deep levels and dlts

Contents

1 Deep levels and DLTS 2

1.1 Deep states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 Transient response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1.1 The trap signature . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.1.2 Electron trap and hole trap . . . . . . . . . . . . . . . . . . . . . 51.1.1.3 Majority and minority carrier traps . . . . . . . . . . . . . . . . 5

1.2 Basis of transient depletion experiment for a majority carrier trap . . . . . . . . . 61.2.1 Rectangular transient charge model . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1.1 The trapped charge spatial distribution . . . . . . . . . . . . . . 81.2.1.2 Charge transient . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Deep Level Transient Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3.1 The two-point subtraction rate window principle . . . . . . . . . . . . . . 111.3.2 Di�culties a�ecting DLTS experiments . . . . . . . . . . . . . . . . . . . . 121.3.3 DLTS & quantum wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.3.1 A brief overview of quantum wells . . . . . . . . . . . . . . . . . 131.3.3.2 DLTS probe of a Schottky barrier with a single quantum well . . 14

1

Chapter 1

Deep levels and DLTS

1.1 Deep states

An electron and a hole in a semiconductor may recombine, if an electron transition from theconduction band to the valence band occurs, or be generated if the electronic transition is in theopposite direction. We may classify the Generation-Recombination processes in various ways:we speak of direct or band-band GR if the carrier transition occurs in a single process, of indirector trap assisted GR if the carrier transition is via energy levels within the bandgap. If the energyinvolved in the transition is given by/transfered to lattice phonons it is non-radiative or thermal,it is radiative or optical if the energy is balanced through a photon; �nally let's hint to Augermechanism, in which energy is losed or acquired by another carrier. Radiative and non-radiativerecombination processes compete between each other, but clearly in light-emitting devices onetries to favour the former process.

Figure 1.1:Emission and capture processes at a deep level:(1) hole emission to the valence band,(2) electron capture from the conduction band,(3) hole capture from the valence band,

(4) electron emission to the valence band.

On the left we show the four possible non-radiative processes via a generic trap level Et.Since the transition probability is exponen-tially related to the energy jump, it is a�ectedby the level position within the bandgap. Ob-viously if the level is next to the valence bandthe (1) and (3) processes are more probablethan the (2) and (4) ones; if the level is nextto the conduction band it happens vice versa;only when the level is close the midgap all theprocesses have the same probability. Becauseof generation of an electron-hole couple impliesthe occuring of processes (1) and (4), while itsrecombination the processes (2) and (3), thenwe speak of GR centre when the level is nearbymidgap situated. The most important indirect

GR mechanism is the non-radiative or Shockley-Read-Hall (SRH) one; there also exist indirectradiative mechanisms. We will not treat extensively the SRH theory; let's only say that, in adoped semiconductor, if the deep level is at or close to the midgap, in low level injection condition(: the excess carriers concentrations are� than the majority carrier concentration), set that σ isthe carrier capture section, < v >is the carrier average thermal velocity then the minority carrierlifetime is τ ∼ τmin0 ≡ 1

Nt<vmin>σmin, i.e. it depends on the deep level density Nt and minority

carrier characteristic quantities; the recombination rate is limited by the minority carrier capturerate. If σn < vn >= σp < vp > then the non-radiative lifetime is minimized if trap level is at orclose to the midgap energy: deep levels are e�ective recombination centers if they are near themiddle of the gap.

2

CHAPTER 1. DEEP LEVELS AND DLTS 3

Defects in the crystal structure are the most common cause for deep levels within the bandgap,and so for non-radiative recombination. These defects include unwanted foreign atoms, nativedefects, dislocations, and any complexes of defects, foreign atoms, or dislocations; in compoundsemiconductors, native defects include interstitials, vacancies, and antisite defects. All suchdefects form one or several energy levels within the forbidden gap of the semiconductor. Owingto the promotion of non-radiative processes, such deep levels or traps are called luminescencekillers.

1.1.1 Transient response

Figure 1.2:Electron and hole capture and emission processes for

an electron trap in a n-type semiconductor

Let's consider a deep state with energy Et andconcentration Nt for a n-type semiconductor;the four basic processes which determine itspopulation dynamic are the capture and emis-sion (from and to the relative band) for bothcarriers.

