PHYS 4315 Lab THE CLASSICAL HALL EFFECT measured by the lock-in amplifier Introduction to the lock-in amplifier Introduction to the classical Hall effect Measuring the classical Hall effect on n-type and p-type Ge NOTE: It will be useful to read through this lab guide first. Pay attention to section 5: Data analysis and content of reports, so that you can plan your data acquisition and subsequent data analysis and literature research accordingly. The Lock-in Amplifier Introduction It is often necessary to detect a miniscule signal, typically a voltage signal, embedded in a large amount of background noise. The lock-in was invented for that reason: to detect faint electrical signals buried in electrical noise. Since practically all measurements are ultimately reduced to an electrical signal, the lock-in is an important instrument, and is widely used in many forms. Noise is characterized by a frequency spectrum (figure below), which tells you what the magnitude is of the noise when measured at a particular frequency, or more accurately in a narrow bandwidth around a certain frequency. Broadly speaking, noise may be categorized into one of three groups: interference, 1/f noise, and Johnson noise. Once we understand these three types of noise, we can see how lock-in detection may be used to reduce them. The idea of the lock-in is quite simple: from among all the frequencies in a frequency spectrum polluted by noise, listen to one particular frequency (and phase), in a narrow bandwidth, at which you know your experiment is driven. It is akin to humans recognizing one voice from among a loud mishmash of acoustic noise. Interference This is likely the first type of noise that comes to mind, and simply put, it is noise due to human- made sources: 60 Hz, 120 Hz, 180 Hz noise from power lines, television and radio signals in MHz and GHz, digital switching noise in electronics, switching phenomena in computer and lighting power supplies, etc. all contribute to interference. This noise tends to occur in a narrow bandwidth around specific frequencies. Fortunately, interference is relatively easy to avoid simply by choosing a frequency other than one of these known frequencies and performing the measurement in a manner that uses knowledge of the interference frequency and of the driving frequency. The solution could be as simple as using a band pass filter. Or as we will see, the lock in amplifier provides a solution as well. 1/f noise It turns out many noise sources have a frequency spectrum that approximately varies as 1/f (large amplitudes occur relatively infrequently, smaller amplitudes more frequently). This noise originates from a variety of sources. For our purposes we need not concern ourselves with why the 1/f relationship exists, but only to accept that it does and choose a frequency for our measurement so that we minimize this noise. At a glance it appears that this can be accomplished by choosing a high frequency, but in actuality we need only to use a frequency such that the 1/f noise is much less than the ‘white’ Johnson noise (below). This happens around 1 kHz in many cases, but is dependent on experimental factors such as temperature and measurement bandwidth. Johnson noise Johnson noise, also called thermal noise, is always present since the apparatus is always at some temperature above absolute zero. This noise can be derived by considering a long lossless transmission line in series with a resistor, applying equipartition and equating the power flowing through the line with the power dissipated by the resistor. This yields the result: = √4 with R the resistance of the resistor, k the Boltzmann constant, T the temperature in Kelvin, and the frequency range (bandwidth) over which the measurement is sensitive. For instance, if you use a voltmeter with a low-pass filter that measures signals from DC to 30 Hz, then = 30 Hz. Johnson noise is independent of frequency and is therefore called white noise (the color white is composed of all frequencies). Note that it is proportional to bandwidth. Note also that you can normalize to bandwidth by expressing the noise in units of Vrms/√Hz, “volts per root hertz”, a unit that one often encounters when studying electrical noise. Clearly, we could reduce T and possibly R to minimize noise, and indeed we should. However often that is not a practical approach. Another recource then is to reduce the bandwidth of the measurement. This is where the power of the lock in amplifier comes in handy. The lock-in amplifier From the above, it is apparent that to reduce noise in an electrical measurement we should narrow the bandwidth of frequencies over which we perform the measurement, and we should pick certain measurement frequencies over others as being less polluted by noise. As we will see, a lock in amplifier in essence acts as a narrow band-pass filter that tracks a reference frequency. It automatically centers itself on the reference frequency, even if this frequency drifts a little throughout the experiment. With a lock-in amplifier, you drive an experiment with an AC signal at a fixed reference frequency and phase (the reference signal). You then detect the response of the system at that same frequency or at one of its harmonics, and at a fixed phase w.r.t. to the