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Effective mass (solid-state physics)

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From Wikipedia, the free encyclopedia

Mass of a particle when interacting with other particles

For negative mass intheoretical physics, seeNegative mass.

Insolid state physics, a particle'seffective mass (often denoted ${\textstyle m^{*}}$ ) is themass that itseems to have when responding to forces, or the mass that it seems to have when interacting with other identical particles in athermal distribution. One of the results from theband 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 avacuum. The effective mass is a quantity that is used to simplify band structures by modeling the behavior of afree 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.

Forelectrons orelectron holes in a solid, the effective mass is usually stated as a factor multiplying therest mass of an electron,m_e (9.11 × 10⁻³¹ kg). This factor is usually in the range 0.01 to 10, but can be lower or higher—for example, reaching 1,000 in exoticheavy fermion materials, or anywhere from zero to infinity (depending on definition) ingraphene. 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.

Simple case: parabolic, isotropic dispersion relation

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At the highest energies of the valence band in many semiconductors (Ge, Si, GaAs, ...), and the lowest energies of the conduction band in some semiconductors (GaAs, ...), the band structureE(k) can be locally approximated as

E(\mathbf {k} )=E_{0}+{\frac {\hbar ^{2}\mathbf {k} ^{2}}{2m^{*}}}

whereE(k) is the energy of an electron atwavevectork in that band,E₀ is a constant giving the edge of energy of that band, andm^* is a constant (the effective mass).

It can be shown that the electrons placed in these bands behave as free electrons except with a different mass, as long as their energy stays within the range of validity of the approximation above. As a result, the electron mass in models such as theDrude model must be replaced with the effective mass.

One remarkable property is that the effective mass can becomenegative, when the band curves downwards away from a maximum. As a result of thenegative mass, the electrons respond to electric and magnetic forces by gaining velocity in the opposite direction compared to normal; even though these electrons have negative charge, they move in trajectories as if they had positive charge (and positive mass). This explains the existence ofvalence-band holes, the positive-charge, positive-massquasiparticles that can be found in semiconductors.^[1]

In any case, if the band structure has the simple parabolic form described above, then the value of effective mass is unambiguous. Unfortunately, this parabolic form is not valid for describing most materials. In such complex materials there is no single definition of "effective mass" but instead multiple definitions, each suited to a particular purpose. The rest of the article describes these effective masses in detail.

Intermediate case: parabolic, anisotropic dispersion relation

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Constant energy ellipsoids in silicon near the six conduction band minima. For each valley (band minimum), the effective masses arem_ℓ = 0.92m_e ("longitudinal"; along one axis) andm_t = 0.19m_e ("transverse"; along two axes).^[2]

In some important semiconductors (notably, silicon) the lowest energies of the conduction band are not symmetrical, as the constant-energy surfaces are nowellipsoids, rather than the spheres in the isotropic case. Each conduction band minimum can be approximated only by

E\left(\mathbf {k} \right)=E_{0}+{\frac {\hbar ^{2}}{2m_{x}^{*}}}\left(k_{x}-k_{0,x}\right)^{2}+{\frac {\hbar ^{2}}{2m_{y}^{*}}}\left(k_{y}-k_{0,y}\right)^{2}+{\frac {\hbar ^{2}}{2m_{z}^{*}}}\left(k_{z}-k_{0,z}\right)^{2}

wherex,y, andz axes are aligned to the principal axes of the ellipsoids, andm^*
_x,m^*
_y andm^*
_z are the inertial effective masses along these different axes. The offsetsk_0,x,k_0,y, andk_0,z reflect that the conduction band minimum is no longer centered at zero wavevector. (These effective masses correspond to the principal components of the inertial effective mass tensor, described later.^[3])

In this case, the electron motion is no longer directly comparable to a free electron; the speed of an electron will depend on its direction, and it will accelerate to a different degree depending on the direction of the force. Still, in crystals such as silicon the overall properties such as conductivity appear to be isotropic. This is because there are multiplevalleys (conduction-band minima), each with effective masses rearranged along different axes. The valleys collectively act together to give an isotropic conductivity. It is possible to average the different axes' effective masses together in some way, to regain the free electron picture. However, the averaging method turns out to depend on the purpose:^[4]

