Quinn J.J., Yi K.-S. Solid State Physics: Principles and Modern Applications

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200 7 Semiconductors

physics from the late 60’s till the present time. Some basic problems that 435

were investigated include: 436

• Transport along the layer 437

surface electron density n

and relaxation time τ as a function of the gate 438

voltage V

, cyclotron resonance, localization, magnetoconductivity, and 439

Hall eﬀect. 440

• Transport perpendicular to the layer 441

optical absorption; Raman scattering, coupling to optical phonons, intra 442

and intersubband collective modes. 443

• Many-body eﬀects on subband structure and on eﬀective mass and eﬀective 444

g-value. 445

7.6.1 Superlattices 446

By novel growth techniques like molecular beam epitaxy (MBE) novel struc- 447

tures can be grown almost one atomic layer at a time. The requirements for 448

such growth are 449

1. The lattice constants of the two materials must be rather close. Otherwise, 450

large strains lead to many crystal imperfections. 451

2. The materials must form appropriate bonds with one another. 452

One very popular example is the GaAs–AlAs system shown in Fig. 7.16. A 453

single layer of GaAs in an AlAs host would be called a quantum well.A454

periodic array of such layers is called a superlattice. It can be thought of as a 455

new material with a supercell in real space that goes from one GaAs to AlAs 456

interface to the next GaAs to AlAs interface. 457

7.6.2 Quantum Wells 458

If a quantum well is narrow, it will lead to quantized motion and subbands 459

just as the MOS surface inversion layer did (see, for example, Fig. 7.17). For 460

Fig. 7.16. Schematic diagram of the GaAs–AlAs superlattice system

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7.6 Surface Space Charge Layers 201

(v)

(c)

Fig. 7.17. Schematic diagram of the subbands formation in a quantum well of the

GaAs–AlAs structure

FREE CARRIERS

Fig. 7.18. Schematic diagram of a quantum well in a modulation doped

GaAs/Ga

1−x

As quantum well

the subbands in the conduction band we have 461

(c)

(k)=ε

(c)

¯h

∗



+ k



. (7.89)

462

The band oﬀsets are diﬃcult to predict theoretically, but they can be mea- 463

sured. 464

7.6.3 Modulation Doping 465

The highest mobility materials have been obtained by growing modulation 466

doped GaAs/Ga

1−x

As quantum wells. In these materials, the donors are 467

located in the GaAlAs barriers, but no closer than several hundred Angstroms 468

to the quantum well. The bands look as shown in Fig. 7.18. A typical sample 469

structure would look like GaAlAs with N

donors/cm

 pure GaAlAs of 470

20 nm thick // GaAs of 10 nm thick // pure GaAlAs of 20 nm thick // GaAlAs 471

with N

donors/cm

(see, e.g., Fig. 7.19). Because the ionized impurities are 472

rather far away from the quantum well electrons, ionized impurity scattering 473

is minimized and very high mobilities can be attained. 474

7.6.4 Minibands 475

When the periodic array of quantum wells in a superlattice has very wide 476

barriers, the subband levels in each quantum well are essentially unchanged 477

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202 7 Semiconductors

nm nm

Fig. 7.19. Schematic layered structure of a typical modulation doped GaAs/

1−x

As quantum well system

(see, for example, Fig. 7.20). However, a new periodicity has been introduced, 478

so we have a quantum number k

that has to do with the eigenvalues of the 479

translation operator. 480

(z)=e

(z) (7.90)

This looks just like the problem of atomic energy levels that give rise to

481

band structure when the atoms are brought together to form a crystal. For 482

very large values of the barrier width, no tunneling occurs, and the minibands 483

are essentially ﬂat as is shown in Fig. 7.21. The supercell in real space extends 484

from z =0toz = a. The ﬁrst Brillouin zone in k-space extends −

≤ k

≤

. 485

The minibands ε

) are ﬂat if the barriers are so wide that no tunneling 486

from one quantum well to its neighbor is possible. When the barriers are nar- 487

rower and tunneling can take place, the ﬂat bands become k

-dependent. One 488

can easily show that in tight binding calculation one would get bands with 489

sinusoidal shape as shown in Fig. 7.22 below. Of course, the same band struc- 490

ture would result from taking free electrons moving in a periodic potential 491

Fig. 7.20. Schematic subband alignments in a superlattice of supercell width a

Fig. 7.21. Schematic miniband alignment in a superlattice of very large barrier

width

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7.7 Electrons in a Magnetic Field 203

Fig. 7.22. Miniband structure in a superlattice of very narrow barrier width

V (r)=



2π

. (7.91)