reference (you lock into the reference). The technique is versatile and powerful. Imagine that you want to detect the weak return signal of a radar signal bounced off our moon. You can detect the returning electromagnetic wave, reflected off the moon, over the noise if you chop the outgoing electromagnetic wave with a given frequency and with a known phase (you can do that by rotating a metal disc with a cutout in front of the emitter horn), and then listen in your detector for a return signal at that frequency and at an appropriate phase. A signal you detect at another chopping frequency is not likely to originate from the chopped electromagnetic wave you sent out. Lore has it that a similar experiment was one step towards the invention of the lock-in, when Bob Dicke, a young physicist working on the radar effort during WW2 at the Radiation Lab at MIT, decided to try an experiment and invented or refined the lock-in in the process (he went on to make major contributions to atomic physics, cosmology, astrophysics, etc.). If everything is set up correctly with a lock-in, the meaningful signal is the one that is modulated at the reference frequency, or at one of its harmonics. The lock-in will then allow you to reject a lot of the noise that would otherwise creep into the measurement. How does a lock-in work Modern lock-in amplifiers are sophisticated pieces of equipment. However, at their core, they consist of 3 main parts. These are a preamp to amplify the incoming signal to be detected (along with the noise at this point), a signal mixer to multiply the input signal and the reference waveform, and a low-pass filter. Let’s imagine we have a signal of the form () = (1) and a reference of the form () = (2 + ). Then after the signal mixer (which multiplies the inputs) we have: = (1)(2 + ) = 2 {[(1 + 2) + ] + [(1 2) ]} The sum (1 + 2) corresponds to a high frequency. The low-pass filter is designed to filter out this high frequency. The low-pass filter will pass the low-frequency signal at (1 2) however. So we can drop the first term, and the voltage output will be: = 2 [(1 2) ] At this point you may notice the importance of the low-pass filter. If a simple one-stage low-pass RC filter is used, we see a bandwidth of 1/RC with a drop off of 6 dB per octave of frequency. If a two-stage low-pass filter is used, the drop off is 12 dB per octave, and so on for more stages (with modern lock-ins you typically can choose 6 dB, 12 dB, 18 dB, 24 dB). Concerning the difference frequency (1 2) , this signal is transmitted as long as the signal frequency is sufficiently close to the reference frequency. Let’s suppose ω1 = ω2. Notice that the phase difference, if any, attenuates the signal, = 2 [(1 2) ] = 2 [(1 2)][] + 2 [(1 2)][] But since [(1 2)]~0 and [(1 2)]~1 we can describe the output as = 2 () After scaling by a trivial factor 2, the output has the familiar form of a phasor. A lock-in can simultaneously detect both the signal component in-phase (φ = 0) with the reference signal, and the signal component 90o out-of-phase (φ = π/2) with the reference signal. These can be called resp. “X” and “Y”, like the components of a complex number X + iY. The output of a lockin is a phasor X + iY. In the complex plane also, i denotes a π/2 phase shift. A modern lock-in also allows you to phase-shift the reference signal, so to change φ. For instance you know that φ = 0 (as written above) if the X-signal is maximized and the Y-signal minimized. Quantifying noise reduction A one-stage low-pass filter with time constant RC will pass frequencies below = 1 . In fact frequencies at = 1 will be attenuated by 3 dB, so by a factor 0.71 (since 20 log10(0.71) = 3), and higher frequencies will be attenuated more. So with the lock-in scheme described above, we are allowing signal frequencies from 1 = 2 1 to 1 = 2 + 1 . This yields = 2 and hence the bandwidth of the lock-in measurement is = 1 . We now see that the effect of e.g. Johnson noise on the output signal of a lock-in depends on the time constant of the low-pass filter as: ∝ 1 It is apparent that one can reduce the effects of noise by increasing the time constant of the low pass filter. However, this comes at the expense of increasing the time the measurement takes to stabilize since you are effectively integrating the signal over a longer period of time. Alternatively, you can think of increasing the time constant as decreasing the bandwidth of a band pass filter with center frequency ω2 that tracks the reference signal. With the lock-in filter setting of 6 dB, the bandwidth of the lock-in measurement is = 1 . It follows that with the setting of 12 dB, the bandwidth is half that, with a setting of 18 dB, one third that, and with a setting of 24 dB, one quarter that. How do you actually change the RC time constant on a lock-in The lock-in will have a knob for preselected RC time constants, e.g. RC = 100 ms, 300 ms, 1 s etc. Note that you should select the low-pass filter cut-off frequency 1/RC to be much lower than the measurement frequency ω1 = ω2 , otherwise the sum frequency ω1 + ω2 may not be fully filtered out and the output voltage will oscillate. The SR830 lock-in Hall effect voltage signals are typically small, in the mV to nV range. Hence often a lock-in amplifier is used to