For calculation of the total density of states and the total carrier density, via thegeometric mean combined with a degeneracy factorg which counts the number of valleys (in silicong = 6):^[3]
$m_{\text{density}}^{*}={\sqrt[{3}]{g^{2}m_{x}m_{y}m_{z}}}$
(This effective mass corresponds to the density of states effective mass, described later.)
For the per-valley density of states and per-valley carrier density, the degeneracy factor is left out.
For the purposes of calculating conductivity as in the Drude model, via theharmonic mean
$m_{\text{conductivity}}^{*}=3\left[{\frac {1}{m_{x}^{*}}}+{\frac {1}{m_{y}^{*}}}+{\frac {1}{m_{z}^{*}}}\right]^{-1}$
Since the Drude law also depends on scattering time, which varies greatly, this effective mass is rarely used; conductivity is instead usually expressed in terms of carrier density and an empirically measured parameter,carrier mobility.

General case

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In general the dispersion relation cannot be approximated as parabolic, and in such cases the effective mass should be precisely defined if it is to be used at all.Here a commonly stated definition of effective mass is theinertial effective mass tensor defined below; however, in general it is a matrix-valued function of the wavevector, and even more complex than the band structure. Other effective masses are more relevant to directly measurable phenomena.

Inertial effective mass tensor

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A classical particle under the influence of a force accelerates according toNewton's second law,a =m⁻¹F, or alternatively, the momentum changes according to⁠d/dt⁠p =F. This intuitive principle appears identically in semiclassical approximations derived from band structure when interband transitions can be ignored for sufficiently weak external fields.^[5]^[6]The force gives a rate of change incrystal momentump_crystal:

\mathbf {F} ={\frac {\operatorname {d} \mathbf {p} _{\text{crystal}}}{\operatorname {d} t}}=\hbar {\frac {\operatorname {d} \mathbf {k} }{\operatorname {d} t}},

whereħ =h/2π is thereduced Planck constant.

Acceleration for a wave-like particle becomes the rate of change ingroup velocity:

\mathbf {a} ={\frac {\operatorname {d} }{\operatorname {d} t}}\,\mathbf {v} _{\text{g}}={\frac {\operatorname {d} }{\operatorname {d} t}}\left(\nabla _{k}\,\omega \left(\mathbf {k} \right)\right)=\nabla _{k}{\frac {\operatorname {d} \omega \left(\mathbf {k} \right)}{\operatorname {d} t}}=\nabla _{k}\left({\frac {\operatorname {d} \mathbf {k} }{\operatorname {d} t}}\cdot \nabla _{k}\,\omega (\mathbf {k} )\right),

where∇_k is thedel operator inreciprocal space. The last step follows from usingthe chain rule for a total derivative for a quantity with indirect dependencies, because the direct result of the force is the change ink(t) given above, which indirectly results in a change inE(k)=ħω(k). Combining these two equations yields

\mathbf {a} =\nabla _{k}\left({\frac {\mathbf {F} }{\hbar }}\cdot \nabla _{k}\,{\frac {E(\mathbf {k} )}{\hbar }}\right)={\frac {1}{\hbar ^{2}}}\left(\nabla _{k}\left(\nabla _{k}\,E(\mathbf {k} )\right)\right)\cdot \mathbf {F} =M_{\text{inert}}^{-1}\cdot \mathbf {F}

using thedot product rule with a uniform force (∇_kF=0). $\nabla _{k}\left(\nabla _{k}\,E(\mathbf {k} )\right)$ is theHessian matrix ofE(k) in reciprocal space. We see that the equivalent of the Newtonian reciprocalinertial mass for a free particle defined bya =m⁻¹F has become a tensor quantity

M_{\text{inert}}^{-1}={\frac {1}{\hbar ^{2}}}\nabla _{k}\left(\nabla _{k}\,E(\mathbf {k} )\right).

whose elements are

\left[M_{\text{inert}}^{-1}\right]_{ij}={\frac {1}{\hbar ^{2}}}\left[\nabla _{k}\left(\nabla _{k}\,E(\mathbf {k} )\right)\right]_{ij}={\frac {1}{\hbar ^{2}}}{\frac {\partial ^{2}E}{\partial k_{i}\partial k_{j}}}\,.

