Treinamento : Testes Paramétricos em Semicondutores Setembro 2012. Cyro Hemsi Engenheiro de Aplicação. Section 3 – Basic I/V Component Measurement. Agenda. Section #3 – Basic I/V Component Measurement Making Accurate Resistance Measurements Characterizing Diodes & Transistors.
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Engenheiro de Aplicação
Section 3 – Basic I/V Component Measurement
Understanding Resistivity and Resistance
When characterizing materials we typically want to know the resistivity, which is a material property that is independent of the dimensions of the device being measured.
Understanding “Ohms per Square”
You may sometimes see the term “Ohms per Square”. This is simply another way of specifying resistivity. As long as the thickness of a material is constant, it has the same measured resistance for any sized square of the material.
Van derPauw Structures are Typically Used to Measure Resistivity On-Wafer
Note that these structures are inherently designed to be measured using Kelvin measurements.
- Kelvin measurement technique for surface resistivity ρmeasurement
V = I x R, correct? NO!
V = I x R(T) Resistance depends on temperature!
This is referred to as the Joule self-heating effect.
Resistance measurement is a tricky balance between two factors:
To keep the resistor from heating up and the resistance value from changing, we need to keep the current (= power) low.
Small currents imply that we need to measure smaller voltages, which in-turn requires more voltage measurement resolution capability.
The Importance of Thermal Resistance
- We cannot arbitrarily increase
the power (= current) into a
Resistor to compensate for
SiO2 thermal conductivity
(h ´ 71) oC-cm2/W, or
0.007 oC-cm2/W for h = 1 mm
SiO2 thermal resistance max power dissipation
Max power dissipation
At or near room temperature the resistance of a Cu or Al metal line changes by about 0.35%/oC. This allows us to compute the maximum amount of power that can be dissipated to produce a 0.1% change in resistance for a Cu or Al metal square 10 mm by 10 mm.
To achieve 0.1% accuracy in a copper structure with an equivalent resistance of 10 mW per square (1 mm thick film) we have the following:
Need 1 mV of voltage measurement resolution!
R = (VM1 – VM2)/IF
Force Current Twice
(Both Ways), i.e. +/- IF
(SMU 3)Modified Kelvin Resistance Measurement
* transient voltage pulse created when
reed relay switches open and close
“Abrupt” Semiconductor Junction:
Holes and electrons flow into the n and p regions (respectively) until the force of diffusion is exactly balanced by the electric field created by the fixed charge.
Ex: “p- and n- doped” silicon
electric field created by the fixed charges
The one-dimensional (x-axis) equations defining current flow in a semiconductor are shown below.
Where J is the current density of electrons (n) and holes (p)
q is the electron charge
Ex is the electric field in the x-dimension
m is the mobility of electrons (n) and holes (p)
D is the diffusion constant for electrons (n) and holes (p)
n is the electron density
p is the hole density
Current flow in a semiconductor consists of two parts: - a drift current proportional to the applied electric field and
- a diffusion current proportional to the spatial first derivative of the free carrier density.
The Einstein relationship relates the ratios of the mobility and diffusion constants as shown below.
q is the magnitude of the electron charge (1.602 ´ 10-19 Coulomb)
k is Boltzmann’s constant (1.38 ´ 10-23 J/K)
T is the absolute temperature [deg K]
m is the mobility of electrons (n) and holes (p)
D is the diffusion constant for electrons (n) and holes (p)
The general form of Poisson’s equation relates the second derivative of the electric potential to the total space charge density (r). Since we know that in a semiconductor this has to be related to the densities of mobile and fixed charge, we can write this as follows.
Where Nd is the donor density concentration (dopation)
Na is the acceptor density concentration
eSi is the permittivity of silicon
The depletion approximation assumes that the semiconductor is divided into distinct regions which are either completely neutral or completely depleted of mobile carriers. This allows us to re-write the above equation as:
We can integrate Poisson’s equation to get the electric field in both the p and n regions as shown below.
Where xp is the width of the space charge in the p region
xn is the width of the space charge in the n region
Graphically, these equations have the appearance shown on the right.
Electric field in both the p and n regions
We know that the electric field has to be continuous at x = 0.
This gives us the result shown below.
This equation shows an important characteristic of pn junctions: the width of the depletion region varies inversely with the magnitude of the dopant concentration. In other words, higher dopant concentrations result in narrower space charge regions.
