semiconductor heterostructures quantum wellsszczytko/ldsn/3_ldsn_2015_qwells.pdf · 2015. 11....
TRANSCRIPT
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Infinite square quantum well
2015-11-27 2
𝜓 𝑥, 𝑡 =2
𝐿sin 𝑘𝑛𝑥 𝑒−𝑖𝜔𝑡
Inside the quantum well:
𝑘𝑛 =𝑛𝜋
𝐿
𝜀𝑛 =ℏ2𝑘𝑛
2
2𝑚=ℏ2𝑛2𝜋2
2𝑚𝐿2
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Infinite square quantum well
2015-11-27 3
𝜓 𝑥, 𝑡 =2
𝐿sin 𝑘𝑛𝑥 𝑒−𝑖𝜔𝑡
Inside the quantum well:
𝜀𝑛 = 𝐸𝑐 +ℏ2𝑘𝑛
2
2𝑚= 𝐸𝑐 +
ℏ2𝑛2𝜋2
2𝑚0𝑚∗𝐿2
𝑘𝑛 =𝑛𝜋
𝐿
𝐸𝑐
𝜀1 = 𝐸𝑔 +ℏ2𝜋2
2𝑚0𝑚∗𝐿2
𝜀2 = 𝐸𝑐 +2ℏ2𝜋2
𝑚0𝑚∗𝐿2
𝜀3 = 𝐸𝑐 +9ℏ2𝜋2
2𝑚0𝑚∗𝐿2
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Infinite square quantum well
2015-11-27 4
𝜓 𝑥, 𝑡 =2
𝐿sin 𝑘𝑛𝑥 𝑒−𝑖𝜔𝑡
Inside the quantum well:
𝜀𝑛 = 𝐸𝑣 +ℏ2𝑘𝑛
2
2𝑚= 𝐸𝑣 −
ℏ2𝑛2𝜋2
2𝑚0𝑚∗𝐿2
𝑘𝑛 =𝑛𝜋
𝐿
𝐸𝑣𝜀1 = 𝐸𝑣 −
ℏ2𝜋2
2𝑚0𝑚∗𝐿2
𝜀2 = 𝐸𝑣 −2ℏ2𝜋2
𝑚0𝑚∗𝐿2
𝜀3 = 𝐸𝑣 −9ℏ2𝜋2
2𝑚0𝑚∗𝐿2
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Bandgap engineering
2015-11-27 5
How can we change the heterostructure band structure:• selecting a material (eg., GaAs / AlAs)• controlling the composition• controlling the stress
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2015-11-27 6
Figure 5. Schematic diagrams depicting the evolution of the conduction band structure in the transverse direction: (a) double heterostructure, (b) separate confinementheterostructure (SCH), (c) graded-index separateconfinement heterostructure (GRIN-SCH), (d) single quantum well heterostructure (QWH), and (e) multiplequantum well (MQW).
The development of the semiconductor laser diode after the first demonstration in 1962 J J Coleman Semicond. Sci. Technol. 27 (2012) 090207
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Vapor-phase (VPE, CVD)
wzrost z fazy gazowej dzięki reakcjom chemicznym prekursorów na powierzchni,
często dzielony ze względu na źródłowe gazy na wodorkową VPE i
metalorganiczną VPE (MOCVD); prędkości wzrostu >10 -20 nm/min.
Liquid-phase (LPE)
wzrost z fazy ciekłej na podłożu w temperaturach niższych od temperatury
topnienia hodowanego materiału. Półprzewodnik jest rozpuszczony w cieczy
innego materiału, wzrost w warunkach bliskich równowagi roztworu i depozycji;
prędkości wzrostu 0.1 to 1 μm/min.
Molecular-beam (MBE)
Materiał źródłowy podgrzewany w komórkach produkuje strumień cząsteczek. W
wysokiej próżni (10-8 Pa) cząsteczki docierają do podłoża i osadzają się na nim;
prędkości wzrostu < 1 monowarstwa/s (1 μm/h).
2015-11-27 7
Jak się robi heterostruktury?
