text:
SPSS TUTORIALS BASICS ANOVA REGRESSION FACTOR CORRELATION

# SPSS Dummy Variable Regression Tutorial

Using categorical predictors in multiple regression requires dummy coding. So how to use such dummy variables and how to interpret the resulting output? This tutorial walks you through.

## Example Data

All examples in this tutorial use staff-dummies.sav, partly shown below.

Our data file already contains dummy variables for representing Contract Type. Two options for creating such dummy variables in other data files are

## Analysis I - T-Test as Dummy Regression

Let's first examine if monthly salary is related to sex. Two options for finding this out are

These analyses come up with the same results. Comparing these is the first step towards understanding dummy variable regression. Let's first run our t-test from the syntax below.

*Independent samples t-test: salary by sex.

t-test groups sex(1 0)
/variables salary.

Gross monthly salary for females is $421.09 higher than for males. Also note that males are coded 0 while females are coded 1. The significance level for this mean difference is 0.004: we'll probably reject the null hypothesis that the population mean salaries are equal between men and women. A 95% confidence interval suggests a likely range for the population mean difference. It runs from$134.52 through $707.67. Let's now rerun this analysis as regression with a single dummy variable. ## Example I - Single Dummy Predictor In SPSS, we first navigate to Analyze Regression Linear and fill out the dialogs as shown below. Completing these steps results in the syntax below. Let's run it. *Regression: salary by single dummy variable (sex). REGRESSION /MISSING LISTWISE /STATISTICS COEFF OUTS CI(95) R ANOVA /CRITERIA=PIN(.05) POUT(.10) /NOORIGIN /DEPENDENT salary /METHOD=ENTER sex. ## Dummy Variable Regression Output I Note that the constant is the mean salary for male respondents. The b-coefficient for sex is the mean salary difference between male and female respondents. This is equal to the average increase in salary associated with a 1-unit increase in sex: from male (coded 0) to female (coded 1). This makes sense because the regression equation is $$Salary' = 2731 + 421 \cdot Sex$$ so for all males we predict a gross monthly salary of $$Salary' = 2731 + 421 \cdot 0 = 2731$$ and for all females we predict $$Salary' = 2731 + 421 \cdot 1 = 3152$$ These predicted salaries are simply the mean salaries for male and female respondents. Finally, note that the significance level and confidence interval for the b-coefficient are identical to their counterparts for the mean difference in the t-test results. ## Analysis II - ANOVA as Dummy Regression Let's now see if salary is related to contract type (freelance, temporary or permanent). Precisely, we'll test the null hypothesis that the population mean salaries are equal across all 3 contract types. Two options for testing this hypothesis are: • ANOVA and • dummy variable regression. As we'll see, the b-coefficients obtained from our regression approach are identical to simple contrasts from ANOVA: the mean for a designated reference category is compared to the mean for each other category. These ANOVA results can be replicated from the syntax below. *ANOVA: salary by type of contract. unianova salary by contract /contrast (contract) = simple(1) /print descriptive etasq. ## Results Since p < 0.05, we reject the null hypothesis that all population means are equal. The effect size, eta squared is 0.125. This is between medium (0.06) and large (0.14). The mean difference between employees on a permanent versus a temporary contract (the reference category) is$465.94.
The p-value and confidence interval indicate that this mean difference is “significantly” different from zero, the null hypothesis for this comparison.

In a similar vein, the mean salaries for employees on a freelance versus a temporary contract are compared (not shown here).

## Example II - Multiple Dummy Predictors

We'll navigate to Analyze Regression Linear and fill out the dialogs as shown below.

We need to choose one reference category and not enter it as a predictor: for representing k categories, we always enter (k - 1) dummy variables.
Competing these steps generates the syntax below.

*Regression with 2 dummy variables representing type of contract.

REGRESSION
/MISSING LISTWISE
/STATISTICS COEFF OUTS CI(95) R ANOVA
/CRITERIA=PIN(.05) POUT(.10)
/NOORIGIN
/DEPENDENT salary
/METHOD=ENTER contract_2 contract_3.

## Dummy Variable Regression Output II

Note that r-squared is equal to the ANOVA eta squared we saw earlier. This is always the case: both measures indicate the proportion of variance in the dependent variable accounted for by the independent variable(s).
R-squared for the entire model (containing only 2 dummy variables) is statistically significant. In fact, the entire regression ANOVA table is identical to the one obtained from an actual ANOVA.
The constant is the mean salary for our reference category: employees having a temporary contract. These respondents score zero on both dummy variables in our model. For them, the regression equation boils down to

$$Salary' = 2675.8 + 465.94 \cdot 0 + 1087.4 \cdot 0 = 2675.8$$

The mean salary difference between employees on permanent (dummy) versus temporary (reference) contracts is $321.14 if we control for working experience. Since p < 0.05, this mean difference is statistically significant. Let's now rerun the exact same analysis as an ANCOVA from the syntax below. *ANCOVA for salary by contract, controlling for experience (years). unianova salary by contract with expn /contrast (contract) = simple(1) /print descriptive etasq. ## Results Partial eta squared for “corrected model” is equal to the regression r-squared. The output also contains effect sizes for both predictors separately. Note that 0.361 and 0.082 add up to 0.443, somewhat larger than 0.440 for the entire model. This is because these effects partially overlap: experience is associated with contract type. The mean salary difference between employees on permanent versus temporary contracts is$321.14 if we correct for working experience. This difference was seen as a b-coefficient in the previous dummy regression output.
Unsurprisingly, the p-value and confidence interval are identical to their dummy regression counterparts as well.

## Is Dummy Variable Regression Useless?

Many textbooks propose dummy variable regression as the only option for using a combination of quantitative and categorical predictors. However, our last example suggests that ANCOVA may be a better option for this scenario. Why? Well,

• ANCOVA does not require adding (technically redundant) dummy variables to your data.
• ANCOVA comes up with a single effect size (partial eta squared) for the entire categorical predictor. This is more useful than effect sizes for separate dummy variables because we never add them separately to a regression model.
• Testing moderation effects between quantitative and categorical predictors is fairly easy via ANCOVA but rather complicated via regression.

## Final Notes

First off, note that the analyses in this tutorial skipped some important steps:

• we didn't inspect any frequency distributions to see if our data look plausible;
• we did't see if there's any missing values in our data;
• we didn't evaluate any model assumptions (normality, linearity, and so on).

We encourage you to thoroughly examine such issues whenever you're working on real-world data files.

Right, so that should do for dummy regression in SPSS. For a handy overview of the output from all 6 analyses, click here. Did you find this tutorial (not) helpful? Do you agree or disagree with us? Please let us know by throwing a comment below.

# Tell us what you think!

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# THIS TUTORIAL HAS 4 COMMENTS:

• ### By YY Ma on April 8th, 2021

A very instructive post!
I have learnt much from it.

Good

• ### By MWETEGYEREZE FRANK on May 1st, 2021

It's really interesting

• ### By N M on May 18th, 2021

Thank you so much.Been looking high and low to figure out dummy variable regression interpretation. So would you say that in regression its ideally best not to add multiple categorical variables in the same model?


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The vertical axis is linear. . / In probability theory and statistics, the Gumbel distribution (Generalized Extreme Value distribution Type-I) is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. Gumbel has shown that the maximum value (or last order statistic) in a sample of a random variable following an exponential distribution minus natural logarithm of the sample size [6] approaches the Gumbel distribution closer with increasing sample size. {\displaystyle F} 0.3665 / If the family and parameter specification is stored in a BiCop() μ − 9 = BB7 copula The shape of the chi-square distribution depends on the number of degrees of freedom. 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text:
# Thread: [SOLVED] Induced matrix norms

1. ## [SOLVED] Induced matrix norms

Consider the matrix

$
A = \left[\begin{matrix}1 & -2\\1 & 3\end{matrix}\right]
$

Consider the matrix norm $||\cdot||_{ind}$ induced by the $1-norm$ on the domain and the $\infty-norm$ on the range. Applied to A, we have that:
a) $||A||_{ind}=5$
b) $||A||_{ind}=\sqrt{15}$
c) $||A||_{ind}=4$
d) $||A||_{ind}=3$

thanks again.

2. ## I think I found the solution

I found a paper by Drakakis, K.
International Journal of Algebra, Vol. 3, 2009, no. 5, 231 - 240

$||A||_{(1,\infty)}=^{max}_{(i,j)|\in\{1,.....,m\}\ times\{1,.....,n\}}|a_{ij}|$

I used this formula and came up with the answer 3.
If there are no objections I will mark this one solved.