492

During the past 20 years, there has been an enormous explosion in the study 493

(both experimental and theoretical) of optical and transport properties of 494

quantum wells, superlattices, quantum wires, and quantum dots. One of the 495

most exciting developments was the observation by Klaus von Klitzing of 496

the quantum Hall eﬀect in a 2DEG in a strong magnetic ﬁeld. Before we 497

give a very brief description of this work, we must discuss the eigenstates of 498

free electrons in two dimensions in the presence of a perpendicular magnetic 499

ﬁeld. 500

7.7 Electrons in a Magnetic Field 501

Consider a 2DEG with n

electrons per unit area. In the presence of a dc 502

magnetic ﬁeld B applied normal to the plane of the 2DEG, the Hamiltonian 503

of a single electron is written by 504

H =



p +



. (7.92)

Here, p =(p

)andA(r) is the vector potential whose curl gives 505

B =(0, 0,B), i.e., B = ∇×A. There are a number of diﬀerent possible 506

choices for A(r) (diﬀerent gauges) that give a constant magnetic ﬁeld in the 507

z-direction. For example, the Landau gauge chooses A =(0,Bx,0) giving us 508



ˆx

∂

∂x



× (ˆyBx)=Bˆz. Another common choice is A =

(−y, x,0); this is 509

called the symmetric gauge. Diﬀerent gauges have diﬀerent eigenstates, but 510

the observables have to be the same. 511

Let us look at the Schr¨odinger equation in the Landau gauge. (H − E) 512

Ψ = 0 can be rewritten as 513







− E



Ψ(r)=0. (7.93)

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204 7 Semiconductors

Because H is independent of the coordinate y,wecanwrite 514

Ψ(x, y, z)=e

iky

ϕ(x). (7.94)

Substituting this into the Schr¨odinger equation gives

515



mω



x +

¯hk

mω



− E



ϕ(x)=0. (7.95)

516

Here, of course, ω

is the cyclotron frequency. If we deﬁne ˜x = x +

¯hk

mω

, 517

∂

∂x

∂

∂ ˜x

and the Schr¨odinger equation is just the simple harmonic oscillator 518

equation. Its solutions are as follows: 519

=¯hω



n +



,n=0, 1, 2,...

(x, y, z)=e

iky



x +

¯hk

mω



. (7.96)

520

The energy is independent of k, so the density of states (per unit length) is a 521

series of δ-functions, as is shown in Fig. 7.23. 522

g(ε) ∝





ε − ¯hω



n +



. (7.97)

The constant of proportionality for a ﬁnite size sample of area L

mω

2π¯h

, 523

so that the total number of states per Landau level is 524

= L



mω

2π¯h



hc/e

. (7.98)

525

For a sample of area L

,eachLandaulevel can accommodate N

electrons (we 526

have omitted spin) and N

is the magnetic ﬂux through the sample divided 527

Fig. 7.23. Density of states for electrons in a dc magnetic ﬁeld

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7.7 Electrons in a Magnetic Field 205

by the quantum of magnetic ﬂux

. We note that the degeneracy of each 528

Landau level can also be rewritten by N

2πλ

in terms of the magnetic 529

length λ =

¯hc

. 530

7.7.1 Quantum Hall Eﬀect 531

If we make contacts to the 2DEG, and send a current I in the x-direction, 532

then we expect 533

∝

and σ

∝

. (7.99)

Here, V

is the applied voltage in the direction of I and V

is the Hall voltage. 534

In the simple classical (Drude model) picture, we know that 535

= σ

1+(ω

τ)

and σ

= −σ

= −

τσ

1+(ω

τ)

, (7.100)

where σ

. In the limit as τ →∞we have σ

→∞, σ

→ 0, and 536

→−

. 537

In the absence of scattering, g(ε)isaseriesofδ-functions. With scattering, 538

the δ-functions are broadened as shown in Fig. 7.24. We know that, when one 539

Landau level is completely ﬁlled and the one above it is completely empty, 540

there can be no current, because to modify the distribution function f

(ε) 541

would require promotion of electrons to the next Landau level. There is a 542

gap for doing this, and at T = 0 there will be no current. If we plot σ

versus 543

≡ ν,theﬁlling factor



= N



we expect σ

to go to zero at 544

any integer values of ν as shown in Fig. 7.25. 545

Our understanding of the integral quantum Hall eﬀect is based on the 546

idea that within the broadened δ-functions representing the density of states, 547

we have both extended states and localized states as shown in Fig. 7.26. The 548

quantum Hall eﬀect was very important because it led to 549

1. A resistance standard ρ =

. 550

2. Better understanding of Anderson localization. 551

3. Discovery of the fractional quantum Hall eﬀect. 552

Fig. 7.24. Density of states for electrons in a dc magnetic ﬁeld in the presence of

scattering

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206 7 Semiconductors

Fig. 7.25. Conductivities as a function of the Landau level ﬁlling factor.