Depletion region width
1) When no voltage is applied to the pn junction a barrier exists to current flow.
2) As we apply a positive voltage to the p-region we reduce the built-in
electric field of the pn junction.
3) At some point the electric field is reduced
enough to allow current to flow .
A forward diode characterization plotting both Id (blue) and log Id (orange).
The reverse breakdown characteristics of a diode.
Here we assume that the voltage applied to the drain (Vd) is small relative to the gate voltage (Vg) . The current is defined by the charge in the channel divided by the transit time:
Voltage to invert the channel
Vg creates a conductive channel
Where mn is the electron mobility
W is the channel width
L is the channel length
Cox is the capacitance per unit area in accumulation
As Vd increases it is no longer negligible relative to Vg, and the space charge region looks like a trapezoid. We make the approximation that the average channel voltage is Vg – Vd/2.
If we plot out the previous equation for different values of Vd, then we get a family of parabolas:
However, the slope of the curves for Vd > Vg - Vt is negative and this does not make any physical sense as it would correspond to a negative transconductance.
When Vd becomes greater than (Vg - Vt) the inversion layer becomes pinched off at the drain. However, as the electrons approach the drain there is no barrier to stop them and they are rapidly accelerated by the high electric field into the drain region. In this situation the current is determined by the rate at which electrons reach the edge of the depletion region, which is in-turn determined by the maximum electron drift velocity. Since the maximum drift velocity is (at least to first order) insensitive to Vd, once Vd reaches the value of (Vg - Vt) the drain current becomes saturated (constant).
In saturation, the drain current is given by the equation shown on the right.
The constants shown in the equation on the previous slide are usually combined into a single constant, beta:
In the case where the transistor source terminal is not at ground it is a correct assumption that (at least to first order) an increase in Vs will have the effect of reducing the influence of the gate voltage. Therefore, it is reasonable to modify the equation for Id in saturation for the case where Vs ¹ 0 as follows:
MOSFET Vt Characterization
Extrapolating a plot of the square root of Id back to the x-axis is one way of determining the transistor Vt.
Applying a negative charge on the bulk will induce positive charge to accumulate near the bulk contact. This causes Vt to increase with decreasing values of Vb (which is commonly known as the “body effect”). Similarly, applying a positive charge on the bulk will cause Vt to decrease.
Negative values of Vb shift the curve to right which implies an increase in the threshold voltage (Vt). Similarly, positive values of Vb shift the curve to the left which implies a decrease in the threshold voltage.
Consider the case where we have the emitter of the NPN transistor at ground and we apply a positive voltage to the base. We are also assuming the transistor collector is at a voltage much greater than that of the base (VC >> VB). If the base to emitter voltage (VBE) is sufficiently large enough to forward bias the base to emitter junction, then current will flow in the transistor as shown below.
As can be seen from this equation, the base to emitter voltage does not need to change much in order to have large changes in the collector current. Therefore, once the NPN transistor has a sufficiently large VBE to become actively biased the magnitude of the collector current is limited by the amount of current entering into the base of the transistor.
A Gummel plot is usually used to determine the value of beta for various values of collector current. One method to perform a Gummel plot on an NPN transistor is to set the collector and base to zero volts and to sweep the emitter negatively. The values of base current and collector current are then measured and plotted on a log scale
As this plot shows, the value of the transistor beta is actually dependent upon the large-signal value of the collector current, although it is approximately constant over a fairly wide range of collector current values.
The flyback method is the most common means used to determine Re. The SMU connected to the collector terminal is placed in current force mode with a very small force value (almost zero amps), making it essentially a high-impedance voltmeter. The SMU connected to the base terminal is also set to current force mode and the collector-emitter voltage (Vce) is measured and plotted versus the base current (Ib).
The slope of the line in the flyback region can be determined by creating a regression line at predefined points using the built-in EasyEXPERT analysis functions. The slope of this line is in units of amps/volt; this is the inverse of resistance (which has units of volts/amp). Therefore, the inverse of the regression line slope gives us the value of emitter resistance (Re).
Power Bipolar Transistor Breakdown Characterization using the B1505A (1)
As in the case of power MOSFETs, the breakdown voltages of power bipolar devices are also very important parameters. The junction breakdown voltages of power BJTs are defined as follows.
BVcbo – The collector-base breakdown voltage with the emitter open
BVceo – The collector-emitter breakdown voltage with the base open
BVebo – The emitter-base breakdown voltage with the collector open