electrons holes exp[ (CBM E F )/k B T ] = exp[ ½E g /k B T ] = e

Transcription

1 Charge Carriers in Semiconductors Charge carriers = carriers = electrons and holes An undoped ( intrinsic ) semiconductor has very few electrons in the conduction band, because a typical band gap of 1 ev is large compared to the thermal energy k B T 25 mev (at room temperature). That makes the probability extremely low for having an electron excited across the band gap. To make this argument quantitative, one has to determine the occupancy at the conduction band minimum, which is determined by the Fermi-Dirac distribution f(e). Since f(e) contains the Fermi level E F (see Lecture 14), one needs to know where in the band gap E F is located. The position of E F is determined by charge neutrality. The density of electrons in the conduction band has to match the density of holes in the valence band. That puts E F close to mid-gap for intrinsic semiconductors. The figure illustrates how to determine the Fermi level position from charge neutrality. electrons holes To calculate the density of electrons and holes (dn/de and dp/de in the figure), multiply the density of states D(E) by the Fermi-Dirac function f(e) and [1 f(e)], respectively. If the density of states in the conduction band D C (E) is smaller than that in valence band, the Fermi level has to move slightly upwards to compensate the decrease in D C (E) by an increase in f(e), as shown in Panel b on the right. The difference is small, since f(e) varies very rapidly with E near the band edges. For intrinsic silicon (E g 1.1 ev), with E F mid-gap, the probability of having an electron excited into the conduction band minimum (CBM) at room temperature is roughly: 0.55eV / ev exp[ (CBM E F )/k B T ] = exp[ ½E g /k B T ] = e e (Approximate the Fermi-Dirac function by exp[ (E E F )/k B T ] for (E E F )» k B T. )

2 Holes in Semiconducturs = Antiparticles Creating an electron-hole pair with a photon (which has negligible momentum): E k e E e CBM E g E photon E F =0 E h k h VBM uncompensated momentum G/2 0 +G/2 k Hole = (N 1) Electrons + N Counter-Charges (Ions) Use conservation laws to determine the quantum numbers of a hole: Quantum Hole Electron Conservation Law Number: h e Charge q +e e q h + q e = q photon = 0 Energy E (E-VBM) +(E-CBM) E h + E e + E g = E photon >0 >0 Momentum ħk k e +k e k h + k e = k photon 0 Holes are the equivalent of antiparticles in quantum field theory. In quantum electrodynamics, for example, positrons are sometimes characterized as missing electrons with negative energy that are moving backwards in time. We would simply say that they are holes. Likewise, the vacuum of quantum electrodynamics is actually a semiconductor. The Dirac equation for electrons has solutions for both positive energy (the conduction band) and negative energy (the valence band, called the Dirac sea in particle physics). The band gap of the vacuum is 2m e c 2 1 MeV, which corresponds to the minimum energy required to create an electron-positron pair by a photon.

3 Doping To achieve a significant electron density in the conduction band, one has to bring the Fermi level closer to the conduction band minimum to allow thermal excitations of an electron into the conduction band. That is achieved by adding a donor, i.e. an atom that has an occupied level close to the conduction band minimum. The Fermi level has to lie above the donor level (at low temperature), since the level is occupied. Likewise, on uses an acceptor atom to push the Fermi level close to the valence band and create holes. An acceptor has an empty level close to the valence band maximum. Donors are typically one row to the right of the semiconductor in the Periodic Table (e.g. P, As, Sb for Si), while acceptors are one row to the left (B for Si). E E E F Donor E F Acceptor D(E) D(E)

4 Calculating Donor / Acceptor Levels The H atom is used to model the extra electron orbiting a donor atom (or the hole orbiting an acceptor atom). The Rydberg constant Ry e 4 m e and the Bohr radius a 0 [e 2 m e ] 1 (Lect. 20, p. 3) need to be adapted to a solid by substituting e 2 e 2 /ε and m e m eff m e. With typical values of 10 and m eff 0.1, the donor binding energy E d is reduced from 1Ry = 13.6 ev by a factor of m eff / That makes it comparable to the thermal energy k B T 25 mev and allows thermal excitation of electrons from the donor level into the conduction band (or from the valence band into the acceptor level). E d = Ry m eff / 2 a d = a 0 /m eff E d T[K] E [mev] to n= (dotted line). are extrapolated

5 Fermi Level E F and Carrier Density n versus Temperature and Doping As the temperature increases, E F moves towards the middle of the gap. Low T Temperature T High T High T Low T Intrinsic Saturation Freeze-out

6 Room Temp.

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8 Mobility versus Temperature for Various Materials The mobility is defined as the electron velocity divided by the electric field (Lect. 25). High mobility is important for fast and efficient electronics. To optimize the mobility, one has to avoid scattering of the electrons. Phonon scattering can be reduced by freezing out phonons at low temperature. To avoid scattering by defects and impurities one needs to perfect crystals. However, dopants also act as defects. They can be eliminated by modulation doping, where the dopants are spatially separated from the carriers (Lect. 33, p. 1,2). Silicon has a mobility of about 10 3 cm 2 /Vs. Electrons tend to have higher mobility than holes, because the effective mass of the conduction band minimum is often smaller than that of the valence band maximum (see the preceding page). Cu Pentacene Si