Let a deep centre be exposed to a �ux ofn < vn >electrons per unit area per unit time:if nt deep states out of Nt are occupied byelectrons at any instant, in a successive timeinterval ∆t the number of electrons capturedby the pt ≡ Nt−nt unoccupied states is ∆nt =

σn < vn > n(Nt − nt)∆t, for σn the electron capture cross section; so the electron capture

rate per unoccupied state is

cn ≡1

Nt − nt· ∆nt

∆t= σn < vn > n (1.1)

and an analogue expression holds for the hole capture rate per occupied state cp. Due torecombination of electrons and holes at the deep centre, the carrier capture rate is related tothe minority carrier lifetime: for a p-type semiconductor 1

τn≡ 1

n ·∆nt∆t , thus the non radiative

lifetime associated with electron capture at the deep state is τnnrad = σn < vn > (Nt − nt).The electron occupancy of the deep state is determined by the competing processes of emis-

sion and capture: electrons are emitted/holes are captured at the nt occupied states, holes areemitted/electrons are captured at the pt unoccupied states, and the net change rate of electronoccupancy is

dntdt

= (cn + ep)(Nt − nt)− (cp + en)nt (1.2)

for en,ep the electron, hole emission rates.Let's determine solutions to the above, so called, rate equation when Nt �the net doping

density Nd − Na, and thus the free carriers concentrations n, p are not in�uenced by the GRprocesses and can be considered constant. If we set the boundary condition nt(t = 0) ≡ nt(0),then its general solution is

nt(t) = nt(∞)− [nt(∞)− nt(0)] exp

(− tτ

)set that the steady state occupancy is

nt(∞)

Nt=

a

a+ b(1.3)

τ ≡ 1a+b is the time constant, de�ned the parameters a ≡ cn + ep, b ≡ en + cp. The initial

occupancynt(0)Nt

is usually set experimentally: when it di�ers from the equilibrium value nt(∞)Nt

,then the occupancy relaxes exponentially to this latter with rate constant a+b; two special casesare:


V for nt(0) = Nt, then nt(t) decays

nt(t) =Nt

a+ b

[a+ b exp

(− tτ

)](1.4)

V for nt(0) = 0, then nt(t) increases with time

nt(t) = nt(∞)

[1− exp

(− tτ

)](1.5)

We must be careful and distinguish between the rate of change of nt, and the emission and capturerates: for instance, when the only process is that of electron emission, then the instantaneousrate of emission of electrons from the trap is −dnt

dt = ennt(t), time dependent, while en is theemission rate �per trapped electron�, i.e. the probability per unit time that a single electron isemitted.

1.1.1.1 The trap signature

At thermal equilibrium, it holds:

V the steady state condition: dntdt = 0,

V the principle of detailed balance: the rates for a process and its inverse must be equaland balance in detail (so that it does not occur a net electron transfer from a band toanother): {

ennt = cn(Nt − nt)ep(Nt − nt) = cpnt

By combining the above requirements with the rate Equation (4.2), we �nd that the thermalequilibrium trap occupancy is

cncn + en

=n̂tNt

=ep

ep + cp(1.6)

on the other hand, at thermal equilibrium, the electron occupancy is ruled by Fermi-Diracstatistics, and so n̂t

Nt= 1

1+g exp(Et−EFkT

) , where g ≡ g0g1

is the degeneracy ratio, g0 being the

degeneracy of the trap level Et when it is electronless and g1 that when the trap level is occupiedby one electron. Thus we can relate the capture and emission rates for both carriers:

encn

= g exp

(Et − EFkT

)=cpep

(1.7)

Since g is of order of unity, then, roughly speaking, if EF > Et ⇒ cn > en, ep > cp, while ifEF < Et ⇒ cn < en, ep < cp ; if EF = Et ⇒ cn ∼ en, ep ∼ cp: strictly speaking en = cn whenEF = Et − kT ln g.

Because of Equation (4.1), the capture rate cn is dependent on the doping density; theanalogue is valid for cp too. Instead, the emission rates and the capture cross sections areintrinsic characteristics of the trap.

V We remember that for a non degenerate semiconductor holds the Boltzmann approxima-

tion: n = NC exp(−EC−EF

kT

), p = NV exp

(−EF−EV

kT

).

