What is Effective mass: Definition and 65 Discussions

In solid state physics, a particle's effective mass (often denoted


{\textstyle m^{*}}
) is the mass that it seems to have when responding to forces, or the mass that it seems to have when interacting with other identical particles in a thermal distribution. One of the results from the band theory of solids is that the movement of particles in a periodic potential, over long distances larger than the lattice spacing, can be very different from their motion in a vacuum. The effective mass is a quantity that is used to simplify band structures by modeling the behavior of a free particle with that mass. For some purposes and some materials, the effective mass can be considered to be a simple constant of a material. In general, however, the value of effective mass depends on the purpose for which it is used, and can vary depending on a number of factors.
For electrons or electron holes in a solid, the effective mass is usually stated in units of the rest mass of an electron, me (9.11×10−31 kg). In these units it is usually in the range 0.01 to 10, but can also be lower or higher—for example, reaching 1,000 in exotic heavy fermion materials, or anywhere from zero to infinity (depending on definition) in graphene. As it simplifies the more general band theory, the electronic effective mass can be seen as an important basic parameter that influences measurable properties of a solid, including everything from the efficiency of a solar cell to the speed of an integrated circuit.

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  1. M

    I Effective mass from the Lagrangian

    Hello! I have the following Lagrangian: $$L = \frac{1}{2}mv^2+fv$$ where ##v = \dot{x}##, where x is my coordinate and f is a function of v only (no explicit dependence on t or x). What I get by solving the Euler-Lagrange equations is: $$\frac{d}{dt}(mv+f+\frac{\partial f}{\partial v} v) =...
  2. Bastion

    B Experimental evidence for effective mass increasing with speed

    I am new to this forum and in line with its rules I am trying not to repeat a question already asked. I find that Jeremyfiennes original question I.e Experimental evidence for effective mass increasing with speed” is the question I want to ask but I am not satisfied with the answers given so...
  3. ergospherical

    I Effective mass in terms of electron states

    I'm trying to figure out the second order extension of the "trick" used on page 92 (https://www.damtp.cam.ac.uk/user/tong/aqm/solid3.pdf) for the calculation of the effective mass matrix ##m^{\star}_{ij} = \hbar^2 (\partial^2 E/ \partial k_i \partial k_j)^{-1}## on page 94. I think for this one...
  4. J

    B Experimental evidence for effective mass increasing with speed

    What is the best recent experimental evidence for effective mass increase (momentum/velocity) with speed, with experimental details? Searching the web all I find is either very old ones (early 1900s). Or vague generalized statements like (Wikipedia) "Many additional experiments concerning the...
  5. chem_

    Band theory: effective mass and Hall's coefficient

    Consider the following scenario. A material has the E-k band scheme as shown in the figure (extended scheme of zones). Could anyone give me a suggestion regarding the following : Electrical character of the material with the temperature. Sign of the Hall coefficient. Sign of the effective mass...
  6. J

    I Changing the effective mass of an electron using electric potentials?

    The Dirac equation for an electron in the presence of an electromagnetic 4-potential ##A_\mu##, where ##\hbar=c=1##, is given by $$\gamma^\mu\big(i\partial_\mu-eA_\mu\big)\psi-m_e\psi=0.\tag{1}$$ I assume the Weyl basis so that $$\psi=\begin{pmatrix}\psi_L\\\psi_R\end{pmatrix}\hbox{ and...
  7. C

    Using Effective Mass to find the accelerations of 3 masses connected by pulleys

    Since 2nd pulley accelerates downward with the same acceleration of m1, and because the string around pulley has a constant length, it must be that a2=−a1+ar and a3=−a1−ar, where ar is the relative acceleration between the pulley and m2. From the above 4 equations, we are supposed to determine...
  8. B

    B What are the different ways to create negative effective mass

    Also can negative effective mass be used as a replacement in a warp drive? Also what is the most realistic warp drive theory?
  9. Z

    A Effective mass of the electron for Si

    https://ecee.colorado.edu/~bart/book/effmass.htm#short Looks like the effective mass for holes in Si can either be 0.57 or 0.81, according to the link above. Is there a temperature regime where one effective mass should be used instead of the other? Is anyone able to explain in layman's terms...
  10. Error404!