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Bandgap engineering
2015-11-27 8
W jaki sposób możemy zmieniać strukturę pasmową heterostruktury:• wybierając materiał• kontrolując skład• kontrolując naprężenie
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Semiconductor heterostructures
2015-11-27 9
Investigation of high antimony-content gallium arsenic nitride-gallium arsenic antimonide heterostructures for long wavelength application
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Bandgap engineering
2015-11-27
10
Valence band offset (Anderson’s rule)
Valence band offset:
(powinowactwo)
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Bandgap engineering
2015-11-27 11
Valence band offset (Anderson’s rule)
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Bandgap engineering
2015-11-27 12
Valence band offset
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Bandgap engineering
2015-11-27 13
How can we change the heterostructure band structure:• selecting a material (eg., GaAs / AlAs)• controlling the composition• controlling the stress
Vegard’s law:the empirical heuristic that the lattice parameter of a solid solution of two constituents is approximately equal to a rule of mixtures of the two constituents' (A and B) lattice parameters at the same temperature:
𝑎 = 𝑎𝐴 1 − 𝑥 + 𝑎𝐵𝑥
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Bandgap engineering
2015-11-27 14
Vegard’s law:Relationship to band gaps of „binarycompound”:
𝐸 = 𝐸𝐴 1 − 𝑥 + 𝐸𝐵𝑥 − 𝑏𝑥(1 − 𝑥)
b – so-called „bowing” (curvature) of the energy gap
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How can we change the heterostructure band structure:• selecting a material (eg., GaAs / AlAs)• controlling the composition• controlling the stress
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Bandgap engineering
2015-11-27 15
Valence band offsetHow can we change the heterostructure band structure:• selecting a material (eg., GaAs / AlAs)• controlling the composition• controlling the stress
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Semiconductor heterostructures
2015-11-27 16
http://beta.globalspec.com/reference/45139/203279/chapter-iii-optical-properties
Thin Solid Films 433 (2003) 22–26
Quaternary compounds
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Semiconductor heterostructures
2015-11-27 17
Quinternary barriers push room-temperature operation of GaSb-based type-I lasers further into mid-infrared
http://beta.globalspec.com/reference/45139/203279/chapter-iii-optical-properties
Quinternary compounds
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Comments on the conduction band
2015-11-27 18
Depending on the semiconductor the bottom of the conduction band may be constructd from a differnent valleys – the same heterostructure may be a well in a one band (eg. Γ) and a barrierin another (eg. 𝑋).
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Heterostruktury półprzewodnikowe
2015-11-27 19
http://www.springermaterials.com/docs/pdf/10681719_656.html
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Bandgap engineering
2015-11-27 20
How can we change the heterostructure band structure:• selecting a material (eg., GaAs / AlAs)• controlling the composition• controlling the stress
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Bandgap engineering
2015-11-27 21
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How can we change the heterostructure band structure:• selecting a material (eg., GaAs / AlAs)• controlling the composition• controlling the stress
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Bandgap engineering
2015-11-27 22
How can we change the heterostructure band structure:• selecting a material (eg., GaAs / AlAs)• controlling the composition• controlling the stress
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Bandgap engineering
2015-11-27 23
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How can we change the heterostructure band structure:• selecting a material (eg., GaAs / AlAs)• controlling the composition• controlling the stress
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Bandgap engineering
2015-11-27 24
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How can we change the heterostructure band structure:• selecting a material (eg., GaAs / AlAs)• controlling the composition• controlling the stress
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Bandgap engineering
2015-11-27 25
Pro
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How can we change the heterostructure band structure:• selecting a material (eg., GaAs / AlAs)• controlling the composition• controlling the stress
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Bandgap engineering
2015-11-27 26
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How can we change the heterostructure band structure:• selecting a material (eg., GaAs / AlAs)• controlling the composition• controlling the stress
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Bandgap engineering
2015-11-27 27
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How can we change the heterostructure band structure:• selecting a material (eg., GaAs / AlAs)• controlling the composition• controlling the stress uniaxial biaxial
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Bandgap engineering
2015-11-27 28
Effects of biaxial strain: decrease of the degeneracy of the valence band and change of the effective masses in the GaxIn1-xAs / GaxIn1-xAsyyP1-y material system.
S.L. Chuang, Phys. Rev. B 43, p. 9649 (1991). 9, 10
The presence of the well changes the symmetry of the crystal (eg. quantum wells in the direction of [001] corresponds to an uniaxial pressure applied perpendicular to the layer). You have to solve the 𝒌𝒑 perturbation theory (Chemla 1983):
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Potencjał harmoniczny
2015-11-27 29
−ℏ2
2𝑚
𝑑2
𝑑𝑧2+1
2𝑚𝜔0
2𝑧2 𝜓 𝑧 = 𝐸𝜓 𝑧 𝜀 =𝐸
ℏ𝜔0𝜉 =
𝑚𝜔0
ℏ𝑧
𝑑
𝑑𝜉− 𝜉
𝑑
𝑑𝜉+ 𝜉 A e
𝜉2
2 = −2𝜀0 + 1 A e𝜉2
2 = 0 ⇒ −2𝜀0 + 1 = 0 ⇒ 𝜀0 =1
2
𝜀𝑛 = 𝑛 +1
2
𝐸𝑛 = ℏ𝜔0 𝑛 +1
2
𝜓𝑛 𝑧 = 𝐴𝑛𝐻𝑛𝑚𝜔0
ℏ𝑧 exp −
𝑚𝜔0
2ℏ𝑧2
𝐻𝑛 - wielomiany Hermite’a
𝐴𝑛 = 2𝑛𝑛!𝜋ℏ
𝑚𝜔
−1/2
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Comments on the valence band
2015-11-27 30
𝐸3/2 𝑘 = 𝐴𝑘2 ± 𝐵2𝑘4 + 𝐶2 𝑘𝑥2𝑘𝑦
2 + 𝑘𝑦2𝑘𝑧
2 + 𝑘𝑧2𝑘𝑥
2
From 𝒌𝒑 perturbation theory :010
00100ത1
0ത10
0ത1ത1
01ത1
0ത11
011
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Comments on the valence band
2015-11-27 31
𝐸3/2 𝑘 = 𝐴𝑘2 ± 𝐵2𝑘4 + 𝐶2 𝑘𝑥2𝑘𝑦
2 + 𝑘𝑦2𝑘𝑧
2 + 𝑘𝑧2𝑘𝑥
2 1/2
X.X. Wei,Y. Yu, Y. Bi; Int. J. of Mechanical Sciences Vol. 50, 1499–1509 (2008),
Fig. 7. The 3D constant-energy surfaces of the heavy hole (HH) band, the light hole (LH) band and the split-off (SO) band and their corresponding conductivity masses for unstrained silicon.