Unlike shallow hydrogenic levels, deep state centres may be occupied by more than one electron.Fynally, through equations (4.1) and (4.7), we derive

en = σn < vn > g exp

(−EC − Et

kT

)(1.8)


and by similar proceding

ep = σp < vp >1

gexp

(−Et − EV

kT

)(1.9)

We may determine the temperature dependence of the emission rates; in the case of electronemission:

V < vn >=√

3kTm∗ , for m

∗ the conduction band e�ective mass,

V NC = 2MC

(2πm∗kT

h2

)3/2, for MC the number of conduction band minima,

V if we assume σ(T ) = σ∞ exp(−∆Eσ

kT

),

then

en(T ) ≡ γT 2σna exp

(−EnakT

)(1.10)

A plot of ln enT 2 versus 1

T , called trap signature, is a straight line; experimental data for mosttraps �t the above curve over many orders of magnitude of en. Let's observe that the apparantactivation energy Ena ≡ EC−Et+∆Eσ is not the trap level, and that the apparent capturecross section σna is∝ to the extrapolated (T =∞) value of the capture cross section through thedegeneracy factor; both the identi�cations are valid only if EC −Et is temperature independent.We may catalogue the traps according the Ena and σna values.

1.1.1.2 Electron trap and hole trap

Because of Equations (4.8)&(4.9), we �nd that en = ep if

Et = E1 ≡ EFi +kT

2ln

(σp < vp >

σn < vn >

)(1.11)

We remember that the intrinsic Fermi level is EFi ≡ EC+EV2 + kT

2 ln(NVNC

), so the energy E1

is located near the midgap energy; we call the deep level an electron trap if it is en > ep, i.e.it is located in the upper part of the gap (Et > E1), while it is an hole trap if it is en < ep, i.e.it is located in the lower part of the gap.

1.1.1.3 Majority and minority carrier traps

Let's make a short resume of what just seen:

V the relation en T ep, which establish if the trap is electron-kind or hole-kind, is determinedby the position of the trap level Et respect to the energy E1(Equation (4.11)),

V the ratio ec depends on the relative positions of Et, EF (Equation (4.7)). But, if we assume

that Nt � the net doping density then EF is controlled by the doping, i.e. it is close to:

� the conduction band edge for a n-type semiconductor ⇒ EF > Et,

� the valence band edge for a p-type semiconductor⇒ EF < Et.

By combining the two points above, we �nd that:

V an electron trap (en � ep) in a n-type semiconductor (cn > en, ep > cp) is such thatcn > en � ep > cp ⇒ nt(t) relaxes with time constant τ = 1

cn+en, and the equilibrium

occupancy is nt(∞)Nt

= cncn+en

; if cn � en, then at the equilibrium n̂t = Nt.


V a hole trap (en � ep) in a p-type semiconductor (cn < en, ep < cp) is such that cn < en �ep < cp ⇒ nt(t) relaxes with time constant τ = 1

cp+ep, and the equilibrium occupancy is

nt(∞)Nt

=ep

ep+cp; if cp � ep, then at the equilibrium n̂t = 0⇒ p̂t = Nt.

Figure 1.3:Majority carrier traps

Let's generalize: we speak of majority carrier (electron for n-type material, hole for p-typeone) trap when emaj � emin and Et is close to the majority carrier band edge; the majoritycarrier equilibrium occupancy is equal to Nt if cmaj � emaj . By analogy, we speak of minority

carrier trap for:

V an electron trap in a p-type semiconductor; now only cn is negligible (because cn ∝ n)

⇒ nt(t) relaxes with time constant τ = 1ep+en+cp

, and the equilibrium occupancy is nt(∞)Nt

=ep

ep+en+cp⇒ the relaxation process involves the exchange of carriers with both bands, and

the equilibrium value p̂t = Nt is reached when both hole capture and electron emissiondominate over hole emission.

V an hole trap in a n-type semiconductor; nt(t) relaxes with time constant τ = 1cn+ep+en

,

and the equilibrium occupancy is nt(∞)Nt

=cn+ep

cn+ep+en⇒ the equilibrium value n̂t = Nt is

reached when electron capture and hole emission dominate over electron capture.