    A Effective Mass of Electron for White Tin

    For some work of mine i need the plasma resonance frequency for tin nanoparticle, i searched various research papers and found different efffective masses tin for different orbits. here i attached screen shot from W A Roger and S B Woods (Cyclotron effective masses in white tin). What value is...
  11. F

    Can't find the effective mass of a 2-dimensional semiconductor

    This is the situation: you have the band structure of a two-dimensional semiconductor E=E(k). Both, valence band and condcution band. You use the definition of effective mass: (m)^(-1)=(d2E/dkidkj), but both bands are in such a way that the 2x2 matrix that you obtain has zero determinant. So...
  12. LesterTU

    How to evaluate the effective mass tensor (band structure)

    Homework Statement The energy-band dispersion for a 3D crystal is given by $$E(\mathbf k) = E_0 - Acos(k_xa) - Bcos(k_yb) - Ccos(k_zc)$$ What is the value of the effective mass tensor at ## \mathbf k = 0 ##? Homework Equations The effective mass tensor is given by $$ \left( \frac{1}{m^*}...
  13. A

    I What is zero and infinite effective mass?

    What is effective mass? What is zero and infinite effective mass?
  14. Ranku

    I Mass of Holes: Measuring Physically

    Holes have an effective mass. How is the mass of holes measured physically?
  15. SchroedingersLion

    A Understanding Effective Mass in Crystal Potentials

    Greetings, I have come across the effective mass in my solid state physics lectures, and I have troubles understanding it. I have uploaded the part of the script (handwritten 4,5 pages) that I don't understand. At page 3, it is shown that the effective mass in the vicinity is negative. What...
  16. B

    How to calculate relative effective mass?

    Homework Statement A simplified E versus k curve for an electron in the conduction band is given. the value of a is 10 Å. Determine the relative effective mass m* / m0. Homework Equations [/B] m* = ħ2 / (d2E /dk2) m0 = 9.11 E-31kg 3. The Attempt at a Solution The second derivative must...
  17. B

    A Zeeman energy with an effective mass

    Hi all, Say we have a model in a solid state system where we have an effective mass, \begin{equation}m^*,\end{equation}some fraction of the free electron mass. Now we apply an external magnetic field. Then the Zeeman energy splitting should be: \begin{equation} E_{Zeeman}=\pm 1/2 g \mu_B B...
  18. N

    B Does Graphene Have Infinite Mass?

    If yes, because graphene has infinite effective mass, has it infinite mass as well? http://physics.stackexchange.com/questions/52125/effective-mass-and-fermi-velocity-of-electrons-in-graphene
  19. bluejay27

    I How can you derive the effective mass?

    I have seen that the energy of the band structure is defined in the following form. How do you obtain this form of the equation? Additionally, how can you obtain the effect mass of the electron? Do you just manipulate algebraically the equation of the energy of the band structure and solve it...
  20. A

    I Exploring Effective Mass and Spin-Orbit Interactions in Semiconductor Physics

    Often I see people using an effective mass model to describe electrons in the bottom of the conduction band. Spin orbit is then included as a perturbation in this effective mass model. But what is the justification for using this sort of model? Would the correct way not be to start from the full...
  21. A

    I Understanding the Derivation of Effective Mass Approximation in Semiconductors

    I have noticed that in a lot of theoretical modelling of semiconductors you assume that the electrons living in the bottom of the conduction band obey a free particle Hamiltonian: H = p^2/2m* , where m* is the effective mass in the conduction band and p^2 is the usual differential operator. I...
  22. A

    I Kinetic energy with effective mass

    I want to write the kinetic energy operator as a matrix within a finite element approach for electrons moving in a crystal with some effective mass that is a function of position. Now usually we have: K = -ħ2/2m d2/dx2 such that the second order derivative of a wavefunction maybe written as...
  23. A

    I Effective mass position dependent

    I have some trouble understanding what to do when the effective mass in a heterostructure depends on position. Suppose for example that you have two materials, which are put together such that in one region the kinetic energy is described by one effective mass and in the other region another...
  24. AwesomeTrains

    Effective mass dependency on the donator atom?