From 𝒌𝒑 perturbation theory :
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Comments on the valence band
2015-11-27 32
X.X. Wei,Y. Yu, Y. Bi; Int. J. of Mechanical Sciences Vol. 50, 1499–1509 (2008),
Fig. 8. The 3D constant-energy surfaces of the heavy hole (HH) band, the light hole (LH) band and the split-off (SO) band and the corresponding conductivity masses for silicon under the uniaxial compression P ¼ 500kN. .
𝐸3/2 𝑘 = 𝐴𝑘2 ± 𝐵2𝑘4 + 𝐶2 𝑘𝑥2𝑘𝑦
2 + 𝑘𝑦2𝑘𝑧
2 + 𝑘𝑧2𝑘𝑥
2 1/2
From 𝒌𝒑 perturbation theory :
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Comments on the valence band
2015-11-27 33
𝐸3/2 𝑘 = 𝐴𝑘2 ± 𝐵2𝑘4 + 𝐶2 𝑘𝑥2𝑘𝑦
2 + 𝑘𝑦2𝑘𝑧
2 + 𝑘𝑧2𝑘𝑥
2 1/2
𝛾1 = −2𝑚
ℏ2𝐴 𝛾2 = −
𝑚
ℏ2𝐵 𝛾3 =
𝑚
ℏ21
3𝐶2 + 𝐵2
Luttinger dimensionless parameters:
𝐸𝑙,ℎ 𝒌 = −ℏ2
2𝑚𝛾1𝑘
2 ± 4𝛾22𝑘4 + 12 𝛾3
2 − 𝛾22 𝑘𝑥
2𝑘𝑦2 + 𝑘𝑦
2𝑘𝑧2 + 𝑘𝑧
2𝑘𝑥2 + heavy holes
− light holes
Expanding around 𝑘 = 0
𝐸ℎℎ 𝒌 = −ℏ2
2𝑚𝛾1 + 2𝛾2 𝒌⊥
2 + 𝛾1 + 2𝛾2 𝑘𝑧2
𝐸𝑙ℎ 𝒌 = −ℏ2
2𝑚𝛾1 − 2𝛾2 𝒌⊥
2 + 𝛾1 − 2𝛾2 𝑘𝑧2
From 𝒌𝒑 perturbation theory :
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Comments on the valence band
2015-11-27 34
𝐸ℎℎ 𝒌 = −ℏ2
2𝑚𝛾1 + 𝛾2 𝒌⊥
2 + 𝛾1 − 2𝛾2 𝑘𝑧2
𝐸𝑙ℎ 𝒌 = −ℏ2
2𝑚𝛾1 − 𝛾2 𝒌⊥
2 + 𝛾1 + 2𝛾2 𝑘𝑧2
It follows that the heavy holes have a "light" mass 𝑚
𝛾1+𝛾2in the plane 𝑥 − 𝑦 , while light holes
have heavy mass 𝑚
𝛾1−𝛾2
The presence of the well changes the symmetry of the crystal (eg. quantum wells in the direction of [001] corresponds to an uniaxial pressure applied perpendicular to the layer). You have to solve the 𝒌𝒑 perturbation theory (Chemla 1983):
𝐸
𝐽𝑧 = ±3
2
𝐽𝑧 = ±1
2
𝐸1ℎℎ
𝐸1𝑙ℎ
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Comments on the valence band
2015-11-27 35
𝐸ℎℎ 𝒌 = −ℏ2
2𝑚𝛾1 + 𝛾2 𝒌⊥
2 + 𝛾1 − 2𝛾2 𝑘𝑧2
𝐸𝑙ℎ 𝒌 = −ℏ2
2𝑚𝛾1 − 𝛾2 𝒌⊥
2 + 𝛾1 + 2𝛾2 𝑘𝑧2
𝐸
𝐽𝑧 = ±3
2
𝐽𝑧 = ±1
2
𝐸1ℎℎ
𝐸1𝑙ℎ
𝐸
𝐸1ℎℎ
𝐸1𝑙ℎ
ℏ2
2𝑚𝛾1 − 𝛾2
ℏ2
2𝑚𝛾1 + 𝛾2
𝑘𝑥
The presence of the well changes the symmetry of the crystal (eg. quantum wells in the direction of [001] corresponds to an uniaxial pressure applied perpendicular to the layer). You have to solve the 𝒌𝒑 perturbation theory (Chemla 1983):
![Page 36: Semiconductor heterostructures quantum wellsszczytko/LDSN/3_LDSN_2015_QWells.pdf · 2015. 11. 27. · Bandgap engineering 2015-11-27 28 Effects of biaxial strain: decrease of the](https://reader036.vdocument.in/reader036/viewer/2022071403/60f7823ade6f2b209b6a0aa3/html5/thumbnails/36.jpg)
Comments on the valence band
2015-11-27 36
𝐸ℎℎ 𝒌 = −ℏ2
2𝑚𝛾1 + 𝛾2 𝒌⊥
2 + 𝛾1 − 2𝛾2 𝑘𝑧2
𝐸𝑙ℎ 𝒌 = −ℏ2
2𝑚𝛾1 − 𝛾2 𝒌⊥
2 + 𝛾1 + 2𝛾2 𝑘𝑧2
𝐸
𝐽𝑧 = ±3
2
𝐽𝑧 = ±1
2
𝐸1ℎℎ
𝐸1𝑙ℎ
𝐸
𝐸1ℎℎ
𝐸1𝑙ℎ
ℏ2
2𝑚𝛾1 − 𝛾2
ℏ2
2𝑚𝛾1 + 𝛾2
𝑘𝑥
PoLDS 385
The presence of the well changes the symmetry of the crystal (eg. quantum wells in the direction of [001] corresponds to an uniaxial pressure applied perpendicular to the layer). You have to solve the 𝒌𝒑 perturbation theory (Chemla 1983):