Figure 1.4:Minority carrier traps

Again Et is close to the appropriate (: minority carrier) band edge. The distinctive feature ofminority carrier traps is that the equilibrium occupancy is established through interaction withboth bands: it follows that the trap occupancy cannot be perturbed by its equilibrium value bysimply acting on the majority carrier population, as in the case of majority carrier traps.

1.2 Basis of transient depletion experiment for a majority carrier

trap

Let's examine the sequence in a transient depletion experiment for a Schottky barrier on a n-typesemiconductor, the latter containing a donor-like majority carrier trap with energy level Et.

With no bias applied, deep states in neutral material (: x > x0, set that x0 is the depletiondepth at zero bias applied) are occupied because cn > en.


Figure 1.5:Conduction band diagram at zero bias applied

By applying suddenly a reverse bias VR, within the enlarged depletion region the captureprocess of electrons from the conduction band to the trap level is turned o� ; the traps empty,and the emitted electrons are rapidly swept out of the depletion region by the associated �eld.

Figure 1.6:Conduction band diagram at reverse bias applied

We remeber we de�ned the transition distance λ (Equation 3.21) as the di�erence xn − x1,being x1 the point in which the trap level crosses the Fermi level; if the bias applied is so large thatλ � xn, the net space charge density during the emission process is ρ(x) = e [Nd +Nt − nt(t)].We also remember that we can express the depletion depth xn in function of the total potentialacross the junction (Equation (2.6) for Nd → Nd +Nt−nt(t)), then we substitute in the parallelplate expression (Equation (3.6)), �nally we obtain the time dependent capacitance

C ′(t) = A(εse

2

)1/2[Nd +Nt − nt(t)]1/2 V −1/2 ≡ C ′(∞)

[1− nt(t)

Nd +Nt

]1/2

set that C ′(∞) is the steady state (since nt(∞) = 0 within the depletion region) capacitance.In the dilute trap concentrantion limit Nt � Nd (⇒ nt � Nd) the square root term in the aboveequation can be expanded, so that the change in capacitance is

∆C ′(t) ≡ C ′(t)− C ′(∞) ∼ −nt(t)2Nd

C ′(∞)

Because the initial occupancy is nt(0) = Nt , then nt(t) follows a decaying law expressed byEquation (4.4). For a majority carrier trap cmaj > emaj are the only not-negligible capture andemission rates, so a + b = cmaj + emaj ; in our case (: electron trap) the only possible processbeing that of electron emission ⇒ a+ b = b = en. Finally nt(t) = Nt exp (−ent) and thus:

∆C ′(t)

C ′(∞)= − Nt

2Ndexp (−ent) (1.12)

If the reverse bias is removed then the empty traps are re�lled by electron capture; as alreadyunderlined, we use a hybrid depletion approximation, since we neglect the free carriers withinthe depletion region ar regards the space charge density, but we consider them as a trap re�llingmean.


The above equation, which states that the capacitance increases exponentially with time ascarriers are emitted from the trap, is the basis of the transient capacitance techniques and theDLTS ones: according to it, the time constant of the capacitance transient gives the thermalemission rate, and

∆C ′0 ≡ ∆C ′(0) = − Nt

2NdC ′(∞) (1.13)

1.2.1 Rectangular transient charge model

Let's brie�y resume the assumptions made for determining Equation (4.13):

V all the traps �lled prior to the starting of the emission process, a circumstance which cannotapply if the capture cross section is very small,

V the emission process occurs all over the depletion region, while the traps within x2 ≡ x0−λfrom the surface are never �lled (they are always above the Fermi level), and those whichbelong to the transition region xn − λ ≡ x1 < x < xn are never empty (they are alwaysbelow the Fermi level),

V we adopted the depletion approximation, instead of considering the distance over whichthe trap occupancy changes near xn.

In particular, the assumption that the emission occurs all over the depletion region does not a�ectseriously the relationship between the transient time constant and the emission rate, instead itdoes in�uence that between the transient amplitude and the deep states concentration: for trappro�ling experiments it is thus important to understanding the factors controlling the extent ofthe region over which the traps emit.

1.2.1.1 The trapped charge spatial distribution

This model, of the spatial distribution of trapped charge producing the transient response in amajority carrier trap experiment, is in many respects an extension to the depletion approxima-tion; from it we will determine a bettered version for the capacitance transient.