    Hello pf Is the effective mass dependend on the donator atom in a semiconductor? In our experiment we have calculated the effective mass in a germanium semiconductor, doped with an unknown atom. It is 0.39m_e. From the internet we know that the effective mass is 0.33m_e. Is our result...
  25. S

    Hole Mass vs Electron Mass: What's the Difference?

    Why is it opposite to that of an electron?
  26. F

    Question about Fermi's gas effective mass.

    Hi everybody; today I was reading some problems about a metal and the electrons of the conduction band; the man who solved them used the mass of the electron as effective mass (m*). I don't know why he did that; I have investigated but I don't have fount an explanation. Can somebody please...
  27. L

    Effective Mass of Electron in Polyethylene

    Hello friends Can anyone tell me the effective mass of electron in polyethylene and how it depends on the crystalline nature of polymer.
  28. C

    Effective mass, Taylor's theorem

    Hi, I'm starting to study conduction in semiconductor and I have a doubt about the concept of effective mass. Let's suppose to deal with an dispersion relation in two dimensions; basically, the effective masses along the two axes are defined by means of the Taylor's expansion of the function...
  29. R

    Calculating the effective mass

    Homework Statement Homework Equations The Attempt at a Solution I understand you can calculate the effective mass by using the equation above. What I'm unsure about is the "calculate the effective mass of the carriers" part. A donor impurity would be near the conduction band, therefore do...
  30. R

    Why is the effective mass of electron different for metals

    ... and semiconductors? I'm trying to get some basic principles sorted in my head and I can't seem to find a straight answer to this. The lecturer drew a dispersion graph to explain it but I'm still a little confused. I understand since ##m^{*}\propto \frac{1}{\frac{\partial^2 E}{\partial...
  31. M

    Weak Form of the Effective Mass Schrodinger Equation

    Hi, I am numerically solving the 2D effective-mass Schrodinger equation \nabla \cdot (\frac{-\hbar^2}{2} c \nabla \psi) + (U - \epsilon) \psi = 0 where c is the effective mass matrix \left( \begin{array}{cc} 1/m^*_x & 1/m^*_{xy} \\ 1/m^*_{yx} & 1/m^*_y \\ \end{array} \right) I know that...
  32. C

    Is It Possible to Reduce the Effective Mass of a System?

    I just want to know, is a technology designed to reduce the effective mass of a system possible, at least in theory? Within the system, everything acts as it should (it retains all chemical, physical, and structural properties), but outside, it acts as if it had a lower mass. It's not as...
  33. J

    Modal participation factor & effective mass - semi definite system

    i have solved a torsional semi-definite system problem. i have uploaded the solved problem. solved for modal participation factor and effective mass. for a semi definite system we know that one of the modes is rigid mode with natural frequency 0. i think due to the presence of rigid...
  34. H

    Understanding Electron Effective Mass in Crystals

    I have got confused about electron effective mass in crystals. Which of the definitions below are correct? 1) When an electron is in a crystal with periodic potential, it behaves such as a free electron with effective mass m^* 2) when an external force is applied on the electron in a...
  35. S

    What is the Effect of Distributed Mass on the Time Period of a Spring?

    Okay, so I've been trying to figure out what would be the time period of spring of mass 'M' & spring constant K which is hung vertically from a ceiling. Dividing the spring into n equal massless springs of spring constant k=nK which are connected to each other by n small objects of mass...
  36. E

    Can we measure the effective mass of charge carriers using a centrifuge?