![Page 37: Semiconductor heterostructures quantum wellsszczytko/LDSN/3_LDSN_2015_QWells.pdf · 2015. 11. 27. · Bandgap engineering 2015-11-27 28 Effects of biaxial strain: decrease of the](https://reader036.vdocument.in/reader036/viewer/2022071403/60f7823ade6f2b209b6a0aa3/html5/thumbnails/37.jpg)
2015-11-27 37
Energy bands
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Energy bands
2015-11-27 38
Example:
D. Wasik.
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Quantum well
2015-11-27 39
http://www.nextnano.de/nextnano3/tutorial/1Dtutorial_AlGaInP_onGaAs.htm
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Finite potential well – square well
2015-11-27 40
Simply example has been already calculated
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Finite potential well – square well
2015-11-27 41
Heterostructures can have different effective masses in different regions:
−ℏ2
2𝑚0𝑚∗
𝑑2
𝑑𝑧2𝜓 𝑧 + V0 z 𝜓 𝑧 = 𝜀𝜓 𝑧
ቤ1
𝑚𝐵
𝑑𝜓
𝑑𝑧z=
a2
= ቤ1
𝑚𝑊
𝑑𝜓
𝑑𝑧z=
a2
Matching conditions at the interface must be modified (equation has to conserve current𝐼𝑊 = 𝐼𝐵 thus 𝑣𝑊 = 𝑣𝐵):
It turns that the conversion 𝑚∗ → 𝑚 𝑧 is not a good solution of the problem (equation ceases to be Hermitean). It has to be done diferently, eg.:
−ℏ2
2𝑚0
𝑑
𝑑𝑧
1
𝑚 𝑧
𝑑
𝑑𝑧𝜓 𝑧 + V0 z 𝜓 𝑧 = 𝜀𝜓 𝑧
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Finite potential well – square well
2015-11-27 42
Inside the well:
𝜓 𝑧, 𝑡 = 𝐶 ቊcos 𝑘𝑛𝑧
sin 𝑘𝑛𝑧𝑒−𝑖𝜔𝑛𝑡
−𝑎
2< 𝑧 <
𝑎
2
The barrier:
𝜓 𝑧 = 𝐷 exp(±𝜅𝑛𝑧)
ℏ2𝜅2
2𝑚 𝑚𝐵= 𝐸𝐵 − 𝐸𝑛 = 𝐵
𝑘𝑛 =1
ℏ2𝑚𝑚𝑊 𝐸𝑛 − 𝐸𝑊
𝜅𝑛 =1
ℏ2𝑚𝑚𝐵 𝐸𝐵 − 𝐸𝑛
−ℏ2
2𝑚0𝑚𝑊
𝑑2
𝑑𝑧2𝜓 𝑧 = 𝐸𝑛 − 𝐸𝑊 𝜓 𝑧
ቤ1
𝑚𝐵
𝑑𝜓
𝑑𝑧z=
a2
= ቤ1
𝑚𝑊
𝑑𝜓
𝑑𝑧z=
a2
Matching conditions:
𝐶𝑘
𝑚𝑊
−sin 𝑘𝑛𝑎
2
cos 𝑘𝑛𝑎
2
= −𝐷𝜅
𝑚𝐵exp 𝑘𝑛
𝑎
2
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Finite potential well – square well
2015-11-27 43
We calculate:
𝐶𝑘
𝑚𝑊
−sin 𝑘𝑛𝑎
2
cos 𝑘𝑛𝑎
2
= −𝐷𝜅𝑛𝑚𝐵
exp 𝜅𝑛𝑎
2
ቚ𝜓𝐵 𝑧z=
a2
= ቚ𝜓𝑊 𝑧z=
a2
ቤ1
𝑚𝐵
𝑑𝜓
𝑑𝑧z=
a2
= ቤ1
𝑚𝑊
𝑑𝜓
𝑑𝑧z=
a2
tan 𝑘𝑛𝑎
2
cot 𝑘𝑛𝑎
2
=𝑚𝑊𝜅𝑛𝑚𝐵𝑘𝑛
=𝑚𝑊
𝑚𝐵
2𝑚0𝑚𝑊𝑉0
ℏ2𝑘𝑛2 − 1
Substituting , and (depends only on 𝑚𝑊)𝜃 =𝑘𝑛𝑎
2𝜃02 =
𝑚0𝑚𝑊𝑉0𝑎2
2ℏ2
ቊtan 𝜃−cot 𝜃
=𝑚𝑊
𝑚𝐵
𝜃02
𝜃2− 1
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Finite potential well – square well
2015-11-27 44
THE SAME mass in the well and in the barrier:
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Finite potential well – square well
2015-11-27 45
THE DIFFERENT mass in the well and in the barrier:
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Finite potential well – square well
2015-11-27 46
THE DIFFERENT mass in the well and in the barrier:
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Finite potential well – square well
2015-11-27 47
THE DIFFERENT mass in the well and in the barrier:
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Harmonic potential
2015-11-27 48
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Harmonic potential
2015-11-27 49
𝜀𝑛 = 𝑛 −1
2ℏ𝜔0𝑉 𝑧 =
1
2𝐾𝑧2 =
1
2𝑚𝜔0
2𝑧2−ℏ2
2𝑚
𝑑2
𝑑𝑧2+ 𝑉(𝑧) 𝜓 𝑧 = 𝜀𝜓 𝑧
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Harmonic potential
2015-11-27 50
𝜙𝑛+1 𝑧 =1
2𝑛𝑛!