Figure 1.7:Energy band diagram for zero applied bias

Let's consider a Schottky barrier on n-typesemiconductor, the latter containing an elec-tron trap level Ent and a hole one Ept , at zeroapplied bias; let the depletion layer edge bex0. Since Equation (4.7), for x > x0 it iscn � en and so the traps are fully occupied.For x < x0, the the free carrier concentrationdecreases with distance from x0(since Equa-tion (3.7) for xn → x0), and so the capturerate (Equation (4.1)⇒ cn(x) ∝ n(x)); insteadthe emission rate does not depend on distance,

thus �nally from the equilibrium trap occupancy Equation (4.6) we desume that n̂t(x, t = 0))also decreases with distance from x0 in the same manner as n(x). Provided that LD � x0, thetransition in trap occupancy may be considered to occur abrutply at depth x2 ≡ x0− λ, i.e. thelocation where the trap level crosses the Fermi level at zero bias applied. From Equation (4.7)we determine that cn(x2) ∼ en, and so again for Equation (4.6) we get n̂t(x2) ∼ Nt

2 ; since bothcn(x), n̂t(x) vary fastly with distance, being the degeneracy factor g ∼ 1, it is usual to consider

cn(x2) = en,n̂t(x2)Nt

= 12 , and for uniformly doped material the two relationships are independent

by the bias applied.


Figure 1.8:Energy band diagram for reverse applied bias

Now let's apply a reverse biasVR to the Schottky barrier, the de-pletion region is enlarged up to xn;again for x < xn the free carrier con-centration decreases with distancefrom xn (Equation (3.7)), and alsothe capture rate. Mutatis mutan-dis, we can repeat all what said be-fore (zero bias case) for x2, now forx1 ≡ xn − λ. In particular, in theregion x < x1, where the trap levelis above the Fermi level, Equation(4.7) implies that cn < en, and so

the traps within relax by electron emission alone, while in the region x > x1 ⇒ cn > en, and sothe traps remain �lled; we regard the transition between the two regions occuring abruptly atx1. We remember that at zero bias applied the traps were �lled for x > x2(see Figure (4.7)). Inconclusion, not in the full width of the depletion region, but in the overlapping of the two aboveregions, x2 < x < x1, it holds that nt(0) = Nt, a = 0, b = en ⇒ nt(t) = Nt exp (−ent).

Figure 1.9:Trapped electron densities at zero and reverse bias applied; the dotted line shows the rectangular

approximation to nt(x, t)

1.2.1.2 Charge transient

Figure 1.10:

Drawing on the trapped chargespatial density (Figure (4.9)), foruniform donor and deep statedensities, we can get the netspace charge density ρ(x, t) =e {N+(x) + [Nt(x)− nt(x, t)]} in thesteady state under reverse bias (Fig-ure (4.10a)), in the steady state (i.e.at the end of the �lling process)with lower bias (Figure (4.10b)),during the emission process (Figure(4.10c)); we remember that N+(xn)is the uncompensated donor densityat xn, de�ned by Equation (3.22).The following steps have alreadybeen seen in past occasions; by suit-ably apply Equation (3.8), we derivea relationship between ρ(x, t) andthe resulting changes in xn(t), V (t) ;provided that Nd, Nt are constants,

we can determine the general equation relating changes per unit time in trap occupancy dnt into


the x2 < x < x1 region, to changes per unit time in depletion depth and bias:

dVRdt

=e

ε

[N+xn

dxndt

+ (Nt − nt(t))x1(t)dx1

dt−(x2

1(t)− x22

)2

dntdt

](1.14)

If the reverse bias is constant⇒ dVR = 0; we take dx1 = dxn (because of Nd is constant, andso it is the transition distance λ, see equation (3.21)) in equation (4.14), and we get a non-linearequation in xn(t). When the trap density Nt is small compared with the net background dopingdensity N+, then N+ ∼ Nd (under the hypothesis the sample is uncompensated), x1 becomese�ectively constant, so we �nally obtain the linear equation:

Ndxndxndt

=x2

1 − x22

2· dntdt⇒ 1

C ′dC ′

dt= − 1

2Nd· x

21 − x2

2

x2n

· dntdt

(1.15)