measure Hall voltages. The lock-in you will use is the SR830, made by Stanford Research Instruments. Most lock-in amplifiers have a similar panel layout and similar controls and input and output connectors, so once you know how to use one of them you can be confident around practically any other lock-in. When you turn on the SR830 lock-in amplifier, it will perform a self-check. Lock-in amplifiers measure in V RMS (root mean square), not V peak-to-peak or V amplitude (half peak-to-peak). That means the sensitivity scale is in V RMS and the Sine Out driver signal (see below) is in V RMS, etc. Output voltages are referenced to line ground (earth ground). The Time Constant section sets the RC time constant. We will use 300 ms. There, under Slope/Oct, you also select the filter that sets the bandwidth of the measurement. We will use 18 dB. The Signal Input section is where you connect the signal to be measured, using BNC cables. In the Signal Input section, push the Input button until A-B is lit up. This makes the lock-in understand it should expect a differential voltage input signal between A and B (“Voltage A – Voltage B”), which is what we will need for the Hall voltage signal. Further, select AC coupling, and select Ground (this connects the outers of the BNCs to earth ground, so that they form a good shield). The Sensitivity section selects the amplification of the signal. In the Sensitivity section, push the arrows until the sensitivity is set to 1 V (full scale, least sensitive to start with). The Reserve section optimizes the distribution of amplification to avoid signal overload in the case of very high noise. We set it at Normal. The Filters section gives you the option to filter out the line frequency (60 Hz) or 2x line frequency (120 Hz). We will not use those filters. The center Display section with Channel One and Channel Two displays, shows the value of the phasor components. Remember that the output of a lock-in is a phasor: it simultaneously detects both the signal component in-phase (φ = 0) with the reference signal, and the signal component 90o out-of-phase (φ = π/2) with the reference signal. These can be called resp. “X” and “Y”, like in a complex number X + iY. We will set the displays to X and Y. You can also set the displays to show magnitude R = (X2 + Y2)1/2 and phase θ (that is what we called φ above), given that you can write X + iY = R eiθ = R cosθ + iR sinθ. The Hall signal of interest will be represented by a voltage value at the CH1 BNC output, set to output X. So, select X as output at the CH1 BNC (in fact if you select X as the choice for the display, you can also choose Display as output). The convention is that the BNC output will give a signal ±10 V at full scale. This means e.g. that if you set the lock-in at the 2 mV sensitivity scale, and your signal has a magnitude of 2 mV (RMS) and zero phase, the CH1 BNC will output +10 V. Still with the lock-in at a 2 mV scale, and supposing your signal has a magnitude of 2 mV and π phase, the CH1 BNC will output -10 V. Supposing your signal has a magnitude of 1 mV and π phase, the CH1 BNC will output -5 V. Etc. It will be important to keep this scaling in mind when you electronically acquire the data. The output voltages are referenced to line ground (earth ground). You should not have to use the CH2 BNC output. The Auto section allows you to tune the lock-in more rapidly than doing it manually. We will not use the Gain and Reserve selections. Yet the Phase selection can be useful. See also more about the phase under the description of the Reference section. Auto Phase will set φ (called θ on this lock-in) such that the X signal will be maximally positive. If you know that physically Lock-in amplifiers typically do not “autoscale” their sensitivity setting. Autoscaling would not be compatible with their typical use. This means you have to constantly watch the X and Y readings and choose the proper scale manually. The proper scale is the setting just larger than the signal, so you obtain maximum resolution and avoid overload (OVL signal lights up). Before connecting or disconnecting the A and B inputs, put the sensitivity to 1 V (full scale, least sensitive) to prevent damage to the delicate input circuitry. Before turning the lock-in off, also put the sensitivity to 1 V. you expect your X to be positive, then the Auto Phase operation nullifies the effects of small stray phases you may have, e,g, due to capacitance of coax cables, inductance of long cables etc. But Auto Phase can also lead to confusion by artificially make an X signal that you expect to be negative into a positive signal. For instance, in the Hall effect experiment you likely don’t know the expected sign of the Hall differential voltage signal, so X could very well be negative in a physically meaningful way. In that case check that the phase remains fairly close to zero, within a few degrees (see below, Reference section). Setup section and Interface section: not used in this experiment. The Reference section allows control of the phase φ (called θ on this lock-in), the frequency of both the driver/measured signal (ω1/2π) and the reference signal (ω2/2π), control and output of a handy driver AC voltage signal, input for an external reference voltage signal etc. The