𝑚𝜔0
ℏ𝜋
1/4
exp −𝑚𝜔0𝑧
2
2ℏ𝐻𝑛
𝑚𝜔0
ℏ𝑧
𝐻0 𝑡 = 1𝐻1 𝑡 = 𝑡𝐻2 𝑡 = 4𝑡2 − 2𝐻3 𝑡 = 8𝑡3 − 12𝑡
𝜙1 𝑧 2 =𝑚𝜔0
𝜋ℏexp −
𝑚𝜔0𝑧2
ℏ
The ground state is a Gauss package – does the packet diffuse?
Wavefunction
Hermite polynomials
𝜀𝑛 = 𝑛 −1
2ℏ𝜔0−
ℏ2
2𝑚
𝑑2
𝑑𝑧2+1
2𝑚𝜔0
2𝑧2 𝜓 𝑧 = 𝜀𝜓 𝑧
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Harmonic potential
2015-11-27 51
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Triangular well
2015-11-27 52
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Triangular well
2015-11-27 53
−ℏ2
2𝑚
𝑑2
𝑑𝑧2+ 𝑒𝐹𝑧 𝜓 𝑧 = 𝜀𝜓 𝑧
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Triangular well
2015-11-27 54
−ℏ2
2𝑚
𝑑2
𝑑𝑧2+ 𝑒𝐹𝑧 𝜓 𝑧 = 𝜀𝜓 𝑧
𝑑2
𝑑𝑧2𝜓 𝑧 =
2𝑚
ℏ2𝑒𝐹𝑧 − 𝜀 𝜓 𝑧
Transformation:
The equation reduces to Stokes or Airy equation:
𝑑2
𝑑𝑧2𝑓 𝑧 = 𝑧𝑓 𝑧
Its two independent solutions the Airy functions 𝐴𝑖(𝑧) and 𝐵𝑖(𝑧). The solutions of the equation are the zeros of a function 𝐴𝑖(𝑧) (after some rearrangements).
Substituting: 𝑧0 =ℏ2
2𝑚𝑒𝐹
1/3
𝜀0 = 𝑒𝐹𝑧0 =𝑒𝐹ℏ 2
2𝑚
1/3
, ҧ𝑧 =𝑧
𝑧0, ҧ𝜀 =
𝜀
𝜀0
𝑑2
𝑑𝑧2𝜓 𝑧 =
2𝑚
ℏ2𝑒𝐹𝑧 − 𝜀 𝜓 𝑧
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Triangular well
2015-11-27 55
𝑉 𝑧 = 𝑒𝐹𝑧 𝜀𝑛 = 𝑐𝑛𝑒𝐹𝑧 2
2𝑚
1/3
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WKB approximation
11/27/2015 56
WKB approximation (Wentzel – Krammers – Brillouin) – for slowly changing potential
It is also known as the LG or Liouville–Green method or JWKB and WKBJ, where the "J" stands for Jeffreys or phase integral method or semi-classical approximation.
−ℏ2
2𝑚
𝑑2
𝑑𝑥2+ 𝑉 𝑥 𝜓 𝑥 = 𝐸𝜓 𝑥
What is slowly varing potential? It is for sure 𝑉 𝑥 = 𝑉0 = 𝑐𝑜𝑛𝑠𝑡. The solution for this potential is a plane wave 𝜓 𝑥 = 𝑒𝑖𝑘𝑥 - phase of the wavefunction 𝑘 𝑥 = 𝑘 = 𝑐𝑜𝑛𝑠𝑡 is constant in the
whole space 𝑘2 =2𝑚
ℏ𝐸 − 𝑉0
Let's define 𝑘2 𝑥 =2𝑚
ℏ𝐸 − 𝑉(𝑥) - we want the phase 𝑘 𝑥 to be slowly varying in space, i.e.