Now we specify that we determine xn by measurement of the high frequency capacitance, i.e.xn = εA

C′(∞)⇒ dxn = − εA

C′2dC′ ⇔ dxn

xn= −dC′

C′ . Finally, by integrating Equation (4.15), by taking

C ′ to be e�ectively constant (for Nt � Nd), we come to the following form of the capacitancetransient:

∆C ′(t)

C ′(∞)= −1

2· x

21 − x2

2

x2n

· nt(t)− nt(∞)

Nd≡ ∆C ′(0)

C ′(∞)exp

(− tτ

)(1.16)

We know that, for majority carrier-electron trap, the occupancy transient during the emissionprocess is governed by nt(t) = Nt exp (−ent) ⇒ τ ≡ 1

en, being the initial occupancy such that

nt(0) = Nt ⇒ the steady state one nt(∞) = 0, so

∆C ′(0)

C ′(∞)≡ −1

2· x

21 − x2

2

x2n

· Nt

Nd(1.17)

In conclusion, what we have just showed is that in the dilute limit (Nt � Nd) the capacitancetransient is exponential, with time constant 1

en, and aplitude ∝ to Nt. Equations (4.16) and

(4.17) are exact within the depletion approximation, withNt, Nd uniform, set that Vbi is constant.When the bias applied is so large that x1 ∼ xn ⇔ λ� xn, and set that x2 � xn, then equation(4.17) reduces to the simpler form (4.13).

1.3 Deep Level Transient Spectroscopy

This technique is used to observe thermal emission from majority carrier traps by means ofcapacitance transients.

Figure 1.11:Repetetive �lling and reverse bias pulse sequence, to which

corresponds the diode capacitance transient

The negative biased test diodeis pulsed (for a short time intervaltfill) with positive tension Vfill, con-sequently the depletion region is re-duced from widthW to thatW ′ andthe traps within ∆W ≡W −W ′ are�lled with majority carriers. The pa-rameter tfill a�ects the �lling level ofthe traps, while the relative magni-tude of VR +Vbi, Vfill the probed re-gion, because the whole (VR + Vbi ∼Vfill) or only a section (VR + Vbi >Vfill) of W is thus repleted; in theformer case the instrument is prob-ably overloaded with the high zero


bias capacitance. When the bias applied is restored to the negative value VR (for a time inter-val tR), the trapped carriers are emitted with rate en, and we consequently observe a transientcapacitance:

C ′(t) = C ′(∞) + ∆C ′0 exp

(− tτ

)(1.18)

This sequence is repeated periodically, with time period tP ≡ tfill + tR; during the DLTSmeasurement, the temperature is slowly increased.

1.3.1 The two-point subtraction rate window principle

Figure 1.12:Exempli�cation of the principle of the two-point subtraction

rate window: above the capacitance transient sampled at times

t1, t2, below the weighting function w(t)

At the core of the DLTS technique isthe principle of the two-point sub-traction rate window, whose im-plementation we illustrate throughthe use of the double box-car or stan-dard dual-gate signal averager sys-tem. Every capacitance signal, fol-lowing the periodical bias pulse, issampled by two gates set at times t1and t2 from the onset of the tran-sient (t = 0), which is equivalent tointegrate the product of the signaland the weight function w(t). The�rst gate t1 should be set so to avoidthe overload recovery part of the ca-pacitance signal. With a small gatewidth ∆t� τ , under the hypothesis

that transients are exponential, the box-car steady signal output is

S(τ) ∝ C ′(t1)− C ′(t2) = ∆C ′0

[exp

(− t1τ

)− exp

(− t2τ

)](1.19)

Figure 1.13:Double box-car signal output

As we can desume from the Equation (4.19), and observe from Figure (4.13), the signal


output is zero for τ � t1 or τ � t2, not negligible only if τ ∼ t1, t2. For t1 < t2, S(τ) < 0 formajority carrier traps, S(τ) > 0 for minority carrier ones.