Bucephalus


text:
BREAKING NEWS
Electron mobility

Summary

In solid-state physics, the electron mobility characterises how quickly an electron can move through a metal or semiconductor when pulled by an electric field. There is an analogous quantity for holes, called hole mobility. The term carrier mobility refers in general to both electron and hole mobility.

Electron and hole mobility are special cases of electrical mobility of charged particles in a fluid under an applied electric field.

When an electric field E is applied across a piece of material, the electrons respond by moving with an average velocity called the drift velocity, ${\displaystyle v_{d}}$. Then the electron mobility μ is defined as

${\displaystyle v_{d}=\mu E.}$

Electron mobility is almost always specified in units of cm2/(Vs). This is different from the SI unit of mobility, m2/(Vs). They are related by 1 m2/(V⋅s) = 104 cm2/(V⋅s).

Conductivity is proportional to the product of mobility and carrier concentration. For example, the same conductivity could come from a small number of electrons with high mobility for each, or a large number of electrons with a small mobility for each. For metals, it would not typically matter which of these is the case, since most metal electrical behavior depends on conductivity alone. Therefore mobility is relatively unimportant in metal physics. On the other hand, for semiconductors, the behavior of transistors and other devices can be very different depending on whether there are many electrons with low mobility or few electrons with high mobility. Therefore mobility is a very important parameter for semiconductor materials. Almost always, higher mobility leads to better device performance, with other things equal.

Semiconductor mobility depends on the impurity concentrations (including donor and acceptor concentrations), defect concentration, temperature, and electron and hole concentrations. It also depends on the electric field, particularly at high fields when velocity saturation occurs. It can be determined by the Hall effect, or inferred from transistor behavior.

Introduction

Drift velocity in an electric field

Without any applied electric field, in a solid, electrons and holes move around randomly. Therefore, on average there will be no overall motion of charge carriers in any particular direction over time.

However, when an electric field is applied, each electron or hole is accelerated by the electric field. If the electron were in a vacuum, it would be accelerated to ever-increasing velocity (called ballistic transport). However, in a solid, the electron repeatedly scatters off crystal defects, phonons, impurities, etc., so that it loses some energy and changes direction. The final result is that the electron moves with a finite average velocity, called the drift velocity. This net electron motion is usually much slower than the normally occurring random motion.

The two charge carriers, electrons and holes, will typically have different drift velocities for the same electric field.

Quasi-ballistic transport is possible in solids if the electrons are accelerated across a very small distance (as small as the mean free path), or for a very short time (as short as the mean free time). In these cases, drift velocity and mobility are not meaningful.

Definition and units

The electron mobility is defined by the equation:

${\displaystyle v_{d}=\mu _{e}E.}$
where:

• E is the magnitude of the electric field applied to a material,
• vd is the magnitude of the electron drift velocity (in other words, the electron drift speed) caused by the electric field, and
• µe is the electron mobility.

The hole mobility is defined by a similar equation:

${\displaystyle v_{d}=\mu _{h}E.}$
Both electron and hole mobilities are positive by definition.

Usually, the electron drift velocity in a material is directly proportional to the electric field, which means that the electron mobility is a constant (independent of the electric field). When this is not true (for example, in very large electric fields), mobility depends on the electric field.

The SI unit of velocity is m/s, and the SI unit of electric field is V/m. Therefore the SI unit of mobility is (m/s)/(V/m) = m2/(Vs). However, mobility is much more commonly expressed in cm2/(V⋅s) = 10−4 m2/(V⋅s).

Mobility is usually a strong function of material impurities and temperature, and is determined empirically. Mobility values are typically presented in table or chart form. Mobility is also different for electrons and holes in a given material.

Derivation

Starting with Newton's Second Law:

${\displaystyle a=F/m_{e}^{*}}$
where:

• a is the acceleration between collisions.
• F is the electric force exerted by the electric field, and
• ${\displaystyle m_{e}^{*}}$ is the effective mass of an electron.

Since the force on the electron is −eE:

${\displaystyle a=-{\frac {eE}{m_{e}^{*}}}}$

This is the acceleration on the electron between collisions. The drift velocity is therefore:

${\displaystyle v_{d}=a\tau _{c}=-{\frac {e\tau _{c}}{m_{e}^{*}}}E,}$
where ${\displaystyle \tau _{c}}$ is the mean free time

Since we only care about how the drift velocity changes with the electric field, we lump the loose terms together to get

${\displaystyle v_{d}=-\mu _{e}E,}$
where ${\displaystyle \mu _{e}={\frac {e\tau _{c}}{m_{e}^{*}}}}$

Similarly, for holes we have

${\displaystyle v_{d}=\mu _{h}E,}$
where ${\displaystyle \mu _{h}={\frac {e\tau _{c}}{m_{h}^{*}}}}$ Note that both electron mobility and hole mobility are positive. A minus sign is added for electron drift velocity to account for the minus charge.

Relation to current density

The drift current density resulting from an electric field can be calculated from the drift velocity. Consider a sample with cross-sectional area A, length l and an electron concentration of n. The current carried by each electron must be ${\displaystyle -ev_{d}}$, so that the total current density due to electrons is given by:

${\displaystyle J_{e}={\frac {I_{n}}{A}}=-env_{d}}$
Using the expression for ${\displaystyle v_{d}}$ gives
${\displaystyle J_{e}=en\mu _{e}E}$
A similar set of equations applies to the holes, (noting that the charge on a hole is positive). Therefore the current density due to holes is given by
${\displaystyle J_{h}=ep\mu _{h}E}$
where p is the hole concentration and ${\displaystyle \mu _{h}}$ the hole mobility.

The total current density is the sum of the electron and hole components:

${\displaystyle J=J_{e}+J_{h}=(en\mu _{e}+ep\mu _{h})E}$

Relation to conductivity

We have previously derived the relationship between electron mobility and current density

${\displaystyle J=J_{e}+J_{h}=(en\mu _{e}+ep\mu _{h})E}$
Now Ohm's Law can be written in the form
${\displaystyle J=\sigma E}$
where ${\displaystyle \sigma }$ is defined as the conductivity. Therefore we can write down:
${\displaystyle \sigma =en\mu _{e}+ep\mu _{h}}$
which can be factorised to
${\displaystyle \sigma =e(n\mu _{e}+p\mu _{h})}$

Relation to electron diffusion

In a region where n and p vary with distance, a diffusion current is superimposed on that due to conductivity. This diffusion current is governed by Fick's Law:

${\displaystyle F=-D_{e}\nabla n}$
where:

• F is flux.
• De is the diffusion coefficient or diffusivity
• ${\displaystyle \nabla n}$ is the concentration gradient of electrons

The diffusion coefficient for a charge carrier is related to its mobility by the Einstein relation:

${\displaystyle D_{e}={\frac {\mu _{e}k_{\mathrm {B} }T}{e}}}$
where:

Examples

Typical electron mobility at room temperature (300 K) in metals like gold, copper and silver is 30–50 cm2/ (V⋅s). Carrier mobility in semiconductors is doping dependent. In silicon (Si) the electron mobility is of the order of 1,000, in germanium around 4,000, and in gallium arsenide up to 10,000 cm2/ (V⋅s). Hole mobilities are generally lower and range from around 100 cm2/ (V⋅s) in gallium arsenide, to 450 in silicon, and 2,000 in germanium.[1]

Very high mobility has been found in several ultrapure low-dimensional systems, such as two-dimensional electron gases (2DEG) (35,000,000 cm2/(V⋅s) at low temperature),[2] carbon nanotubes (100,000 cm2/(V⋅s) at room temperature)[3] and freestanding graphene (200,000 cm2/ V⋅s at low temperature).[4] Organic semiconductors (polymer, oligomer) developed thus far have carrier mobilities below 50 cm2/(V⋅s), and typically below 1, with well performing materials measured below 10.[5]

List of highest measured mobilities [cm2/ (V⋅s)]
Material Electron mobility Hole mobility
AlGaAs/GaAs heterostructures 35,000,000[2]
Freestanding Graphene 200,000[4]
Carbon nanotubes 79,000[6][7]
Crystalline silicon 1,400[1] 450[1]
Polycrystalline silicon 100
Metals (Al, Au, Cu, Ag) 10-50
2D Material (MoS2) 10-50
Organics 8.6[8] 43[9]
Amorphous silicon ~1[10]

Electric field dependence and velocity saturation

At low fields, the drift velocity vd is proportional to the electric field E, so mobility μ is constant. This value of μ is called the low-field mobility.