(a) Longitudinal conductivity σ

,(b) Hall conductivity σ

Fig. 7.26. Scattering eﬀects on the density of states and conductivity components

in an integral quantum Hall state

7.8 Amorphous Semiconductors 553

Except for introducing donors and acceptors in semiconductors, we have essen- 554

tially restricted our consideration to ideal, defect-free inﬁnite crystals. There 555

are two important aspects of order that crystals display. The ﬁrst is short 556

range order. This has to do with the regular arrangement of atoms in the 557

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7.8 Amorphous Semiconductors 207

vicinity of any particular atom. This short range order determines the local 558

bonding and the crystalline ﬁelds acting on a given atom. The second aspect 559

is long range order. This is responsible for the translational and rotational 560

invariance that we used in discussing Bloch functions and band structure. 561

It allowed us to use Bloch’s theorem and to deﬁne the Bloch wave vector k 562

within the ﬁrst Brillouin zone. 563

In real crystals, there are always 564

• Surface eﬀects associated with the ﬁnite size of the sample. 565

• Elementary excitations (dynamic perturbations like phonons, magnons, 566

etc.) 567

• Imperfections and defects (static disorder). 568

For an ordered solid, one can start with the perfect crystal as the zeroth 569

approximation and then treat static and dynamic perturbations by per- 570

turbation theory. For a disordered solid this type of approximation is not 571

meaningful. 572

7.8.1 Types of Disorder 573

We can classify disorder by considering some simple examples in two dimen- 574

sions that we can represent on a plane. 575

Perfect Crystalline Order Atoms in perfect crystalline array (see Fig. 7.27a). 576

Compositional Disorder Impurity atoms (e.g. in an alloy) are randomly dis- 577

tributed among crystalline lattice sites (See Fig. 7.27(b).) 578

Positional Disorder Some separations and some bond angles are not perfect 579

(See Fig. 7.27(c).) 580

Topological Disorder Fig. 7.27d shows some topological disorder. 581

Because we cannot use translational invariance and energy band concepts, 582

it is diﬃcult to evaluate the eigenstates of a disordered system. What has 583

been found is that in disordered systems, some of the electronic states can 584

be extended states and some can be localized states. An extended state 585

is one in which, if |Ψ(0)|

is ﬁnite, |Ψ(r)|

remains ﬁnite for r very large. 586

A localized state is one in which |Ψ(r)|

falls oﬀ very quickly as r becomes large 587

(usually exponentially). There is an enormous literature on disorder and local- 588

ization (starting with a classic, but diﬃcult, paper by P.W. Anderson

in the 589

1950s). 590

7.8.2 Anderson Model 591

The Anderson model described a system of atomic levels at diﬀerent sites n 592

and allowed for hopping from site n to m. The Hamiltonian is written by 593

H =



†

+ T





†

(7.101)

P.W. Anderson, Phys. Rev. 109, 1492 (1958).

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208 7 Semiconductors

Fig. 7.27. Various types of disorder. (a) Atoms in perfect crystalline array.

(b) Impurity atoms are randomly distributed among crystalline lattice sites.

rings, but some ﬁve and sixfold rings leaving dangling bonds represent topological

disorder

Fig. 7.28. Basic assumption of energy level distribution on diﬀerent sites in the

Anderson model

This is just the description of band structure in terms of an atomic level 594

ε on site n where the periodic potential gives rise to the hopping term.In 595

tight binding approximation, we would restrict T , the hopping term to near- 596

est neighbor hops, and that is what the prime on





in the second-term 597

means. 598

In Anderson model, it was assumed that ε

the energy on site n was not 599

a constant, but that it was randomly distributed over a range w (see, for 600

example, Fig. 7.28). Anderson showed that if the parameter

,whereB is 601

the band width (caused by and proportional to T )satisﬁed

≥ 5, the state 602

at E = 0 (the center of the band) is localized, while if

≤ 5 it is extended. 603

7.8.3 Impurity Bands 604

Impurity levels in semiconductors form Anderson-like systems. In these sys- 605

tems, the energy E is independent of n; it is equal to the donor energy ε

. 606

However, the hopping term T is randomly distributed between certain limits, 607

since the impurities are randomly distributed. Sometimes (when two impuri- 608

ties are close together) it is easy to hop and T is large. Sometimes, when they 609

are far apart, T is small. 610

7.8.4 Density of States 611

Although the eigenvalues of the Anderson Hamiltonian can not be calculated 612

in a useful way, it is possible to make use of the idea of density of states. In 613

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7.8 Amorphous Semiconductors 209

Fig. 7.29. Density of states of an ordinary crystal and that of a disordered material

Fig. 7.29, we sketch the density of states of an ordinary crystal and then the 614

density of states of a disordered material. In the latter, the tails on the density 615

of states usually contain localized states, while the states in the center of the 616

band are extended. The energies E



and E

are called mobility edges.They 617

separate localized and extended states. When E

is in the localized states, 618

there is no conduction at T = 0. The ﬁeld of amorphous materials, Anderson 619

localization, and mobility edges are of current research interest, but we do not 620

have time to go into greater detail. 621