Charge carrier density in metals and semiconductors

Transcription

1 Charge carrier density in metals and semiconductors 1. Introduction The Hall Effect Particles must overlap for the permutation symmetry to be relevant. We saw examples of this in the exchange energy in atoms. Does the same hold true for solids? The answer depends on the carrier concentration. Exchange energy is large in metals with large free electron densities, and small in semiconductors. The mobile carrier density can be found with the Hall Effect. This is a potential difference VV HH that appears perpendicular to the direction of current flow in an applied magnetic field, due to the Lorentz force. The drift velocity approximation, valid for small magnetic fields, adds to the forces on a charged particle moving in crossed EE and BB fields, a force due to collisions in a solid. This force is proportional to its velocity vv = μμμμ, where μμ is the mobility of the particle. The free-particle solution is not a good starting point in small magnetic fields and with high scattering rates, because charges cannot complete even a small fraction of a cyclotron orbit before being scattered (ωω cc ττ 1). We must solve the problem from the beginning. For a EE xx field, a BB zz field and an induced EE yy field (figure 1(a)) we get {nnnnee xx + II yy BB zz mm II eeττ xx = 0, nnnnee yy II xx BB zz mm ee ddpp yy dddd = 0 = eeee yy eebb zz vv xx pp yy ττ. eeττ ee II yy = 0}. Alternatively, ddpp xx dddd = 0 = eeee xx + eebb zz vv yy pp xx ττ and Setting II yy = 0 in the steady-state gives {nnnnee xx mm eeττ ee II xx = 0, nnnnee yy II xx BB zz = 0} or ρρ EE xx = mm (no effect of BB II xx ee 2 nnττ zz ), ρρ HH EE yy = BB zz, and tan φφ ee II xx nnnn HH = EE yy = BB zz eeττ EE xx mm ee The free-particle limit corresponds to ττ ee and EE yy = 0 condition. Then, nnnnee xx + II yy BB zz = 0 nnnnvv yy = II yy = nnnnee xx BB zz vv yy = EE xx BB zz and II xx BB zz = 0 II xx = 0, as expected. The Hall coefficient is defined as RR HH VV HHtt = 1, where tt is the thickness of the material. IIII nnnn Measuring VV HH (BB) with a variable applied field BB and fitting for the slope is the standard method to find the density of mobile charge carriers and the sign of their charge. For instance, the slope is much smaller in Cu than in doped Si, and its sign is different for nn doped (with electrons) and pp doped Si (with holes) because the sign of mobile charge carriers is different. Page 1

2 BB zz 1 4 EE yy (a) EE xx 2 3 (b) Fig.1: (a) Applied (EE xx, BB zz ) and induced (EE yy ) fields. (b) Geometry and contact definitions in practice. The current is injected across two contacts, and the voltage measured across two different contacts. The current flow will depend on the sample shape and current and voltage contact positions, and therefore, so will the measured resistance RR = VV II. 2. Measuring resistivity of thin films You measured the resistivity of a thin wire (B-E statistics) and a thick disk (Meissner Effect). The samples of this experiment are in the form of a thin film. How can we measure its resistivity? In ideal conditions, a thin film sample on a completely insulating substrate has a resistance RR = ρρρρ wwww, where ρρ is the resistivity of the material, ll the film length, ww and tt are the width and thickness of the film. The quantities that are not well-known are ρρ and tt. Because of this, instead of solving for ρρ, we solve for ρρ = RRRR. We can measure RR, ll, ww, but a further tt ll simplification is useful and is often chosen to make the comparison of results from different labs easier. When ww = ll, the quantity on the RHS does not depend on the in-plane physical dimensions and is called the sheet resistance R, with units of Ω/ssssssssssss. This is the intrinsic electrical transport parameter of the thin film. The sheet resistance R can be measured with four equidistant electrodes if the film is large enough so that the boundary effects are small or with four corner electrodes if the film is smaller. Then, it can be shown that R = ππ ln 2 VV 2ππ and R = II ln 2 VV, respectively (in both cases, II the result does not depend on the distance between electrodes). These standard electrode configurations are also applied in measurements on thick samples. We will use the second configuration (figure 1(b)). Page 2

3 3. Experiments In principle, measuring VV and II once would be enough in an ideal case. In practice, to account for sample shape and differences between contact placement and quality, a series of measurements is necessary. We will be using a lock-in method for higher sensitivity 3.1. Measuring the sheet resistance In this case, the current is injected between two neighboring contacts. Measure the 8 combinations RR aaaa,cccc (RR 12,34, RR 23,41, RR 34,12, RR 41,23 ) as well as the ones with contacts inverted (RR 21,43, RR 32,14, RR 43,21, RR 14,32 ). Then, average RR aaaa,cccc with RR bbbb,dddd (1 st consistency check, should be satisfied within 5%) and average RR 12,34 with RR 34,12 and RR 23,41 with RR 41,23 (reciprocity theorem, 2 nd consistency check). We end up with two resistances RR 12,34 = RR AA and RR 23,41 = RR BB, which go into the van der Pauw equation ee ππrr AA RRss + ee ππrr BB RRss = 1. It can be shown that its solution RR ss of this equation is the intrinsic sheet resistance of the film Mobile carrier density in semiconductors and metals Consistent with the Hall Effect model geometry of crossed II and VV HH directions, measurements are made across the current contacts (VV 13 and II 24 etc.). Start with a lightly-doped semiconductor (pp-doped Ge or nn doped Si wafer) that have a small carrier density and a large Hall voltage Apply the field in one direction (call this positive, PP) and measure VV 13PP with II 24. Invert both the current and voltage leads and measure VV 31PP with II 42 (the reciprocity theorem requires that these should be close, 1 st consistency check). Interchange the voltage and current leads to obtain VV 24PP with II 13 and VV 42PP with II 31. Repeat for the reversed magnetic field Calculate VV CC = VV 24PP VV 24NN, VV DD = VV 42PP VV 42NN, VV EE = VV 13PP VV 13NN, VV FF = VV 31PP VV 31NN 10 The charge carrier density is then (if holes) pp ss = 8 8 IIII qq 2(VV CC +VV DD +VV EE +VV FF ) Then, switch to the Cu thin foil, which will have a smaller voltage because of the larger carrier density A measurement of the sheet resistivity and carrier density allows finding the mobility μμ = 1 eenn ss RR ss of the charge carriers and the scattering time ττ = mm ee μμ. Find the distance between scattering events and compare to the distance between electrons obtained from their density for metals and semiconductors. Page 3

4 4. Conclusion The different mobile carrier concentrations obtained show why classical Boltzmann statistics works well for semiconductors, while quantum statistics must be used for metals. The Hall voltage VV HH 1 and the measurements are easier in thin samples because the tt current density is 1 for the same injected current II. tt The classical Hall Effect is the simplest of a family of Hall Effects that includes the anomalous Hall Effect (dependence of VV HH on sample magnetization), Integer and Fractional Quantum Hall Effects. Page 4

5 Phys-602 Quantum Mechanics Laboratory I Charge carrier density in metals and semiconductors lab report Name Date Page 5