𝑑𝑘
𝑑𝑥≪ 𝑘2
(such condition).
We are looking for the solution 𝜓 𝑥 = 𝑒𝑖𝜒 𝑥 where 𝜒 𝑥 is the phase of the wavefunction.
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WKB approximation
11/27/2015 57
−ℏ2
2𝑚
𝑑2
𝑑𝑥2+ 𝑉 𝑥 𝜓 𝑥 = 𝐸𝜓 𝑥
Let’s define 𝑘2 𝑥 =2𝑚
ℏ𝐸 − 𝑉(𝑥) - slowly varing in real space 𝑘 𝑥 .
We are looking for the solutuion 𝜓 𝑥 = 𝑒𝑖𝜒 𝑥 where 𝜒 𝑥 its the phase of the wavefunction. Inserting into Schrodinger equation:
𝜒′ 𝑥 2 − 𝑖𝜒′′ 𝑥 =2𝑚
ℏ𝐸 − 𝑉 𝑥 ≡ 𝑘2 𝑥 - this is rigorous.
Zero-order WKB approximation assumes 𝜒′ 𝑥 2 ≫ 𝜒′′ 𝑥 czyli 𝜒′′ 𝑥 ≈ 0
𝜒′ 𝑥 2 = 𝑘2 𝑥 czyli𝜒 𝑥 = ±න
𝑥
𝑘 𝑥′ 𝑑𝑥′
Usually we expand more𝜒′ 𝑥 2 = 𝑘2 𝑥 + 𝑖𝜒′′ 𝑥 = 𝑘2 𝑥 + 𝑖 𝜒′ 𝑥 ′ ≈ 𝑘2 𝑥 ± 𝑖 𝑘 𝑥 ′
Thus:
𝜒′ 𝑥 ≈ ±𝑘 𝑥 1 +𝑖𝑘′ 𝑥
𝑘2 𝑥≈± 𝑘 𝑥 +
𝑖𝑘′ 𝑥
2𝑘 𝑥
WKB approximation (Wentzel – Krammers – Brillouin) – for slowly changing potential
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WKB approximation
2015-11-27 58
Typically, the WKB method continues into
𝜒′ 𝑥 2 = 𝑘2 𝑥 + 𝑖𝜒′′ 𝑥 = 𝑘2 𝑥 + 𝑖 𝜒′ 𝑥 ′ ≈ 𝑘2 𝑥 ± 𝑖 𝑘 𝑥 ′
Thus:
𝜒′ 𝑥 ≈ ±𝑘 𝑥 1 +𝑖𝑘′ 𝑥
𝑘2 𝑥≈± 𝑘 𝑥 +
𝑖𝑘′ 𝑥
2𝑘 𝑥
therefore:
𝜒 𝑥 = ±න𝑥
𝑘 𝑥′ 𝑑𝑥′ +𝑖
2ln 𝑘 𝑥
We get:
𝜓(𝑥) ≈1
𝑘 𝑥exp ±𝑖 න
𝑥
𝑘 𝑥′ 𝑑𝑥′
The term 1/ 𝑘 𝑥 - the density of probability of fast-moving particles is small for large 𝑘 - OK!
WKB approximation (Wentzel – Krammers – Brillouin) – for slowly changing potential
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WKB approximation
2015-11-27 59
𝐸
In the case of turning points (ie. the edges of the barriers well) potential changes rapidly compared with the wavelength 𝑘 𝑥 - WKB approximation is not valid (a rigorous approach avoids it by moving into the complex plane). Another way is to note that the potential is linear for a small region around the turning point Δ𝑥𝐿 - and the Airy functions are the exact solutions. Then the solution must match on both regions near 𝑥𝐿.The problem sounds complicated but fortunately the results are simple (we got additional phase).