This tensor allows the acceleration and force to be in different directions, and for the magnitude of the acceleration to depend on the direction of the force.

For parabolic bands, the off-diagonal elements ofM_inert⁻¹ are zero, and the diagonal elements are constants
For isotropic bands the diagonal elements must all be equal and the off-diagonal elements must all be equal.
For parabolic isotropic bands,M_inert⁻¹ =⁠1/m^*⁠I, wherem^* is a scalar effective mass andI is the identity.
In general, the elements ofM_inert⁻¹ are functions ofk.
The inverse,M_inert = (M_inert⁻¹)⁻¹, is known as theeffective mass tensor. Note that it is not always possible to invertM_inert⁻¹

For bands with linear dispersion $E\propto k$ such as with photons or electrons ingraphene, the group velocity is fixed, i.e. electrons travelling with parallel withk to the force directionF cannot be accelerated and the diagonal elements ofM_inert⁻¹ are obviously zero. However, electrons travelling with a component perpendicular to the force can be accelerated in the direction of the force, and the off-diagonal elements ofM_inert⁻¹ are non-zero. In fact the off-diagonal elements scale inversely withk, i.e. they diverge (become infinite) for smallk. This is why the electrons in graphene are sometimes said to have infinite mass (due to the zeros on the diagonal ofM_inert⁻¹) and sometimes said to be massless (due to the divergence on the off-diagonals).^[7]

Cyclotron effective mass

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Classically, a charged particle in a magnetic field moves in a helix along the magnetic field axis. The periodT of its motion depends on its massm and chargee,

T=\left\vert {\frac {2\pi m}{eB}}\right\vert

whereB is themagnetic flux density.

For particles in asymmetrical band structures, the particle no longer moves exactly in a helix, however its motion transverse to the magnetic field still moves in a closed loop (not necessarily a circle). Moreover, the time to complete one of these loops still varies inversely with magnetic field, and so it is possible to define acyclotron effective mass from the measured period, using the above equation.

The semiclassical motion of the particle can be described by a closed loop in k-space. Throughout this loop, the particle maintains a constant energy, as well as a constant momentum along the magnetic field axis. By definingA to be thek-space area enclosed by this loop (this area depends on the energyE, the direction of the magnetic field, and the on-axis wavevectork_B), then it can be shown that the cyclotron effective mass depends on the band structure via the derivative of this area in energy:

m^{*}\left(E,{\hat {B}},k_{\hat {B}}\right)={\frac {\hbar ^{2}}{2\pi }}\cdot {\frac {\partial }{\partial E}}A\left(E,{\hat {B}},k_{\hat {B}}\right)

Typically, experiments that measure cyclotron motion (cyclotron resonance,De Haas–Van Alphen effect, etc.) are restricted to only probe motion for energies near theFermi level.

Intwo-dimensional electron gases, the cyclotron effective mass is defined only for one magnetic field direction (perpendicular) and the out-of-plane wavevector drops out. The cyclotron effective mass therefore is only a function of energy, and it turns out to be exactly related to the density of states at that energy via the relation $\scriptstyle g(E)\;=\;{\frac {g_{v}m^{*}}{\pi \hbar ^{2}}}$ , whereg_v is the valley degeneracy. Such a simple relationship does not apply in three-dimensional materials.

Density of states effective masses (lightly doped semiconductors)

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Density of states effective mass in various semiconductors^[8]^[9]^[10]^[11]
Group	Material	Electron	Hole
IV	Si (4 K)	1.06	0.59
	Si (300 K)	1.09	1.15
	Ge	0.55	0.37
III–V	GaAs	0.067	0.45
III–V	InSb	0.013	0.6
II–VI	ZnO	0.29	1.21
II–VI	ZnSe	0.17	1.44

In semiconductors with low levels of doping, the electron concentration in the conduction band is in general given by

n_{\text{e}}=N_{\text{C}}\exp \left(-{\frac {E_{\text{C}}-E_{\text{F}}}{kT}}\right)

whereE_F is theFermi level,E_C is the minimum energy of the conduction band, andN_C is a concentration coefficient that depends on temperature. The above relationship forn_e can be shown to apply for any conduction band shape (including non-parabolic, asymmetric bands), provided the doping is weak (E_C −E_F ≫kT); this is a consequence ofFermi–Dirac statistics limiting towardsMaxwell–Boltzmann statistics.