selected parameter will be shown in the display, and can be changed by the knob. We mentioned that a lock-in allows you to phase-shift the reference signal, so to change φ. You can change phase φ by selecting Phase, and turning the knob, or by using the buttons to add or subtract in increments of π /2. We also mentioned that typically with a lock-in amplifier, you drive an experiment with an AC signal at a given frequency ω1/2π and phase relative to a reference signal with frequency (ω2/2π) and phase φ (ω2/2π and φ characterize the reference signal; let’s assign phase zero to the driver signal; then the reference signal has phase φ). You then detect the response of the system at the parameters of the reference signal, namely at frequency ω2/2π, which is usually ω1/2π or one of its harmonics nω1/2π, and at a fixed phase φ relative to the driver signal. Typically the reference signal either is the driver signal or has the same frequency as the driver signal or the reference frequency is a multiple (higher harmonic) of the driver frequency. The pure sinusoidal driver signal is output at the Sine Out BNC. You will use this driver signal to generate a sample current creating the Hall effect in the semiconductor sample. You control the driver signal RMS amplitude via the Ampl button, and its frequency ω1/2π via the Freq button. The Harm # button allows you to set the reference frequency ω2/2π (for the lock-detection process) to the 1st harmonic ω1/2π or to the 2nd harmonic 2ω1/2π or 3rd harmonic 3ω1/2π etc. We will detect at the 1st harmonic ω1/2π (Harm # = 1). You can also feed the lock-in with an unrelated external reference signal (not necessarily purely sinusoidal), at the Ref In BNC, and select how the lock- in will trigger on that signal. That external signal then actually functions as the driver signal: the Sine Out BNC gives a pure sinusoidal signal at the 1st harmonic frequency (fundamental frequency) and at the phase (taken as the zero of φ) of the external signal. Hence the shorthand Ref In is somewhat of a misnomer; it is just that the reference signal is synthesized internally in the lock-in based on this external signal. If you use the internal reference then you select Source as Internal. If you use an external reference signal you unselect Internal. If then the lock-in cannot detect a reference signal, the Unlock sign will light up. We will drive the semiconductor sample using the internal Sine Out signal, and hence we select Source as Internal. We will detect at the 1st harmonic. Note that the setting of phase φ rotates the reference coordinate axes by angle φ (positive CCW) w.r.t the driver coordinate axes. The coordinates X and Y in the Display section are measured w.r.t. the reference coordinate axes. Once you actually use the lock-in in your experiment the actual meaning and action of all these settings will become clearer. The Hall effect in germanium 1. Introduction The classical Hall effect. When an electric current I flows through a conductor which is placed in a magnetic field B, the magnetic field exerts a transverse force F, the Lorentz force, on the moving charge carriers (see the figure below; we will define the geometry and symbols below). This force tends to push the carriers to one edge of the conductor. A charge imbalance is built up between the edges, giving rise to a transverse component of the electric field E. This component (Ex) is normal to both the current density vector j and the applied magnetic field B. The transverse electric field component results in a measurable voltage across the width W of the conductor. The generation of the transverse electric field component and associated voltage is called the (classical) Hall effect. The Hall effect can be used to measure the charge-carrier concentration (n), and to determine the sign of the charge carriers. The Hall effect is also used in sensitive magnetic sensors (the instrument you will use in this lab to measure B is in fact based on a semiconductor Hall sensor). In this lab, you will measure the classical Hall effect in two elemental doped semiconductors, n-type and p-type germanium. Other Hall effects. The classical Hall effect forms the subject of this experiment. The phenomenon carries the name of Edwin Hall, a US physicist who discovered it in 1879. However, other types of Hall effects have quite recently been discovered, some of deep significance as topological quantum phenomena, and sometimes still only partially understood. The Hall effects are part of solid state physics. We know of the integer quantum Hall effect (Nobel prize in Physics 1985), the fractional quantum Hall effect (Nobel prize in Physics 1998), the spin Hall effect, the quantum spin Hall effect (measured in 2007), the anomalous Hall effect, and the quantum anomalous Hall effect (discovered in 2013). The integer quantum Hall effect is presently used in metrology, as it allows a very precise measurement of the ratio e2/h of fundamental constants (e2/h has units of electrical conductance and is the quantum of electrical conductance). 