edetc/ LED/band.htm. Hole migration image from www.madlab.org/electrnx/ lesson4.html. Germanium electron density image adapted from Physics, by Jay Orear. Other images from http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html, and http://cfa-www.harvard.edu/newtop/previous/071700.html. RCA tube image from http://cfa-www.harvard.edu/newtop/previous/071700.html. All other material copyright © 2002,2006,2008,2012,2015 Steven R. Majewski. All rights reserved. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 313 and Astronomy 3130 at the University of Virginia.