The concept of effective mass is useful to model the temperature dependence ofN_C, thereby allowing the above relationship to be used over a range of temperatures. In an idealized three-dimensional material with a parabolic band, the concentration coefficient is given by

\quad N_{\text{C}}=2\left({\frac {2\pi m_{\text{e}}^{*}kT}{h^{2}}}\right)^{\frac {3}{2}}

In semiconductors with non-simple band structures, this relationship is used to define an effective mass, known as thedensity of states effective mass of electrons. The name "density of states effective mass" is used since the above expression forN_C is derived via thedensity of states for a parabolic band.

In practice, the effective mass extracted in this way is not quite constant in temperature (N_C does not exactly vary asT^3/2). In silicon, for example, this effective mass varies by a few percent between absolute zero and room temperature because the band structure itself slightly changes in shape. These band structure distortions are a result of changes in electron–phonon interaction energies, with the lattice'sthermal expansion playing a minor role.^[8]

Similarly, the number of holes in the valence band, and thedensity of states effective mass of holes are defined by:

n_{\text{h}}=N_{\text{V}}\exp \left(-{\frac {E_{\text{F}}-E_{\text{V}}}{kT}}\right),\quad N_{\text{V}}=2\left({\frac {2\pi m_{\text{h}}^{*}kT}{h^{2}}}\right)^{\frac {3}{2}}

whereE_V is the maximum energy of the valence band. Practically, this effective mass tends to vary greatly between absolute zero and room temperature in many materials (e.g., a factor of two in silicon), as there are multiple valence bands with distinct and significantly non-parabolic character, all peaking near the same energy.^[8]

Determination

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Experimental

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Traditionally effective masses were measured usingcyclotron resonance, a method in which microwave absorption of a semiconductor immersed in a magnetic field goes through a sharp peak when the microwave frequency equals the cyclotron frequency $\scriptstyle f_{c}\;=\;{\frac {eB}{2\pi m^{*}}}$ . In recent years effective masses have more commonly been determined through measurement ofband structures using techniques such asangle-resolved photoemission spectroscopy (ARPES) or, most directly, thede Haas–van Alphen effect. Effective masses can also be estimated using the coefficient γ of the linear term in the low-temperature electronicspecific heat at constant volume $\scriptstyle c_{v}$ . The specific heat depends on the effective mass through the density of states at theFermi level and as such is a measure of degeneracy as well as band curvature. Very large estimates of carrier mass from specific heat measurements have given rise to the concept ofheavy fermion materials. Since carriermobility depends on the ratio of carrier collision lifetime $\tau$ to effective mass, masses can in principle be determined from transport measurements, but this method is not practical since carrier collision probabilities are typically not known a priori. Theoptical Hall effect is an emerging technique for measuring the freecharge carrier density, effective mass and mobility parameters in semiconductors. The optical Hall effect measures the analogue of the quasi-static electric-field-induced electrical Hall effect at optical frequencies in conductive and complex layered materials. The optical Hall effect also permits characterization of the anisotropy (tensor character) of the effective mass and mobility parameters.^[12]^[13]

Theoretical

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A variety of theoretical methods includingdensity functional theory,k·p perturbation theory, andothers are used to supplement and support the various experimental measurements described in the previous section, including interpreting, fitting, and extrapolating these measurements. Some of these theoretical methods can also be used forab initio predictions of effective mass in the absence of any experimental data, for example to study materials that have not yet been created in the laboratory.

Significance

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The effective mass is used in transport calculations, such as transport of electrons under the influence of fields or carrier gradients, but it also is used to calculate the carrier density anddensity of states in semiconductors. These masses are related but, as explained in the previous sections, are not the same because the weightings of various directions and wavevectors are different. These differences are important, for example inthermoelectric materials, where high conductivity, generally associated with light mass, is desired at the same time as highSeebeck coefficient, generally associated with heavy mass. Methods for assessing the electronic structures of different materials in this context have been developed.^[14]