2. Background of the classical Hall effect The relationship between the current I flowing in an electrically conducting sample (metal, semiconductor, electrolytic solution, plasma) and the applied voltage V is, in the absence of an applied magnetic field, approximated by Ohm’s law, V = I R. Here R is the resistance of the sample. This assumes a linear relationship between voltage and current, which is indeed often found if voltage and current are sufficiently small. Often two-point current-voltage (I-V) measurements are used to find R. Two-point means that the electrical contacts to the sample, used to apply the current are the same as the contacts used to measure the Ohmic voltage drop. In four-point measurements (which you will use in this experiment), the current and voltage contacts are not the same, which allows more flexibility in the measurements and allows us to eliminate the effects of the contact itself. The resistance R that is measured depends on materials parameters and on sample geometry. Consider positively charged particles, of charge q = +e (with e the elemental charge) drifting in a conductor with a drift velocity vd (an average velocity) under a voltage existing over the conductor. The drift velocity is not the instantaneous velocity of a given particle, rather it is the average velocity of an ensemble of particles left over after their individual random motion has been averaged out. Consider the geometry depicted in the figure on the previous page. For now, let’s omit the magnetic field B. A sample is oriented with its long side, of length L, along y. It has a width W along x and a thickness t along z. A current I is applied, such that there is a current density vector j, mostly along y throughout the sample (see below). The areal cross-section through which I flows is Wt = A, so j = I / (Wt). Let’s assume t is small so that the current density along z, jz ≈ 0. Also assume that L / W is very large, so L / W → ∞. Then the current density along x, jx ≈ 0. Hence the current density vector j will be very predominantly along y and j = jy . The current density j is associated with the drift velocity vd : j = n e vd , where n is the carrier density. The applied current generates an electric field E along y, given by: j = σ E , or E = ρ j , where σ is the electrical conductivity and ρ = 1/σ is the electrical resistivity of the material. Drift velocity and electric field are related by: vd = μ E , where μ is the carrier mobility. Combining some expressions we find: ρ = 1 / (n e μ) and σ = n e μ . So we have now defined two parameters, mobility μ and carrier density n. Let’s relate these parameters to measurable quantities. In the figure we apply a current I as indicated. We can measure the voltage drop between points 1 and 2, and relate this voltage drop to the electric field E // y : V1 – V2 = E L. Combining some expressions we find: R = (V1 – V2) / I = E L / (n e μ A E) = L / (n e μ A) , R = ρ (L / A). We can measure V1 – V2 and I, and from the sample geometry, L and A, we know ρ = 1 / (n e μ). But we don’t know n yet. To know n, we will apply B, oriented along z, as in the figure. Application of B changes the picture considerably. As before, the current density is along y, j = jy. B oriented along z will induce a Lorentz force F, transverse to vd and to j , so along x. F will induce a component of E along x. So now E is no longer purely along y, but will have an x- component. E and j are no longer parallel, and writing E = ρ j with ρ a scalar will no longer work. The resistivity has to become a matrix instead of a scalar. In the geometry of the figure, the so-called magnetoresistivity matrix links the electric field vector E to the current density vector j as follows: = 0 0 00 Here ρ = 1 / (n e μ) is still the scalar resistivity, entering in every (non-zero) element of the matrix. Note that the relation between E and j is still linear, it just has become 3-dimensional. With jz = 0 and jx = 0 we find: Ex = – μ B ρ jy and Ey = ρ jy and Ez = 0. The expression Ey = ρ jy shows that along the y direction nothing has changed from the case B = 0: we found the same relation above. But Ex = – μ B ρ jy shows that jy has now induced a component Ex. This component will lead to the classical Hall effect. We can measure the voltage drop between points 3 and 4, and relate this voltage drop to the electric field component Ex : V3 – V4 = Ex W = VH , where we have introduced the Hall or transverse voltage VH. Also, VH = – μ B ρ jy W = – μ B ρ (I/(Wt)) W = – μ B (1/(n e μ)) (I/t) = – (B/(ne))(1/t) I RH = VH / I = – (B/(ne))(1/t) , which defines the Hall or transverse resistance RH. We can call this quantity a resistance because the relation between Ex and jy is still linear. Note that RH is linear in B and that VH is linear in both B and I. You will experimentally verify these linearities during this lab. The expression Ey = ρ jy shows that we can still define what we called “R , the resistance of the sample”, above. Except, to be careful with words, we will now call R the longitudinal resistance, and use the symbol RL. And we will call V1 – V2 = VL, the longitudinal voltage. As above, we have: RL = (V1 – V2) / I = VL / I = ρ (L / A) = (1/(n e μ)) (L / A) . The longitudinal voltage drop is dissipative in energy, because Ey jy ≠ 0. But the Hall voltage drop is non-dissipative, because Ex jy = 0. Note that in our sample geometry, we found Ex = – μ B ρ jy , with a minus-sign. This means that the actual electric field component