As the electric field is increased, however, the carrier velocity increases sublinearly and asymptotically towards a maximum possible value, called the saturation velocity vsat. For example, the value of vsat is on the order of 1×107 cm/s for both electrons and holes in Si. It is on the order of 6×106 cm/s for Ge. This velocity is a characteristic of the material and a strong function of doping or impurity levels and temperature. It is one of the key material and semiconductor device properties that determine a device such as a transistor's ultimate limit of speed of response and frequency.

This velocity saturation phenomenon results from a process called optical phonon scattering. At high fields, carriers are accelerated enough to gain sufficient kinetic energy between collisions to emit an optical phonon, and they do so very quickly, before being accelerated once again. The velocity that the electron reaches before emitting a phonon is:

${\displaystyle {\frac {m^{*}v_{\text{emit}}^{2}}{2}}\approx \hbar \omega _{\text{phonon (opt.)}}}$
where ωphonon(opt.) is the optical-phonon angular frequency and m* the carrier effective mass in the direction of the electric field. The value of Ephonon (opt.) is 0.063 eV for Si and 0.034 eV for GaAs and Ge. The saturation velocity is only one-half of vemit, because the electron starts at zero velocity and accelerates up to vemit in each cycle.[11] (This is a somewhat oversimplified description.[11])

Velocity saturation is not the only possible high-field behavior. Another is the Gunn effect, where a sufficiently high electric field can cause intervalley electron transfer, which reduces drift velocity. This is unusual; increasing the electric field almost always increases the drift velocity, or else leaves it unchanged. The result is negative differential resistance.

In the regime of velocity saturation (or other high-field effects), mobility is a strong function of electric field. This means that mobility is a somewhat less useful concept, compared to simply discussing drift velocity directly.

Relation between scattering and mobility

Recall that by definition, mobility is dependent on the drift velocity. The main factor determining drift velocity (other than effective mass) is scattering time, i.e. how long the carrier is ballistically accelerated by the electric field until it scatters (collides) with something that changes its direction and/or energy. The most important sources of scattering in typical semiconductor materials, discussed below, are ionized impurity scattering and acoustic phonon scattering (also called lattice scattering). In some cases other sources of scattering may be important, such as neutral impurity scattering, optical phonon scattering, surface scattering, and defect scattering.[12]

Elastic scattering means that energy is (almost) conserved during the scattering event. Some elastic scattering processes are scattering from acoustic phonons, impurity scattering, piezoelectric scattering, etc. In acoustic phonon scattering, electrons scatter from state k to k', while emitting or absorbing a phonon of wave vector q. This phenomenon is usually modeled by assuming that lattice vibrations cause small shifts in energy bands. The additional potential causing the scattering process is generated by the deviations of bands due to these small transitions from frozen lattice positions.[13]

Ionized impurity scattering

Semiconductors are doped with donors and/or acceptors, which are typically ionized, and are thus charged. The Coulombic forces will deflect an electron or hole approaching the ionized impurity. This is known as ionized impurity scattering. The amount of deflection depends on the speed of the carrier and its proximity to the ion. The more heavily a material is doped, the higher the probability that a carrier will collide with an ion in a given time, and the smaller the mean free time between collisions, and the smaller the mobility. When determining the strength of these interactions due to the long-range nature of the Coulomb potential, other impurities and free carriers cause the range of interaction with the carriers to reduce significantly compared to bare Coulomb interaction.

If these scatterers are near the interface, the complexity of the problem increases due to the existence of crystal defects and disorders. Charge trapping centers that scatter free carriers form in many cases due to defects associated with dangling bonds. Scattering happens because after trapping a charge, the defect becomes charged and therefore starts interacting with free carriers. If scattered carriers are in the inversion layer at the interface, the reduced dimensionality of the carriers makes the case differ from the case of bulk impurity scattering as carriers move only in two dimensions. Interfacial roughness also causes short-range scattering limiting the mobility of quasi-two-dimensional electrons at the interface.[13]

Lattice (phonon) scattering

At any temperature above absolute zero, the vibrating atoms create pressure (acoustic) waves in the crystal, which are termed phonons. Like electrons, phonons can be considered to be particles. A phonon can interact (collide) with an electron (or hole) and scatter it. At higher temperature, there are more phonons, and thus increased electron scattering, which tends to reduce mobility.

Piezoelectric scattering

Piezoelectric effect can occur only in compound semiconductor due to their polar nature. It is small in most semiconductors but may lead to local electric fields that cause scattering of carriers by deflecting them, this effect is important mainly at low temperatures where other scattering mechanisms are weak. These electric fields arise from the distortion of the basic unit cell as strain is applied in certain directions in the lattice.[13]

Surface roughness scattering

Surface roughness scattering caused by interfacial disorder is short range scattering limiting the mobility of quasi-two-dimensional electrons at the interface. From high-resolution transmission electron micrographs, it has been determined that the interface is not abrupt on the atomic level, but actual position of the interfacial plane varies one or two atomic layers along the surface. These variations are random and cause fluctuations of the energy levels at the interface, which then causes scattering.[13]

Alloy scattering

In compound (alloy) semiconductors, which many thermoelectric materials are, scattering caused by the perturbation of crystal potential due to the random positioning of substituting atom species in a relevant sublattice is known as alloy scattering. This can only happen in ternary or higher alloys as their crystal structure forms by randomly replacing some atoms in one of the sublattices (sublattice) of the crystal structure. Generally, this phenomenon is quite weak but in certain materials or circumstances, it can become dominant effect limiting conductivity. In bulk materials, interface scattering is usually ignored.[13][14][15][16][17]

Inelastic scattering

During inelastic scattering processes, significant energy exchange happens. As with elastic phonon scattering also in the inelastic case, the potential arises from energy band deformations caused by atomic vibrations. Optical phonons causing inelastic scattering usually have the energy in the range 30-50 meV, for comparison energies of acoustic phonon are typically less than 1 meV but some might have energy in order of 10 meV. There is significant change in carrier energy during the scattering process. Optical or high-energy acoustic phonons can also cause intervalley or interband scattering, which means that scattering is not limited within single valley.[13]

Electron–electron scattering

Due to the Pauli exclusion principle, electrons can be considered as non-interacting if their density does not exceed the value 1016~1017 cm−3 or electric field value 103 V/cm. However, significantly above these limits electron–electron scattering starts to dominate. Long range and nonlinearity of the Coulomb potential governing interactions between electrons make these interactions difficult to deal with.[13][14][15]

Relation between mobility and scattering time

A simple model gives the approximate relation between scattering time (average time between scattering events) and mobility. It is assumed that after each scattering event, the carrier's motion is randomized, so it has zero average velocity. After that, it accelerates uniformly in the electric field, until it scatters again. The resulting average drift mobility is:[18]

${\displaystyle \mu ={\frac {q}{m^{*}}}{\overline {\tau }}}$
where q is the elementary charge, m* is the carrier effective mass, and τ is the average scattering time.

If the effective mass is anisotropic (direction-dependent), m* is the effective mass in the direction of the electric field.

Matthiessen's rule

Normally, more than one source of scattering is present, for example both impurities and lattice phonons. It is normally a very good approximation to combine their influences using "Matthiessen's Rule" (developed from work by Augustus Matthiessen in 1864):

${\displaystyle {\frac {1}{\mu }}={\frac {1}{\mu _{\rm {impurities}}}}+{\frac {1}{\mu _{\rm {lattice}}}}.}$
where µ is the actual mobility, ${\displaystyle \mu _{\rm {impurities}}}$ is the mobility that the material would have if there was impurity scattering but no other source of scattering, and ${\displaystyle \mu _{\rm {lattice}}}$ is the mobility that the material would have if there was lattice phonon scattering but no other source of scattering. Other terms may be added for other scattering sources, for example
${\displaystyle {\frac {1}{\mu }}={\frac {1}{\mu _{\rm {impurities}}}}+{\frac {1}{\mu _{\rm {lattice}}}}+{\frac {1}{\mu _{\rm {defects}}}}+\cdots .}$
Matthiessen's rule can also be stated in terms of the scattering time:
${\displaystyle {\frac {1}{\tau }}={\frac {1}{\tau _{\rm {impurities}}}}+{\frac {1}{\tau _{\rm {lattice}}}}+{\frac {1}{\tau _{\rm {defects}}}}+\cdots .}$
where τ is the true average scattering time and τimpurities is the scattering time if there was impurity scattering but no other source of scattering, etc.