Certain groupIII–V compounds such asgallium arsenide (GaAs) andindium antimonide (InSb) have far smaller effective masses thantetrahedral group IV materials likesilicon andgermanium. In the simplestDrude picture of electronic transport, the maximum obtainable charge carrier velocity is inversely proportional to the effective mass: ${\textstyle {\vec {v}}\;=\;\left\Vert \mu \right\Vert \cdot {\vec {E}}}$ , where ${\textstyle \left\Vert \mu \right\Vert \;=\;{e\tau }/{\left\Vert m^{*}\right\Vert }}$ with ${\textstyle e}$ being theelectronic charge. The ultimate speed ofintegrated circuits depends on the carrier velocity, so the low effective mass is the fundamental reason that GaAs and its derivatives are used instead of Si in high-bandwidth applications likecellular telephony.^[15]

In April 2017, researchers at Washington State University claimed to have created a fluid with negative effective mass inside aBose–Einstein condensate, by engineering thedispersion relation.^[16]

Footnotes

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^Kittel,Introduction to Solid State Physics 8th edition, page 194–196
^Charles Kittel (1996).op. cit. Wiley. p. 202.ISBN 978-0-471-11181-8.
^^a ^bGreen, M. A. (1990). "Intrinsic concentration, effective densities of states, and effective mass in silicon".Journal of Applied Physics.67 (6):2944–2954.Bibcode:1990JAP....67.2944G.doi:10.1063/1.345414.
^"Effective mass in semiconductors". University of Colorado Electrical, Computer and Energy Engineering. Archived fromthe original on 2017-10-20. Retrieved2016-07-23.
^Callaway, Joseph (1976).Quantum Theory of the Solid State. Academic Press.
^Grecchi, Vincenzo; Sacchetti, Andrea (2005). "Bloch Oscillators: motion of wave-packets".arXiv:quant-ph/0506057.
^Chaitanya K. Ullal, Jian Shi, and Ravishankar Sundararamana American Journal of Physics 87, 291 (2019);https://doi.org/10.1119/1.5092453
^^a ^b ^cGreen, M. A. (1990). "Intrinsic concentration, effective densities of states, and effective mass in silicon".Journal of Applied Physics.67 (6):2944–2954.Bibcode:1990JAP....67.2944G.doi:10.1063/1.345414.
^S.Z. Sze,Physics of Semiconductor Devices,ISBN 0-471-05661-8.
^W.A. Harrison,Electronic Structure and the Properties of Solids,ISBN 0-486-66021-4.
^This site gives the effective masses of Silicon at different temperatures.
^M. Schubert,Infrared Ellipsometry on Semiconductor Layer Structures: Phonons, Plasmons and Polaritons,ISBN 3-540-23249-4.
^Schubert, M.; Kuehne, P.; Darakchieva, V.; Hofmann, T. (2016). "The optical Hall effect – model description: tutorial".Journal of the Optical Society of America A.33 (8):1553–68.Bibcode:2016JOSAA..33.1553S.doi:10.1364/JOSAA.33.001553.PMID 27505654.
^Xing, G. (2017). "Electronic fitness function for screening semiconductors as thermoelectric materials".Physical Review Materials.1 (6) 065405.arXiv:1708.04499.Bibcode:2017PhRvM...1f5405X.doi:10.1103/PhysRevMaterials.1.065405.S2CID 67790664.
^Silveirinha, M. R. G.; Engheta, N. (2012)."Transformation electronics: Tailoring the effective mass of electrons".Physical Review B.86 (16) 161104.arXiv:1205.6325.Bibcode:2012PhRvB..86p1104S.doi:10.1103/PhysRevB.86.161104.
^Khamehchi, K.A. (2017). "Negative-Mass Hydrodynamics in a Spin-Orbit–coupled Bose-Einstein Condensate".Physical Review Letters.118 (15) 155301.arXiv:1612.04055.Bibcode:2017PhRvL.118o5301K.doi:10.1103/PhysRevLett.118.155301.PMID 28452531.S2CID 44198065.

References

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Pastori Parravicini, G. (1975).Electronic States and Optical Transitions in Solids.Pergamon Press.ISBN 978-0-08-016846-3. This book contains an exhaustive but accessible discussion of the topic with extensive comparison between calculations and experiment.
S. Pekar, The method of effective electron mass in crystals, Zh. Eksp. Teor. Fiz.16, 933 (1946).