Ex points opposite to what is depicted in the figure. You can draw a Lorentz force diagram using F = q vd x B, to figure out that indeed, for positive charges flowing through the sample, there will be a net positive charge accumulation as indicated in the figure, on the right side of the sample. There will be a net deficit of positive charge (negative charge) on the left side. Ex points from positive to negative charge, hence to the left. So, the Lorentz force diagram and the magnetoresistivity matrix come to the same conclusion. In the experiment corresponding to the figure and assuming positive charge carriers, we would find V3 – V4 < 0. For negatively charged charge carriers, the sign would be the opposite, and we would find V3 - V4 > 0. In your particular setup, depending how you wire up the circuits, you should figure out which Hall voltage sign you expect for positive or negative carriers. You can use the direction of the conventional current, the direction of vector vd and the direction of vector B plus you knowledge of the Lorentz force F to figure this out. Remember that for holes in p-Ge (positive charge carriers) the direction of the conventional current and of vd are the same, but for electrons in n-Ge (negative charge carriers) the direction of the conventional current and of vd are opposite. The sign of the carriers can be determined from the sign of the Hall voltage. By plotting RH vs B, and having knowledge of t, and using RH = VH / I = ± (B/(ne))(1/t), you can determine the carrier density n. 3. Aim of the experiments In this lab, you will measure the Hall resistance of p-doped Ge (p-type Ge or p-Ge, where the charge carriers are majority positive) and n-doped Ge (n-type Ge or n-Ge, where the charge carriers are majority negative). In both cases, you will be able to determine the sign of the charges and the charge carrier density. You will also verify the linearity of RH on B and I. We briefly review the physics of charge carriers in solids to understand the observations that you will make. The theory of electrical conduction in metals, semiconductors and insulators relies on the energy band theory of solids. The presence of the lattice of atoms in the solid leads to “bands” of allowed energy levels and forbidden bands (energy gaps). The allowed energy levels are filled up with electrons, until you run out of electrons. The highest filled energy level is called the Fermi level, if this level falls in a band of allowed energy levels (metal). It may also happen that the highest filled energy level occurs exactly at the edge of a forbidden band, so that the highest energy band is completely full (insulator). The electrical behavior of a material is critically dependent on which case happens. If the highest filled energy level falls in a band of allowed energy levels, such that that band is only partially full, then you only need a tiny amount of energy to raise electrons to a higher empty energy level with an electric field, and you have a metal (a good conductor). Also, the number of available electrons for conduction is just the number of electrons in the highest energy band (and this number is independent of temperature). In contrast, in an insulator or semiconductor the highest energy band (referred to as the valence band) is completely full. There is an energy gap (typically ~ 0.3 – 2.5 eV) between the full valence band and the (almost) empty conduction band. To raise an electron to the next higher empty energy level, you need to overcome this energy gap. At T → 0 K, the conduction band would be empty. At finite temperature, some of the electrons in the valence band get thermally excited to the conduction band. They leave behind positive “holes” in the valence band. The current in an “intrinsic” undoped semiconductor consists of the flow of the electrons in the conduction band and the holes in the valence band. The carrier densities of electrons and holes are very sensitive to temperature. Doped semiconductors are created through the controlled addition of impurities to intrinsic semiconductors. They result in an excess of one type of charge carrier over that present in the intrinsic semiconductor. They come in two types: n-type semiconductors have an excess of negative electrons, while p-type semiconductors have an excess of positive holes. In p-type Ge you will find that at room temperature, it has a Hall resistance that is consistent with a positive charge carrier. At room temperature, the density of holes created by the impurities is larger than that of the intrinsic electrons and holes, but as temperature increases the density of intrinsic holes and electrons can exceed that of the impurity holes. The electrons have a higher μ than the holes in Ge, and the intrinsic electrons have effectively more contribution to the Hall resistance than the holes. Thus, at a certain temperature you may see the Hall resistance change sign. The energy gap in Ge is 0.67 eV. The picture above is somewhat simplified. The energy band structure in solids can give rise to positive charge carriers in metals as well (still called holes), for instance in aluminum. Ultimately, the dynamics of particles in solids are different from vacuum dynamics due to the fact that whereas outer space (vacuum) has complete