Matthiessen's rule is an approximation and is not universally valid. This rule is not valid if the factors affecting the mobility depend on each other, because individual scattering probabilities cannot be summed unless they are independent of each other.[17] The average free time of flight of a carrier and therefore the relaxation time is inversely proportional to the scattering probability.[13][14][16] For example, lattice scattering alters the average electron velocity (in the electric-field direction), which in turn alters the tendency to scatter off impurities. There are more complicated formulas that attempt to take these effects into account.[19]

Temperature dependence of mobility

Typical temperature dependence of mobility[20]
Si Ge GaAs
Electrons ∝T −2.4 ∝T −1.7 ∝T −1.0
Holes ∝T −2.2 ∝T −2.3 ∝T −2.1

With increasing temperature, phonon concentration increases and causes increased scattering. Thus lattice scattering lowers the carrier mobility more and more at higher temperature. Theoretical calculations reveal that the mobility in non-polar semiconductors, such as silicon and germanium, is dominated by acoustic phonon interaction. The resulting mobility is expected to be proportional to T −3/2, while the mobility due to optical phonon scattering only is expected to be proportional to T −1/2. Experimentally, values of the temperature dependence of the mobility in Si, Ge and GaAs are listed in table.[20]

As ${\textstyle {\frac {1}{\tau }}\propto \left\langle v\right\rangle \Sigma }$, where ${\displaystyle \Sigma }$ is the scattering cross section for electrons and holes at a scattering center and ${\displaystyle \left\langle v\right\rangle }$ is a thermal average (Boltzmann statistics) over all electron or hole velocities in the lower conduction band or upper valence band, temperature dependence of the mobility can be determined. In here, the following definition for the scattering cross section is used: number of particles scattered into solid angle dΩ per unit time divided by number of particles per area per time (incident intensity), which comes from classical mechanics. As Boltzmann statistics are valid for semiconductors ${\displaystyle \left\langle v\right\rangle \sim {\sqrt {T}}}$.

For scattering from acoustic phonons, for temperatures well above Debye temperature, the estimated cross section Σph is determined from the square of the average vibrational amplitude of a phonon to be proportional to T. The scattering from charged defects (ionized donors or acceptors) leads to the cross section ${\displaystyle {\Sigma }_{\text{def}}\propto {\left\langle v\right\rangle }^{-4}}$. This formula is the scattering cross section for "Rutherford scattering", where a point charge (carrier) moves past another point charge (defect) experiencing Coulomb interaction.

The temperature dependencies of these two scattering mechanism in semiconductors can be determined by combining formulas for τ, Σ and ${\displaystyle \left\langle v\right\rangle }$, to be for scattering from acoustic phonons ${\displaystyle {\mu }_{ph}\sim T^{-3/2}}$ and from charged defects ${\displaystyle {\mu }_{\text{def}}\sim T^{3/2}}$.[14][16]

The effect of ionized impurity scattering, however, decreases with increasing temperature because the average thermal speeds of the carriers are increased.[12] Thus, the carriers spend less time near an ionized impurity as they pass and the scattering effect of the ions is thus reduced.

These two effects operate simultaneously on the carriers through Matthiessen's rule. At lower temperatures, ionized impurity scattering dominates, while at higher temperatures, phonon scattering dominates, and the actual mobility reaches a maximum at an intermediate temperature.

Disordered Semiconductors

Density of states of a solid possessing a mobility edge, ${\displaystyle E_{C}}$.

While in crystalline materials electrons can be described by wavefunctions extended over the entire solid,[21] this is not the case in systems with appreciable structural disorder, such as polycrystalline or amorphous semiconductors. Anderson suggested that beyond a critical value of structural disorder,[22] electron states would be localized. Localized states are described as being confined to finite region of real space, normalizable, and not contributing to transport. Extended states are spread over the extent of the material, not normalizable, and contribute to transport. Unlike crystalline semiconductors, mobility generally increases with temperature in disordered semiconductors.

Multiple Trapping and Release

Mott later developed[23] the concept of a mobility edge. This is an energy ${\displaystyle E_{C}}$, above which electrons undergo a transition from localized to delocalized states. In this description, termed multiple trapping and release, electrons are only able to travel when in extended states, and are constantly being trapped in, and re-released from, the lower energy localized states. Because the probability of an electron being released from a trap depends on its thermal energy, mobility can be described by an Arrhenius relationship in such a system:

Energy band diagram depicting electron transport under multiple trapping and release.

${\displaystyle \mu =\mu _{0}\exp \left(-{\frac {E_{A}}{k_{B}T}}\right)}$

where ${\displaystyle \mu _{0}}$ is a mobility prefactor, ${\displaystyle E_{A}}$ is activation energy, ${\displaystyle k_{B}}$ is the Boltzmann Constant, and ${\displaystyle T}$ is temperature. The activation energy is typically evaluated by measuring mobility as a function of temperature. The Urbach Energy can be used as a proxy for activation energy in some systems.[24]

Variable Range Hopping

At low temperature, or in system with a large degree of structural disorder (such as fully amorphous systems), electrons cannot access delocalized states. In such a system, electrons can only travel by tunnelling for one site to another, in a process called variable range hopping. In the original theory of variable range hopping, as developed by Mott and Davis,[25] the probability ${\displaystyle P_{ij}}$, of an electron hopping from one site ${\displaystyle i}$, to another site ${\displaystyle j}$, depends on their separation in space ${\displaystyle r_{ij}}$, and their separation in energy ${\displaystyle \Delta E_{ij}}$.

${\displaystyle P_{ij}=P_{0}\exp \left(-2\alpha r_{ij}-{\frac {\Delta E_{ij}}{k_{B}T}}\right)}$

Here ${\displaystyle P_{0}}$ is a prefactor associated with the phonon frequency in the material,[26] and ${\displaystyle \alpha }$ is the wavefunction overlap parameter. The mobility in a system governed by variable range hopping can be shown[25] to be:

${\displaystyle \mu =\mu _{0}\exp \left(-\left[{\frac {T_{0}}{T}}\right]^{-1/(d+1)}\right)}$

where ${\displaystyle \mu _{0}}$ is a mobility prefactor, ${\displaystyle T_{0}}$ is a parameter (with dimensions of temperature) that quantifies the width of localized states, and ${\displaystyle d}$ is the dimensionality of the system.

Measurement of semiconductor mobility

Hall mobility

Hall effect measurement setup for holes
Hall effect measurement setup for electrons

Carrier mobility is most commonly measured using the Hall effect. The result of the measurement is called the "Hall mobility" (meaning "mobility inferred from a Hall-effect measurement").

Consider a semiconductor sample with a rectangular cross section as shown in the figures, a current is flowing in the x-direction and a magnetic field is applied in the z-direction. The resulting Lorentz force will accelerate the electrons (n-type materials) or holes (p-type materials) in the (−y) direction, according to the right hand rule and set up an electric field ξy. As a result there is a voltage across the sample, which can be measured with a high-impedance voltmeter. This voltage, VH, is called the Hall voltage. VH is negative for n-type material and positive for p-type material.

Mathematically, the Lorentz force acting on a charge q is given by

For electrons:

${\displaystyle {\vec {F}}_{Hn}=-q({\vec {v}}_{n}\times {\vec {B}}_{z})}$

For holes:

${\displaystyle {\vec {F}}_{Hp}=+q({\vec {v}}_{p}\times {\vec {B}}_{z})}$

In steady state this force is balanced by the force set up by the Hall voltage, so that there is no net force on the carriers in the y direction. For electron,

${\displaystyle {\vec {F}}_{y}=(-q){\vec {\xi }}_{y}+(-q)[{\vec {v}}_{n}\times {\vec {B}}_{z}]=0}$

${\displaystyle \Rightarrow -q\xi _{y}+qv_{x}B_{z}=0}$

${\displaystyle \xi _{y}=v_{x}B_{z}}$

For electrons, the field points in the −y direction, and for holes, it points in the +y direction.

The electron current I is given by ${\displaystyle I=-qnv_{x}tW}$. Sub vx into the expression for ξy,

${\displaystyle \xi _{y}=-{\frac {IB}{nqtW}}=+{\frac {R_{Hn}IB}{tW}}}$

where RHn is the Hall coefficient for electron, and is defined as

${\displaystyle R_{Hn}=-{\frac {1}{nq}}}$

Since ${\displaystyle \xi _{y}={\frac {V_{H}}{W}}}$

${\displaystyle R_{Hn}=-{\frac {1}{nq}}={\frac {V_{Hn}t}{IB}}}$

Similarly, for holes

${\displaystyle R_{Hp}={\frac {1}{pq}}={\frac {V_{Hp}t}{IB}}}$

From the Hall coefficient, we can obtain the carrier mobility as follows:

{\displaystyle {\begin{aligned}\mu _{n}&=\left(-nq\right)\mu _{n}\left(-{\frac {1}{nq}}\right)\\&=-\sigma _{n}R_{Hn}\\&=-{\frac {\sigma _{n}V_{Hn}t}{IB}}\end{aligned}}}

Similarly,

${\displaystyle \mu _{p}={\frac {\sigma _{p}V_{Hp}t}{IB}}}$

Here the value of VHp (Hall voltage), t (sample thickness), I (current) and B (magnetic field) can be measured directly, and the conductivities σn or σp are either known or can be obtained from measuring the resistivity.