energia cząstki
𝑉(𝑥)
𝑥
𝑥𝐿
WKB approximation (Wentzel – Krammers – Brillouin) – for slowly varying potential
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WKB approximation
2015-11-27 60
𝐸𝜓 𝑥
carrier energy
𝑉(𝑥)
𝑥
𝑥𝐿
𝜓 𝑥 ~2
𝑘 𝑥cos න
𝑥𝐿
𝑥
𝑘 𝑥′ 𝑑𝑥′ −𝜋
4, 𝑥 ≫ 𝑥𝐿
𝜓 𝑥 ~1
𝜅 𝑥exp −න
𝑥𝐿
𝑥
𝜅 𝑥′ 𝑑𝑥′ , 𝑥 ≪ 𝑥𝐿
WKB approximation (Wentzel – Krammers – Brillouin) – for slowly varying potential
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WKB approximation
11/27/2015 61
𝜓 𝑥 ~2
𝑘 𝑥cos න
𝑥𝐿
𝑥
𝑘 𝑥′ 𝑑𝑥′ −𝜋
4, 𝑥 ≫ 𝑥𝐿
Examples:1. „Hard” (infinitely steep) wall – the wave function goes to zero at the boundaries, so an exact number of half-wavelengths must fit between them
න𝑥𝐿
𝑥𝑅
𝑘 𝑥′ 𝑑𝑥′ = 𝑛𝜋
2. „Soft wall” – an allowed state must obey the matching conditions (𝑥 ≫ 𝑥𝐿, above) and at 𝑥𝐿 it
has additional phase −𝜋
4and similarly at 𝑥𝑅 - next −
𝜋
4. Altogether:
න𝑥𝐿
𝑥𝑅
𝑘 𝑥′ 𝑑𝑥′ = 𝑛 −1
2𝜋
𝑘𝑛 =𝑛𝜋
𝐿For : 𝑘 𝑥 = 𝑐𝑜𝑛𝑠𝑡 we have
WKB approximation (Wentzel – Krammers – Brillouin) – for slowly varying potential
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WKB approximation
2015-11-27 62
𝜓 𝑥 ~2
𝑘 𝑥cos න
𝑥𝐿
𝑥
𝑘 𝑥′ 𝑑𝑥′ −𝜋
4, 𝑥 ≫ 𝑥𝐿
3. Triangular well of „hard” and „soft” well:
Eg. Triangular well: 𝑉 𝑥 = 𝑒𝐹𝑥 wtedy 𝑘𝑛 𝑥 =1
ℏ2𝑚 𝐸𝑛 − 𝑉 𝑥 =
1
ℏ2𝑚 𝐸𝑛 − 𝑒𝐹𝑥
න𝑥𝐿
𝑥𝑅
𝑘 𝑥′ 𝑑𝑥′ = 𝑛 −1
4𝜋
න𝑥𝐿
𝑥𝑅
𝑘𝑛 𝑥′ 𝑑𝑥′ = න0
𝐸𝑛/𝑒𝐹 1
ℏ2𝑚 𝐸𝑛 − 𝑒𝐹𝑥′ 𝑑𝑥′ =
2𝑚𝐸𝑛ℏ2
1/2 𝐸𝑛𝑒𝐹
න0
1
1 − 𝑠 𝑑𝑠 =
2𝑚𝐸𝑛ℏ2
1/2 𝐸𝑛𝑒𝐹
න0
1
1 − 𝑠 𝑑𝑠 =2
3
2𝑚𝐸𝑛ℏ2
1/2 𝐸𝑛𝑒𝐹
𝑠 =𝑥
𝑥𝑅
𝐸𝑛 =3
2𝜋 𝑛 −
1
4
2/3𝑒𝐹ℏ 2
2𝑚
1/3
WKB approximation (Wentzel – Krammers – Brillouin) – for slowly varying potential
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Triangular well
2015-11-27 63
𝐸𝑛 =3
2𝜋 𝑛 −
1
4
2/3𝑒𝐹ℏ 2
2𝑚
1/3
WKB approximation (Wentzel – Krammers – Brillouin) – for slowly varying potential
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Triangular well
2015-11-27 64
𝐸𝑛 =3
2𝜋 𝑛 −
1
4
2/3𝑒𝐹ℏ 2
2𝑚
1/3
http://www.phys.unsw.edu.au/QED/research/2D_scattering.htm
WKB approximation (Wentzel – Krammers – Brillouin) – for slowly varying potential
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WKB approximation
2015-11-27 65
𝑘𝑛 𝑥 =1
ℏ2𝑚𝑉 𝑥 =
1
ℏ2𝑚𝑉𝑏 1 −
𝑥
𝑑
2
𝑉 𝑥 = 𝑉𝑏 1 −𝑥
𝑑
2
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Low dimensional structures
2015-11-27 66
Full Hamiltonian in our universe has three spatial dimensions 𝑥, 𝑦, 𝑧, 𝑡 = 𝑅, 𝑡