    Hello you all! What about a slightly exotic idea? Here I propose to measure the "effective" mass of charge carriers by centrifugal force. Electrons in vacuum have a mass, and when moving in a solid an other mass, often called "effective" (as if the vacuum mass were ineffective). Centrifugal...
  37. J

    Electron/Hole Effective Mass vs Solar Cell efficiency

    Hi. I'm doing research relating to solar cells and came about the topic related to effective masses. The wiki article stated how effective mass (solid state materials) influences the efficiency of solar cells, but doing more research I couldn't find any solid examples of an effective mass vs...
  38. H

    Infinite effective mass in metals

    Hello, I have a question about conduction in metals. I guess you all know a common pedagogical picture where an electron bands are drawn as ~ cosine curves in 1D or E_0 - cos(k_x) - cos(k_y) in 2D. Now, in metals , we were told that the Fermi surface passes through the band. Therefore...
  39. GeorgeDishman

    Effective mass of gravitational waves

    Hulse and Taylor's observation of PSR B1913+16 provides indirect evidence for the existence of gravitational waves ("GW") in accordance with GR. Simulations show recoil of merging black holes due to the momentum of the GW they emit during the inspiral. http://www.black-holes.org/explore2.html...
  40. J

    Effective mass of Dirac electron increased by electrostatic potential?

    The Dirac electron in the Higgs vacuum field v and an electromagnetic field with vector potential A_\mu is described by the following equation: i \gamma^\mu \partial_\mu \psi = g v \psi + e \gamma_\mu A^\mu \psi where g is the coupling constant to the Higgs field and e is the coupling...
  41. M

    Effective Mass of Cantilever beam (getting .36m instead of .23m)

    Homework Statement Im trying to find the equivalent mass of a beam so it can be modeled as a hanging point mass problem. Given the v = y_dot equation, y equation, and T Somewhere I am messing up with my math I believe as I am getting .36m instead of what I believe should be .23m...
  42. M

    Derivation of Effective Mass from a Parabola

    There is some fundamental about effective mass that I am misunderstanding about effective masses. I understand the relation E\left(k\right) = E_0 + \frac{1}{2m^*}k^2 But I'm not sure when it's appropriate to fit this to a parabola. I would have thought the fitting is only done when...
  43. M

    Electron Effective Mass and Effective Mass Theory in Semiconductors

    I would like to ask about the reason why the electron "effective mass" was introduced in semiconductors. What is its' usefulness ? And also about the so-called "effective mass theory" used to calculate energies for the shallow defects in semiconducors. What are mean pecularities and...
  44. M

    What are the units for effective mass in the formula m* = (h/2pi)2(d2E/dk2)-1?

    Are the units for effective mass in kg? I have been trying to wrap my head around the concept then realized I wasn't even sure about the units. The formula is: m* = (h/2pi)2(d2E/dk2)-1 the units of h are in J-s, and the units of E are in J so... (J-s)2/J = J-s2 J = kg m2/s2 kg m2 s2/s2...
  45. W

    Effective mass of Si in semiconductor physics

    Hello, I have the problem, from Sze' book, tunneling current, what is m*? I want to konw the value of it. The m* has many reference, how do I select? My interested device is diode that was doped with Boron and Phosphorus for p-n junction. Thank very much.
  46. inflector

    What Does Anisotropic Effective Mass Mean?

    What Does "Anisotropic Effective Mass" Mean? I'm reading "The Defintion of Mach's Principle" by Julian Barbour: http://arxiv.org/abs/1007.3368 from July of this year, and it contains a paragraph (Section 9 bottom of page 23) I do not understand, the beginning of which says: I have...
  47. R

    Can we easily extract effective masses from Si band structure?

    can we estimate effective masses of electron and hole from the band structure if yes how? can somebody explain with reference to Si band structure?
  48. D

    Calculating Effective Mass of InGaAs Alloy

    Does anyone know how to find an effective mass for an alloy such as InGaAs?
  49. 6

    Effective mass of a charge carrier

    What is the effective mass of a charge carrier, numerically? Is it equivalent to the mass of an electron? Is it the same for negative charge carriers as it is for positive charge carriers?
  50. F

    How Do You Calculate the Effective Mass of Electrons in a Kronig-Penney Model?

    Homework Statement Consider electrons in a Kronig-Penney model with the following parameters: a=b=3 angstroms Uo=10.0 eV Calculate the effective mass of the electrons near the start of the first allowed band. You can find this by fitting the E-k results to an equation E=E1+(hbar^2...