Field-effect mobility

The mobility can also be measured using a field-effect transistor (FET). The result of the measurement is called the "field-effect mobility" (meaning "mobility inferred from a field-effect measurement").

The measurement can work in two ways: From saturation-mode measurements, or linear-region measurements.[27] (See MOSFET for a description of the different modes or regions of operation.)

Using saturation mode

In this technique,[27] for each fixed gate voltage VGS, the drain-source voltage VDS is increased until the current ID saturates. Next, the square root of this saturated current is plotted against the gate voltage, and the slope msat is measured. Then the mobility is:

${\displaystyle \mu =m_{\text{sat}}^{2}{\frac {2L}{W}}{\frac {1}{C_{i}}}}$
where L and W are the length and width of the channel and Ci is the gate insulator capacitance per unit area. This equation comes from the approximate equation for a MOSFET in saturation mode:
${\displaystyle I_{D}={\frac {\mu C_{i}}{2}}{\frac {W}{L}}(V_{GS}-V_{th})^{2}.}$
where Vth is the threshold voltage. This approximation ignores the Early effect (channel length modulation), among other things. In practice, this technique may underestimate the true mobility.[28]

Using the linear region

In this technique,[27] the transistor is operated in the linear region (or "ohmic mode"), where VDS is small and ${\displaystyle I_{D}\propto V_{GS}}$ with slope mlin. Then the mobility is:

${\displaystyle \mu =m_{\text{lin}}{\frac {L}{W}}{\frac {1}{V_{DS}}}{\frac {1}{C_{i}}}.}$
This equation comes from the approximate equation for a MOSFET in the linear region:
${\displaystyle I_{D}=\mu C_{i}{\frac {W}{L}}\left((V_{GS}-V_{th})V_{DS}-{\frac {V_{DS}^{2}}{2}}\right)}$
In practice, this technique may overestimate the true mobility, because if VDS is not small enough and VG is not large enough, the MOSFET may not stay in the linear region.[28]

Optical mobility

Electron mobility may be determined from non-contact laser photo-reflectance measurements. A series of photo-reflectance measurements are made as the sample is stepped through focus. The electron diffusion length and recombination time are determined by a regressive fit to the data. Then the Einstein relation is used to calculate the mobility.[29][30]

Terahertz mobility

Electron mobility can be calculated from time-resolved terahertz probe measurement.[31][32] Femtosecond laser pulses excite the semiconductor and the resulting photoconductivity is measured using a terahertz probe, which detects changes in the terahertz electric field.[33]

Time resolved microwave conductivity (TRMC)

A proxy for charge carrier mobility can be evaluated using time-resolved microwave conductivity (TRMC).[34] A pulsed optical laser is used to create electrons and holes in a semiconductor, which are then detected as an increase in photoconductance. With knowledge of the sample absorbance, dimensions, and incident laser fluence, the parameter ${\displaystyle \phi \Sigma \mu =\phi (\mu _{e}+\mu _{h})}$ can be evaluated, where ${\displaystyle \phi }$ is the carrier generation yield (between 0 and 1), ${\displaystyle \mu _{e}}$ is the electron mobility and ${\displaystyle \mu _{h}}$ is the hole mobility. ${\displaystyle \phi \Sigma \mu }$ has the same dimensions as mobility, but carrier type (electron or hole) is obscured.

Doping concentration dependence in heavily-doped silicon

The charge carriers in semiconductors are electrons and holes. Their numbers are controlled by the concentrations of impurity elements, i.e. doping concentration. Thus doping concentration has great influence on carrier mobility.

While there is considerable scatter in the experimental data, for noncompensated material (no counter doping) for heavily doped substrates (i.e. ${\displaystyle 10^{18}\mathrm {cm} ^{-3}}$ and up), the mobility in silicon is often characterized by the empirical relationship:[35]

${\displaystyle \mu =\mu _{o}+{\frac {\mu _{1}}{1+\left({\frac {N}{N_{\text{ref}}}}\right)^{\alpha }}}}$
where N is the doping concentration (either ND or NA), and Nref and α are fitting parameters. At room temperature, the above equation becomes:

Majority carriers:[36]

${\displaystyle \mu _{n}(N_{D})=65+{\frac {1265}{1+\left({\frac {N_{D}}{8.5\times 10^{16}}}\right)^{0.72}}}}$
${\displaystyle \mu _{p}(N_{A})=48+{\frac {447}{1+\left({\frac {N_{A}}{6.3\times 10^{16}}}\right)^{0.76}}}}$

Minority carriers:[37]

${\displaystyle \mu _{n}(N_{A})=232+{\frac {1180}{1+\left({\frac {N_{A}}{8\times 10^{16}}}\right)^{0.9}}}}$
${\displaystyle \mu _{p}(N_{D})=130+{\frac {370}{1+\left({\frac {N_{D}}{8\times 10^{17}}}\right)^{1.25}}}}$

These equations apply only to silicon, and only under low field.