−ℏ2
2𝑚𝛻2 + 𝑉 𝑅 𝜓 𝑅 = 𝐸𝜓 𝑅
For 𝑉 𝑅 = 𝑉(𝑧) we obtain:
−ℏ2
2𝑚
𝜕2
𝜕𝑥2+
𝜕2
𝜕𝑦2+
𝜕2
𝜕𝑧2+ 𝑉 𝑧 𝜓 𝑥, 𝑦, 𝑧 = 𝐸𝜓 𝑥, 𝑦, 𝑧
𝜓 𝑥, 𝑦, 𝑧 = exp 𝑖𝑘𝑥𝑥 exp 𝑖𝑘𝑦𝑦 𝑢(𝑧)
Along directions 𝑥 and 𝑦 mamy ruch swobodny:
We can show (on the blackboard!), that final eigenenergies of the potential 𝑉 𝑧 are:
131
𝜀 = 𝐸 −ℏ2𝑘𝑥
2
2𝑚−ℏ2𝑘𝑦
2
2𝑚−ℏ2
2𝑚
𝑑2
𝑑𝑧2+ 𝑉 𝑧 𝑢 𝑧 = 𝜀𝑢 𝑧
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Low dimensional structures
2015-11-27 67
𝐸𝑛 𝑘𝑥, 𝑘𝑦 = 𝜀𝑛 +ℏ2𝑘𝑥
2
2𝑚+ℏ2𝑘𝑦
2
2𝑚
𝜓𝑘𝑥,𝑘𝑦,𝑛 𝑥, 𝑦, 𝑧 = exp 𝑖𝑘𝑥𝑥 exp 𝑖𝑘𝑦𝑦 𝑢𝑛 𝑧 = 𝜓𝒌,𝑛 𝒓, 𝑧 = exp 𝑖𝒌 ∙ 𝒓 𝑢𝑛 𝑧
𝐸𝑛 𝒌 = 𝜀𝑛 +ℏ2𝒌2
2𝑚
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Low dimensional structures
2015-11-27 68
𝐸𝑛 𝑘𝑥, 𝑘𝑦 = 𝜀𝑛 +ℏ2𝑘𝑥
2
2𝑚+ℏ2𝑘𝑦
2
2𝑚
𝜓𝑘𝑥,𝑘𝑦,𝑛 𝑥, 𝑦, 𝑧 = exp 𝑖𝑘𝑥𝑥 exp 𝑖𝑘𝑦𝑦 𝑢𝑛 𝑧 = 𝜓𝒌,𝑛 𝒓, 𝑧 = exp 𝑖𝒌 ∙ 𝒓 𝑢𝑛 𝑧
𝐸𝑛 𝒌 = 𝜀𝑛 +ℏ2𝒌2
2𝑚
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Finite potential well – square well
2015-11-27 69
THE DIFFERENT mass in the well and in the barrier:
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Low dimensional structures
2015-11-27 70
𝜓 𝑅 = 𝜓𝒌,𝑛 𝒓, 𝑧 = exp 𝑖𝒌 ∙ 𝒓 𝑢𝑛 𝑧
−ℏ2
2𝑚0 𝑚𝑊,𝐵𝛻2 + 𝑉 𝑅 𝜓 𝑅 = 𝐸𝜓 𝑅
Effective mass in the barrier 𝑚𝐵 and in the well 𝑚𝑊
−ℏ2
2𝑚0 𝑚𝑊𝛻2 + 𝐸𝑊 𝜓 𝑅 = 𝐸𝜓 𝑅
For separated wave functions:
−ℏ2
2𝑚0 𝑚𝐵𝛻2 + 𝐸𝐵 𝜓 𝑅 = 𝐸𝜓 𝑅
We got (on the blackboard!):
−ℏ2
2𝑚0 𝑚𝑊
𝑑2
𝑑𝑧2+
ℏ2𝒌2
2𝑚0 𝑚𝑊+ 𝐸𝑊 𝑢𝑛 𝑧 = 𝜀𝑢𝑛 𝑧
−ℏ2
2𝑚0 𝑚𝐵
𝑑2
𝑑𝑧2+
ℏ2𝒌2
2𝑚0 𝑚𝐵+ 𝐸𝐵 𝑢𝑛 𝑧 = 𝜀𝑢𝑛 𝑧
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Low dimensional structures
2015-11-27 71
The particle moves in the well which potentiald depends on 𝒌, in fact 𝑘 = 𝒌
−ℏ2
2𝑚0 𝑚𝑊
𝑑2
𝑑𝑧2+
ℏ2𝒌2
2𝑚0 𝑚𝑊+ 𝐸𝑊 𝑢𝑛 𝑧 = 𝜀𝑢𝑛 𝑧
−ℏ2
2𝑚0 𝑚𝐵
𝑑2
𝑑𝑧2+
ℏ2𝒌2
2𝑚0 𝑚𝐵+ 𝐸𝐵 𝑢𝑛 𝑧 = 𝜀𝑢𝑛 𝑧
𝑉0 𝑘 = 𝐸𝐵 − 𝐸𝑊 +ℏ2𝑘2
2𝑚0
1
𝑚𝐵−
1
𝑚𝑊
The particle gains partially the effective mass of the barrier:
𝐸𝑛 𝑘 = 𝜀𝑛(𝑘) +ℏ2𝑘2
2𝑚0𝑚𝑊≈ 𝜀𝑛(𝑘 = 0) +
ℏ2𝑘2
2𝑚0𝑚𝑒𝑓𝑓
𝑚𝑒𝑓𝑓 ≈ 𝑚𝑊𝑃𝑊 +𝑚𝐵𝑃𝐵
prawdopodobieństwo znalezienia cząstki
Np. w strukturze GaAs-AlGaAs 𝑚𝐵 >𝑚𝑊 więc studnia robi się „płytsza”
energy of the bond state depends on 𝑘
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Low dimensional structures
2015-11-27 72
−ℏ2
2𝑚0 𝑚𝑊
𝑑2
𝑑𝑧2+
ℏ2𝒌2
2𝑚0 𝑚𝑊+ 𝐸𝑊 𝑢𝑛 𝑧 = 𝜀𝑢𝑛 𝑧
−ℏ2
2𝑚0 𝑚𝐵
𝑑2
𝑑𝑧2+
ℏ2𝒌2
2𝑚0 𝑚𝐵+ 𝐸𝐵 𝑢𝑛 𝑧 = 𝜀𝑢𝑛 𝑧
𝑉0 𝑘 = 𝐸𝐵 − 𝐸𝑊 +ℏ2𝑘2
2𝑚0
1
𝑚𝐵−
1
𝑚𝑊
Np. w strukturze GaAs-AlGaAs 𝑚𝐵 >𝑚𝑊 więc studnia robi się „płytsza”
The particle moves in the well which potentiald depends on 𝒌, in fact 𝑘 = 𝒌