Hall Effect Brian Kaster November 4th, 2013 Physics 481

Outline • • • • •

History: Maxwell and Hall Background: A review of physics Theory: Van Der Pauw’s method Experiment: Making the magic happen Uses: Why it all matters

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History • Maxwell stated that the only force which acts on electrons is electromotive force. Therefore, the mechanical force a conductor sees in a magnetic field is due to the conductor itself, and not the electrons • Hall was not convinced. His PhD Thesis (1879) was working on thin gold films in an electromagnet demonstrating that it is the charge which the electromotive force acts on by measuring a voltage transverse the current and applied magnetic field 3

Background • Lorentz Force 𝐹 =𝑞 𝐸+𝑣×𝐵

I

E applied B=0

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Background • Lorentz Force 𝐹 =𝑞 𝐸+𝑣×𝐵

I

E applied B ×

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Background • Lorentz Force in thin film

-

+

B=0

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Background • Lorentz Force in thin film

-

+

B ×

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Background • Lorentz Force in thin film

-

+

B ×

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Theory • Hall Voltage – This is the voltage induced for a given magnetic field and current in a material – In metals, it can be easily calculated using: 𝑉𝐻 =

𝐼𝐵 𝑛𝑞𝑡

where I=current, B=magnetic field, n=carrier density, q=elementary charge, and t=thickness.

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Theory • Hall Coefficient – This is a property of the material itself, and is defined as: 𝐸𝑦 𝑅𝐻 = 𝑗𝑥 𝐵 – This is appropriated for metals, where electrons are the only charge carrier

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Theory • Electrons and Holes – In semiconductors, there are two means to transport charge: • An electron can jump from one available place to another, and flow through the material • A state can be occupied by a neighboring electron, opening a new state which is in turn occupied by another neighbor, causing an effective flow of a missing electron (called a hole)

– Both occur in semiconductors, but one is typically much more prevalent than the other 11

Theory • Mobility – The relative ease of the motion of the charge carrier – Effected by many material properties – Each type of carrier has its own mobility

• Carrier Concentration – The number of carriers in a given unit of space – Typically generated through impurities in materials to generate extra electrons or holes 12

Theory • Semiconductor Hall Coefficient – Redefined to include multiple contributors as: 𝑝𝜇ℎ2 − 𝑛𝜇𝑒2 𝑅𝐻 = 𝑒 𝑝𝜇ℎ + 𝑛𝜇𝑒 2 – Simplifies to metallic hall coefficient in the case of p=0

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Theory • Van Der Pauw Method – In the case of a perfectly square sample, the Hall effect can be calculated simply – Van Der Pauw created a method in 1958 which can be applied to any arbitrary thin film shape – To perform this measurement, we need to be able to supply a current, measure a voltage, and apply a magnetic field – From this measurement we can find the carrier type, concentration, and mobility 14

Theory • Van Der Pauw Contacts – Contacts used to both supply current and measure voltage – Cloverleaf requires extra processing steps – Using these four contacts, all relevant electrical measurements can be made

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Theory • Van Der Pauw Measurements – Sheet Resistivity • Supply a current between two contacts on one side to generate and measure a voltage across the two contacts on the opposite side • Repeat with positive and negative I V currents, and from one side to the other • Use Ohm’s law to determine the horizontal and vertical resistances for the device under test. 16

Theory • Van Der Pauw Measurements – Sheet Resistivity • Using this method, the sheet resistivity can be determined by: −𝜋𝑅𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙

−𝜋𝑅ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙

𝑅𝑠 + 𝑒 𝑅𝑠 = 1 𝑒 • No direct analytical solution (unless Rvertical=Rhorizontal) • Numerical methods used to calculate

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Theory • Van Der Pauw Measurements – Hall Voltage • A number of hall voltages are generated by applying a current across two contacts in opposite corners and measuring the voltage across the other two contacts using both a positive and negative magnetic field I V • Symmetry is utilized to generate two distinctly different values, which B × are differenced to determine the hall voltage

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Theory • Van Der Pauw Measurements – Hall Voltage • The voltages from the four possible contact orientations are used (13, 24, 31, 42), where the values for the positive and negative fields are differenced 𝑉𝑖𝑗 = 𝑉𝑖𝑗,𝑝 − 𝑉𝑖𝑗,𝑛 • The resulting hall voltage is then: 𝑉13 + 𝑉24 + 𝑉31 + 𝑉42 𝑉𝐻 = 8

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Theory • Van Der Pauw Calculations – Sheet Density • Remember our initial definition of hall voltage:

𝑉𝐻 =

𝐼𝐵 𝑛𝑞𝑡

This can be rearranged to determine nt (or ns) 𝑛𝑠 =

𝐼𝐵 𝑞𝑉𝐻

• The sign tells us if the majority carriers are electrons or holes, and the value tells us the concentration per area (pay careful attention to units!)

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Theory • Van Der Pauw Calculations – Mobility • Using the resistivity of a semiconductor, and assuming one carrier plays a dominant role, the mobility can be determined by: 1 𝜇𝑚 = 𝑞𝑛𝑠 𝑅𝑠 • This factor tells you how easily carriers can move through the semiconductor

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Experiment • Electronics used: – – – – – –

Four point multimeter Matrixing switch Electromagnet power supply Electromagnet current reversal switch Magnetic field sensor Temperature Controller

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Experiment • Equipment used: – – – –

Electromagnet Cryocompressor Vacuum insulated sample holder Turbopump

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Experiment • Our Capabilities – – – – – –

Fields up to 1.5 Tesla Currents down to nanoamps Voltages down to nanovolts Temperatures down to 10K Up to 4 samples per run Complete automation of measurements

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Uses • Measurement of doping – Very important to control for p-n junctions used in photovoltaics, the primary devices in MPL – Many studies include doping during growth and the effects of annealing on the carrier concentrations.

• Measurement of mobility – Mobility is a good indication of crystal quality – Mobility is important for generating charge separation used in photovoltaics 25

Uses • Measuring magnetic field – Most magnetic field sensors (including the one used in this experiment) uses a probe with a known hall response to determine the magnitude of the magnetic field

• Many others!

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