References

1. ^ a b c "NSM Archive - Physical Properties of Semiconductors". www.matprop.ru. Retrieved 2020-07-25.
2. ^ a b Umansky, V.; Heiblum, M.; Levinson, Y.; Smet, J.; Nübler, J.; Dolev, M. (2009). "MBE growth of ultra-low disorder 2DEG with mobility exceeding 35×106 cm2 V−1 s−1". Journal of Crystal Growth. 311 (7): 1658–1661. Bibcode:2009JCrGr.311.1658U. doi:10.1016/j.jcrysgro.2008.09.151.
3. ^ Dürkop, T.; Getty, S. A.; Cobas, Enrique; Fuhrer, M. S. (2004). "Extraordinary Mobility in Semiconducting Carbon Nanotubes". Nano Letters. 4 (1): 35. Bibcode:2004NanoL...4...35D. doi:10.1021/nl034841q. S2CID 45010238.
4. ^ a b Bolotin, K; Sikes, K; Jiang, Z; Klima, M; Fudenberg, G; Hone, J; Kim, P; Stormer, H (2008). "Ultrahigh electron mobility in suspended graphene". Solid State Communications. 146 (9): 351–355. arXiv:0802.2389. Bibcode:2008SSCom.146..351B. doi:10.1016/j.ssc.2008.02.024.
5. ^ Nawrocki, Robert (2016). "300‐nm Imperceptible, Ultraflexible, and Biocompatible e‐Skin Fit with Tactile Sensors and Organic Transistors". Advanced Electronic Materials. 2 (4): 1500452. doi:10.1002/aelm.201500452.
6. ^ Dürkop, T.; Getty, S. A.; Cobas, Enrique; Fuhrer, M. S. (2004). "Extraordinary Mobility in Semiconducting Carbon Nanotubes". Nano Letters. 4 (1): 35–39. Bibcode:2004NanoL...4...35D. doi:10.1021/nl034841q. S2CID 45010238.
7. ^ Snow, E. S.; Campbell, P. M.; Ancona, M. G.; Novak, J. P. (2005). "High-mobility carbon-nanotube thin-film transistors on a polymeric substrate". Applied Physics Letters. 86 (3): 033105. Bibcode:2005ApPhL..86c3105S. doi:10.1063/1.1854721. ISSN 0003-6951.
8. ^ He, Tao; Stolte, Matthias; Würthner, Frank (2013-12-23). "Air‐Stable n‐Channel Organic Single Crystal Field‐Effect Transistors Based on Microribbons of Core‐Chlorinated Naphthalene Diimide". Advanced Materials. 25 (48): 6951–6955. doi:10.1002/adma.201303392.
9. ^ Yuan, Yongbo (2014). "Ultra-high mobility transparent organic thin film transistors grown by an off-centre spin-coating method". Nature Communications. 5: 3005. doi:10.1038/ncomms4005. PMID 24398476.
10. ^ Heremans, Paul (2015). "Mechanical and Electronic Properties of Thin‐Film Transistors on Plastic, and Their Integration in Flexible Electronic Applications". Advanced Materials. 28 (22): 4266–4282. doi:10.1002/adma.201504360. PMID 26707947.
11. ^ a b Vladimir Vasilʹevich Mitin; Vi︠a︡cheslav Aleksandrovich Kochelap; Michael A. Stroscio (1999). Quantum heterostructures: microelectronics and optoelectronics. Cambridge University Press. pp. 307–9. ISBN 978-0-521-63635-3. Retrieved 2 March 2011.
12. ^ a b Singh (2008). Electronic Devices And Integrated Circuits. PHI Learning Pvt. Ltd. pp. 77–. ISBN 978-81-203-3192-1. Retrieved 1 March 2011.
13. Ferry, David K. Semiconductor transport. London: Taylor & Francis, 2000. ISBN 0-7484-0865-7 (hbk.), ISBN 0-7484-0866-5 (pbk.)
14. ^ a b c d Ibach, Harald. ; Luth, Hans. Solid-state physics : an introduction to principles of materials science / Harald Ibach, Hans Luth. New York: Springer, 2009. -(Advanced texts in physics) ISBN 978-3-540-93803-3
15. ^ a b Bulusu, A. (2008). "Review of electronic transport models for thermoelectric materials". Superlattices and Microstructures. 44 (1): 1–36. Bibcode:2008SuMi...44....1B. doi:10.1016/j.spmi.2008.02.008..
16. ^ a b c Bhattacharya, Pallab. Semiconductor optoelectronic devices / Pallab Bhattacharya. Upper Saddle River (NJ): Prentice-Hall, 1997. ISBN 0-13-495656-7 (nid.)
17. ^ a b Y. Takeda and T.P. Pearsall, "Failure of Mattheissen's Rule in the Calculation of Carrier Mobility and Alloy Scattering Effects in Ga0.47In0.53As", Electronics Lett. 17, 573-574 (1981).
18. ^ Peter Y. Yu; Manuel Cardona (30 May 2010). Fundamentals of Semiconductors: Physics and Materials Properties. Springer. pp. 205–. ISBN 978-3-642-00709-5. Retrieved 1 March 2011.
19. ^ Antonio Luque; Steven Hegedus (9 June 2003). Handbook of photovoltaic science and engineering. John Wiley and Sons. p. 79, eq. 3.58. ISBN 978-0-471-49196-5. Retrieved 2 March 2011. weblink (subscription only)
20. ^ a b Chapter 2: Semiconductor Fundamentals. Online textbook by B. Van Zeghbroeck]
21. ^ Hook, J. R.; Hall, H. E. (1991-09-05). Solid State Physics. Wiley. ISBN 978-0-471-92804-1.
22. ^ Anderson, P. W. (1958-03-01). "Absence of Diffusion in Certain Random Lattices". Physical Review. 109 (5): 1492–1505. doi:10.1103/PhysRev.109.1492.
23. ^ Mott, N. F. (1967-01-01). "Electrons in disordered structures". Advances in Physics. 16 (61): 49–144. doi:10.1080/00018736700101265. ISSN 0001-8732.
24. ^ Brotherton, S. D. (2013). Introduction to Thin Film Transistors: Physics and Technology of TFTs. Springer International Publishing. p. 143. ISBN 978-3-319-00001-5.
25. ^ a b Electronic Processes in Non-Crystalline Materials. Oxford Classic Texts in the Physical Sciences. Oxford, New York: Oxford University Press. 2012-03-24. ISBN 978-0-19-964533-6.
26. ^ Emin, David (1974-02-11). "Phonon-Assisted Jump Rate in Noncrystalline Solids". Physical Review Letters. 32 (6): 303–307. doi:10.1103/PhysRevLett.32.303.
27. ^ a b c Constance Rost-Bietsch (August 2005). Ambipolar and Light-Emitting Organic Field-Effect Transistors. Cuvillier Verlag. pp. 17–. ISBN 978-3-86537-535-3. Retrieved 1 March 2011.. This reference mistakenly leaves out a factor of 1/VDS in eqn (2.11). The correct version of that equation can be found, e.g., in Stassen, A. F.; De Boer, R. W. I.; Iosad, N. N.; Morpurgo, A. F. (2004). "Influence of the gate dielectric on the mobility of rubrene single-crystal field-effect transistors". Applied Physics Letters. 85 (17): 3899–3901. arXiv:cond-mat/0407293. Bibcode:2004ApPhL..85.3899S. doi:10.1063/1.1812368.
28. ^ a b Constance Rost-Bietsch (August 2005). Ambipolar and Light-Emitting Organic Field-Effect Transistors. Cuvillier Verlag. pp. 19–. ISBN 978-3-86537-535-3. Retrieved 20 April 2011. "Extracting the field-effect mobility directly from the linear region of the output characteristic may yield larger values for the field-effect mobility than the actual one, since the drain current is linear only for very small VDS and large VG. In contrast, extracting the field-effect mobility from the saturated region might yield rather conservative values for the field-effect mobility, since the drain-current dependence from the gate-voltage becomes sub-quadratic for large VG as well as for small VDS."
29. ^ W. Chism, "Precise Optical Measurement of Carrier Mobilities Using Z-scanning Laser Photoreflectance," arXiv:1711.01138 [physics:ins-det], Oct. 2017.
30. ^ W. Chism, "Z-scanning Laser Photoreflectance as a Tool for Characterization of Electronic Transport Properties," arXiv:1808.01897 [cond-mat:mes-hall], Aug. 2018.
31. ^ Ulbricht, Ronald; Hendry, Euan; Shan, Jie; Heinz, Tony F.; Bonn, Mischa (2011). "Carrier dynamics in semiconductors studied with time-resolved terahertz spectroscopy" (PDF). Reviews of Modern Physics. 83 (2): 543–586. Bibcode:2011RvMP...83..543U. doi:10.1103/RevModPhys.83.543. hdl:10871/15671. ISSN 0034-6861.
32. ^ Lloyd-Hughes, James; Jeon, Tae-In (2012). "A Review of the Terahertz Conductivity of Bulk and Nano-Materials". Journal of Infrared, Millimeter, and Terahertz Waves. 33 (9): 871–925. Bibcode:2012JIMTW..33..871L. doi:10.1007/s10762-012-9905-y. ISSN 1866-6892.
33. ^ Evers, Wiel H.; Schins, Juleon M.; Aerts, Michiel; Kulkarni, Aditya; Capiod, Pierre; Berthe, Maxime; Grandidier, Bruno; Delerue, Christophe; van der Zant, Herre S. J.; van Overbeek, Carlo; Peters, Joep L.; Vanmaekelbergh, Daniel; Siebbeles, Laurens D. A. (2015). "High charge mobility in two-dimensional percolative networks of PbSe quantum dots connected by atomic bonds". Nature Communications. 6: 8195. Bibcode:2015NatCo...6.8195E. doi:10.1038/ncomms9195. ISSN 2041-1723. PMC 4598357. PMID 26400049.
34. ^ Savenije, Tom J.; Ferguson, Andrew J.; Kopidakis, Nikos; Rumbles, Garry (2013-11-21). "Revealing the Dynamics of Charge Carriers in Polymer:Fullerene Blends Using Photoinduced Time-Resolved Microwave Conductivity". The Journal of Physical Chemistry C. 117 (46): 24085–24103. doi:10.1021/jp406706u. ISSN 1932-7447.
35. ^ B. L. Anderson and R. L. Anderson, "Fundamentals of Semiconductor Devices, " Mc Graw Hill, 2005
36. ^ Caughey, D.M.; Thomas, R.E. (1967). "Carrier mobilities in silicon empirically related to doping and field". Proceedings of the IEEE. 55 (12): 2192–2193. doi:10.1109/PROC.1967.6123.
37. ^ Del Alamo, J (1985). "Measuring and modeling minority carrier transport in heavily doped silicon". Solid-State Electronics. 28 (1): 47–54. Bibcode:1985SSEle..28...47D. doi:10.1016/0038-1101(85)90209-6.


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# If x is an integer, is y an integer?

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The Official Guide For GMAT® Quantitative Review, 2ND Edition

If x is an integer, is y an integer?

(1) The average (arithmetic mean) of x, y, and y - 2 is x.
(2) The average (arithmetic mean) of x and y is not an integer.

Data Sufficiency
Question: 118
Category: Arithmetic Statistics; Properties of numbers
Page: 161
Difficulty: 600

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SOLUTION

If x is an integer, is y an integer?

(1) The average (arithmetic mean) of x, y and y – 2 is x --> $$\frac{x+y+(y-2)}{3}=x$$ --> $$x+2y-2=3x$$ --> $$y=x+1=integer+integer=integer$$. Sufficient.

(2) The average (arithmetic mean) of x and y is not an integer --> if $$x=1$$ and $$y=2$$ the answer is YES but $$x=1$$ and $$y=\frac{1}{2}$$ the answer is NO. not sufficient.

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Re: If x is an integer, is y an integer? [#permalink]

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28 Feb 2014, 06:25
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Statement 1 is sufficient because if we perform algebra we get :

x+y-2+y/3=x
=2x= 2y-2
=x-1=y so if X is and integer Y is definitely an integer !!!

Statement 2 is insufficient because Y could be an Odd integer which results in a non integer
or Y could be a decimal and will result in a non integer.

Therefore "A" is the answer !!
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Re: If x is an integer, is y an integer? [#permalink]

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02 Mar 2014, 22:28
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[quote="Bunuel"]The Official Guide For GMAT® Quantitative Review, 2ND Edition

If x is an integer, is y an integer?

(1) The average (arithmetic mean) of x, y, and y -2 is x.
(2) The average (arithmetic mean) of x and y is not an integer.

Statement I is sufficient:

(x + y + y - 2) = 3x
x +2y - 2 = 3x
2y = 2x + 2
y = x + 1

If x is an integer then x + 1 will also be an integer

Statement II is insufficient:
x = 1, y = 2 - the average of x and y is not an integer however y is an integer
x = 1, y = 1.5, the average of x and y is not an integer and y is also not an integer

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Re: If x is an integer, is y an integer? [#permalink]

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01 Feb 2018, 11:36
Bunuel wrote:
SOLUTION

If x is an integer, is y an integer?

(1) The average (arithmetic mean) of x, y and y – 2 is x --> $$\frac{x+y+(y-2)}{3}=x$$ --> $$x+2y-2=3x$$ --> $$y=x+1=integer+integer=integer$$. Sufficient.

(2) The average (arithmetic mean) of x and y is not an integer --> if $$x=1$$ and $$y=2$$ the answer is YES but $$x=1$$ and $$y=\frac{1}{2}$$ the answer is NO. not sufficient.

With the limited time constraints, how can you tell if it's best to try out alegebra or to plug in numbers?

I tried to plug in numbers for statement 1, but didn't seem to plug in correctly. I then proceeded to plug in #'s for number #2, and this one was much easier to solve!
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Re: If x is an integer, is y an integer? [#permalink]

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02 Feb 2018, 10:21
OCDianaOC wrote:
Bunuel wrote:
SOLUTION

If x is an integer, is y an integer?

(1) The average (arithmetic mean) of x, y and y – 2 is x --> $$\frac{x+y+(y-2)}{3}=x$$ --> $$x+2y-2=3x$$ --> $$y=x+1=integer+integer=integer$$. Sufficient.

(2) The average (arithmetic mean) of x and y is not an integer --> if $$x=1$$ and $$y=2$$ the answer is YES but $$x=1$$ and $$y=\frac{1}{2}$$ the answer is NO. not sufficient.

With the limited time constraints, how can you tell if it's best to try out alegebra or to plug in numbers?

I tried to plug in numbers for statement 1, but didn't seem to plug in correctly. I then proceeded to plug in #'s for number #2, and this one was much easier to solve!

Hi

I think this comes best with practice. The more we practice various questions and the more you try to apply both algebra and number plug-in methodology in those questions, the more our understanding will increase steadily.
Eg., in this question only substitution works better in statement 2 but algebra works better in statement 1.

Sometimes we will see that when there are only 1/2 variables, substitution works out well and when more variables are given in a question, generally algebra or some conceptual application fits in better towards a solution. (That is not true all the time though)
Re: If x is an integer, is y an integer?   [#permalink] 02 Feb 2018, 10:21
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text:
# scipy.optimize.excitingmixing¶

scipy.optimize.excitingmixing(F, xin, iter=None, alpha=None, alphamax=1.0, verbose=False, maxiter=None, f_tol=None, f_rtol=None, x_tol=None, x_rtol=None, tol_norm=None, line_search='armijo', callback=None, **kw)[source]

Find a root of a function, using a tuned diagonal Jacobian approximation.

The Jacobian matrix is diagonal and is tuned on each iteration.

Warning

This algorithm may be useful for specific problems, but whether it will work may depend strongly on the problem.

Parameters: F : function(x) -> f Function whose root to find; should take and return an array-like object. xin : array_like Initial guess for the solution alpha : float, optional Initial Jacobian approximation is (-1/alpha). alphamax : float, optional The entries of the diagonal Jacobian are kept in the range [alpha, alphamax]. iter : int, optional Number of iterations to make. If omitted (default), make as many as required to meet tolerances. verbose : bool, optional Print status to stdout on every iteration. maxiter : int, optional Maximum number of iterations to make. If more are needed to meet convergence, NoConvergence is raised. f_tol : float, optional Absolute tolerance (in max-norm) for the residual. If omitted, default is 6e-6. f_rtol : float, optional Relative tolerance for the residual. If omitted, not used. x_tol : float, optional Absolute minimum step size, as determined from the Jacobian approximation. If the step size is smaller than this, optimization is terminated as successful. If omitted, not used. x_rtol : float, optional Relative minimum step size. If omitted, not used. tol_norm : function(vector) -> scalar, optional Norm to use in convergence check. Default is the maximum norm. line_search : {None, ‘armijo’ (default), ‘wolfe’}, optional Which type of a line search to use to determine the step size in the direction given by the Jacobian approximation. Defaults to ‘armijo’. callback : function, optional Optional callback function. It is called on every iteration as callback(x, f) where x is the current solution and f the corresponding residual. sol : ndarray An array (of similar array type as x0) containing the final solution. NoConvergence When a solution was not found.

#### Previous topic

scipy.optimize.anderson

#### Next topic

scipy.optimize.linearmixing


text:
# How do you solve the system of equations -6x + 5y = 8 and x + 5y = - 13?

Dec 19, 2017

$x = - 3 \mathmr{and} y = - 2$

#### Explanation:

Let,
$- 6 x + 5 y = 8$ ----- (1)

$x + 5 y = - 13$----(2)

(2)$\implies x = - 13 - 5 y$

Substitute this value of $x$ in (1):

(1)$\implies - 6 \left(- 13 - 5 y\right) + 5 y = 8$

$\implies 78 + 30 y + 5 y = 8$

$\implies 35 y = 8 - 78$

$\implies y = - \frac{70}{35} = - 2$

$\therefore y = - 2$

Now substitute this value in the expression of $x$:

$\implies x = - 13 - 5 y$

$\implies x = - 13 - 5 \left(- 2\right)$

$\implies x = - 13 + 10$

$\therefore x = - 3$

Answer: $x = - 3 \mathmr{and} y = - 2$

Dec 19, 2017

$x = - 3 \mathmr{and} y = - 2$

#### Explanation:

Notice that the $y$ terms both have the same coefficient.

Subtracting the equations will therefore eliminate the $y$ terms.

$\text{ "x" "color(red)(+" "5y)" " =-13" } \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . A$
$- 6 x \text{ "color(red)(+" "5y)" " =8" } \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . B$

Subtract $A - B \text{ }$ (change the signs of $B$)

$\text{ "x" "+" "5y" " =-13" } \textcolor{w h i t e}{\times \times \times \times x} A$
ul(color(blue)(+)6x" "color(blue)(-)" "5y" " =color(blue)(-)8)" "color(white)(xxxxxxxxx)B
" "7xcolor(white)(xxxxxxxxx) =-21" "color(white)(xxxxxxxxxx)C" "larr div 7
$\text{ } x \textcolor{w h i t e}{\times \times \times \times x} = - 3$

Substitute $- 3 \text{ for } x$ in $A$

$- 3 + 5 y = - 13$
$\text{ } 5 y = - 13 + 3$
$\text{ } 5 y = - 10$
$\text{ } y = - 2$

An alternative method is to equate the $y$ terms.
Transpose the equations to isolate $5 y$ in each:

$5 y = - 13 - x \text{ and } 5 y = 8 + 6 x$

We know that $\text{ } 5 y = 5 y$

Therefore: $\text{ "6x+8 = -13-x" } \leftarrow$ now solve for $x$

$6 x + x = - 13 - 8$
$\text{ } 7 x = - 21$
$\text{ } x = - 3$

Then proceed as described above to get $y = - 2$

Check in $B$
$- 6 x + 5 y$
$= - 6 \left(- 3\right) + 5 \left(- 2\right)$
$= 18 - 10$
$= 8 \text{ } \leftarrow$ the answer is the same as the RHS


text:
# grid_tincanopy

0th

Percentile

##### Canopy height model based on a triangulation.

Canopy height model based on a triangulation of first returns. Depending on the inputs this function computes a simple Delaunay triangulation of the first returns with a linear interpolation within each triangle. This function also enables the use of the pit-free algorithm developed by Khosravipour et al. (2014), which is based on the computation of a set of classical triangulations at different heights (see reference).

##### Usage
grid_tincanopy(x, res = 0.5, thresholds = c(0, 2, 5, 10, 15),
max_edge = c(0, 1), subcircle = 0, filter = "-keep_first")
##### Arguments
x

A LAS object

res

numeric. Resolution of the canopy height model.

thresholds

numeric. Set of height thresholds. If thresholds = 0 the algorithm is a strict rasterization of the triangulation of the first returns. However, if an array is passed to the function it becomes the Khosravipour et al. pit-free algorithm.

max_edge

numeric. Maximum edge-length of a triangle in the Delaunay triangulation. If a triangle has an edge greater than this value it will be removed. It is used to drive the pit-free algorithm (see reference) and to trim dummy interpolation on non-convex areas. The first number is the value for the classical triangulation (threshold = 0), the second number is the value for the pit-free algorithm for (thresholds > 0). If max_edge = 0 no trimming will be done.

subcircle

numeric. Radius of the circles. To obtain fewer pits the algorithm can replace each return with a circle composed of 8 points before computing the triangulation (see also grid_canopy).

filter

character. Streaming filter while reading the files (see readLAS). If the input is a LAScatalog the function readLAS is called internally. The user cannot manipulate the lidar data directly but can use streaming filters instead.

##### Value

Returns a data.table of class lasmetrics, which enables easier plotting and RasterLayer casting.

##### Use with a LAScatalog

When the parameter x is a LAScatalog the function processes the entire dataset in a continuous way using a multicore process. Parallel computing is set by default to the number of core available in the computer. A buffer is required. The user can modify the global options using the function catalog_options. lidR support .lax files. Computation speed will be significantly improved with a spatial index.

##### References

Khosravipour, A., Skidmore, A. K., Isenburg, M., Wang, T., & Hussin, Y. A. (2014). Generating pit-free canopy height models from airborne lidar. Photogrammetric Engineering & Remote Sensing, 80(9), 863-872.

##### Aliases
• grid_tincanopy
##### Examples
# NOT RUN {
LASfile = system.file("extdata", "Tree.laz", package="lidR")
las = readLAS(LASfile, Classification = FALSE, Intensity = FALSE, filter = "-drop_z_below 0")

# Basic triangulation and rasterization
chm1 = grid_tincanopy(las, thresholds = 0, max_edge = 0)

# Khosravipour et al. pitfree algorithm
chm2 = grid_tincanopy(las, thresholds = c(0,2,5,10,15), max_edge = c(0, 1.5))

plot(chm1)
plot(chm2)
# }

Documentation reproduced from package lidR, version 1.3.0, License: GPL-3

### Community examples

Looks like there are no examples yet.


text:
## Geometry: Common Core (15th Edition)

$\frac{1}{2}$
$\frac{2-(-3)}{6-(-4)}$ = $\frac{5}{10}$ If you divide a 5 out of the numerator and the denominator, you will have $\frac{1}{2}$. 5 $\div$ 5 = 1 10 $\div$ 5 = 2


text:
Assertion (A): A standing bus suddenly accelerates. If there was no friction between the feet of a passenger and the floor of the bus, the passenger would move back.
Reason (R): In the absence of friction, the floor of the bus would slip forward under the feet of the passenger.

1. (A) is true but (R) is false.
2. (A) is false but (R) is true.
3. Both (A) and (R) are true and (R) is the correct explanation of (A).
4. Both (A) and (R) are true but (R) is not the correct explanation of (A).

Subtopic:  Newton's Laws |
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Select the correct option based on statements below:
 Assertion (A): Newton's law of action and reaction is a consequence of Newton's law of inertia. Reason (R): Newton's law of inertia implies that any body that is not acted upon by external forces cannot change its state of rest or uniform motion.

 1 (A) is true but (R) is false. 2 (A) is false but (R) is true. 3 Both (A) and (R) are true and (R) is the correct explanation of (A). 4 Both (A) and (R) are true but (R) is not the correct explanation of (A).
Subtopic:  Newton's Laws |
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A block of mass $$m$$ is placed on a flat horizontal surface, and the coefficient of friction between the block and the surface is $$\mu$$. A force $$F_A$$ is applied to the block from above, and a force $$F_R$$ is applied to the right. In all situations being considered below, the block remains at rest. Let $$f$$ be the force of friction on the block.

Consider the statements:
(P) $$f$$ increases if $$m$$ is increased.
(Q) $$f$$ increases if $$F_A$$ is increased.
(R) $$f$$ increases if $$F_R$$ is increased.

1.  Only P is true.
2.  Only Q is true.
3.  P, Q are true.
4.  Only R is true.
Subtopic:  Friction |
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 Statement I: (Newton's 1st Law of Motion) Every body continues in its state of rest or of uniform motion in a straight line except in so far as it be compelled by an externally impressed force to act otherwise. Statement II: It is observed that when a car brakes suddenly, the passengers are thrown forward.

 1 Statement I is true, Statement II is true and Statement I is the correct explanation for Statement II. 2 Statement I is true, Statement II is true, Statement I is not the correct reason for Statement II. 3 Statement I is true, Statement II is false. 4 Statement I is false, Statement II is true.
Subtopic:  Newton's Laws |
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A block of mass M lies at rest on a horizontal table.

Statement I: (Newton's 3rd Law) To every action, there is an equal and opposite reaction. Action and reaction forces act on different bodies and in opposite directions.

Statement II: The normal reaction is the reaction force, while the weight is the action.

1. Statement I is true, Statement II is true and Statement I is the correct reason for Statement II.

2. Statement I is true, Statement II is true, Statement I is not the correct reason for Statement II.

3. Statement I is true, Statement II is false.

4. Statement I is false, Statement II is true.

Subtopic:  Newton's Laws |
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Consider the following two statements :

(A) The linear momentum of a particle is independent of the frame of reference.
(B) The kinetic energy of a particle is independent of the frame of reference.

1. Both A and B are true

2. A is true but B is false

3. A is false but B is true

4. both A and B are false

Subtopic:  Newton's Laws |
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Mark the correct statements about the friction between two bodies.
(a) static friction is always greater than kinetic friction.
(b) coefficient of static friction is always greater than the coefficient of kinetic friction.
(c) limiting friction is always greater than kinetic friction.
(d) limiting friction is never less than static friction.
Choose the correct option:
1. (a), (b) and (c)
2. (b), (c) and (d)
3. (a) and (d)
4. (c) and (d)

Subtopic:  Friction |
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In the figure, the coefficient of friction between the floor and body B is 0.1. The coefficient of friction between bodies B and A is 0.2. A force F is applied as shown on B. The mass of A is rn/2 and of B is m.

(a) The bodies will move together if F = O .25 mg
(b) The A will slip with B if F = 0.5 mg
(c) The bodies will move together if F = 0.5 mg
(d) The bodies will be at rest if F = 0.1 mg
(e) The maximum value of F for which the two bodies will move together is 0.45 mg

Which of the following statement(s) is/are true?

1. (a, b, d, e)

2. (a, c, d, e)

3. (b, c, d)

4. (a, b, c)

Subtopic:  Application of Laws | Friction |
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A particle is on a smooth horizontal plane. A force F is applied, whose F-t graph is given.

Consider the following statements.
a. At time ${\mathrm{t}}_{1}$, acceleration is constant.
b. Initially the particle must be at rest.
c. At time ${\mathrm{t}}_{2}$, acceleration is constant.
d. The initial acceleration is zero.
Select the correct statement(s):
1. (a), (c)
2. (a), (b), (d)
3. (c), (d)
4. (b), (c)

Subtopic:  Newton's Laws |
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The figure shows the position-time graph of a particle of mass 4 kg. What is the force on the particle for t > 4 s?  (Consider one-dimensional motion only).

1. 0

2. 40 N

3. 20 N

4. 10 N

Subtopic:  Application of Laws |
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text:
# Revision history [back]

Your callback should only look like this:

def listener_callback(self):
self.destroy_node()

You can check that the node has been killed through command line: ros2 node list

Note that the rclpy.spin(minimal_action_server) will not terminate even if you kill the node

Your callback should only look like this:

def listener_callback(self):
self.destroy_node()

I tested this with the examples_rclpy_minimal_publisher/publisher_member_function. Here for example you can call self.destroy_node() in the timer callback after N messages have been published.

def timer_callback(self):

if self.i > 5:
self.destroy_node()
return

msg = String()
msg.data = 'Hello World: %d' % self.i
self.publisher_.publish(msg)
self.get_logger().info('Publishing: "%s"' % msg.data)
self.i += 1

You can check that the node has been killed through command line: ros2 node list

Note that the rclpy.spin(minimal_action_server) will not terminate even if you kill the node