Figure 1.14:Above deep level spectrum produced by a rate

window with reference time constant τref , below the

correspondent point on the trap Arrhenius plot

Set that τ ≡ 1en, the peak output is deter-

mined by the condition 0 = dS(T )dT = dS(en)

den·

dendT , which is reduced to dS(en)

den= 0 since it

holds dendT 6= 0 ∀T (because of Equation (4.10)),and thus it occurs when

τref ≡t2 − t1ln t2

t1

(1.20)

for τref the reference time constant of the ratewindow t2 − t1, which depends only by t2, t1;preset τref at the beginning of the DLST mea-surement, the peak temperature Tpeak is char-acteristic of the trap, and the emission rateen(Tpeak) is equal to 1

τref. By repeating the

temperature scan with di�erent values of τref ,sets of values (Tpeak, en(Tpeak)) are measured,from which we can generate an Arrhenius plot

of ln(T 2

en

)versus 1

T , and so determine the trap parameters Ena, σna.

Let's observe that Speak ≡ S(τref ) is function only of the ratio β ≡ t2t1, in fact

Speak ∝ ∆C ′0

[exp

(− lnβ

β − 1

)− exp

(−β lnβ

β − 1

)]thus changing τref such that β is constant then the peak height is independent from τref ; becauseof Speak ∝ ∆C ′0, through Equation (4.13) we can estimate the trap concentration; a more correctevalutation takes into account the transition distance λ (Equation (4.17)

Since the transient is not recorded directly, it is possible to work with time constants as shortas a few ms; the repetition time can also be short (∼ 10ms), so that en is e�ectively constant overseveral cycles and thus the output signal is averaged without prolonging the scan time providedthat τref values are not too long.

Alternative rate window methods are those employing the lock-in ampli�er or an exponentialcorrelator system.

1.3.2 Di�culties a�ecting DLTS experiments

The DLTS rate window technique draws on the hypothesis of exponential transient decaying,but this does not occur when:

V Nt is not so small compared with the net doping density N+,

V the thermal emission rate is dependent upon electric �eld, and then en becomes dependentupon position within the depletion region due to the spatial variation of the electric �eld,

V in a semiconductor alloy, the thermal emission rate at a particular trap site is dependentby the local crystal composition, and so its value is not uniquely determined through thesample.

All above mechanisms lead to broadening of DLTS spectrum; by recording spectra for di�erentpairs of t1, t2 values such that τref is constant, the peak should occur at the same temperaturein case of single exponential decay, while this does not happen with non-exponential one.


Also an interfacial layer can a�ect a DLTS measurement; due to a thin oxide which remainswhen the barrier metal is evaporated, or to other surface treatments during the diode fabrication,it can introduce peaks into the spectrum associated with interfacial states or modifying thecapacitance transient originated by traps in the semiconductor bulk.

1.3.3 DLTS & quantum wells

1.3.3.1 A brief overview of quantum wells

Figure 1.15:Example of N-i-P double heterostructure in

direct bias, which operates like the active

region of a light-emitting diode or a

semiconductor laser

Single or double heterostructures can create poten-tial wells in the conduction and/or valence bands,which can con�ne carriers in the direction orthog-onal to the well, leaving them free to move inthe two other directions (i.e. the plane paral-lel to the heterojunction); If the potential wellis very narrow the allowed energy levels of thecon�ned electrons and holes will be quantized:the resulting structure is called a quantum well(QW). Within the QW, assuming the e�ective massapproximation, the carrier wavefunction is solu-tion of a 3D time-independent Schrodinger equa-

tion[− ~2

2m∗∇2 + V (

−→R )]ψ(−→R ) = Eψ(

−→R ); in this

special case V (−→R ) = V (z), which suggests that

ψ(−→R ) ∝ exp (ik · r)u(z), set that k ≡ (kx, ky),

r ≡ (x, y), u(z) is the solution of the 1D equation[− ~2

2m∗∂2z − V (z)

]u(z) = εu(z), ε ≡ E− ~2

2m∗k2.

Let it be un(z) the eigenfunction corresponding to the energy eigenvalue εn for the given V (z),

then the associated solution of the 3D original problem is ψn,k(−→R ) ∝ exp (ik · r)un(z); in

En(k) = εn − ~22m∗k

2 we recognize, for any set n value, the dispersion relation for a free 2Delectron gas with the energy ground state shifted to εn: we call every parabola electric sub-

band. Let's note that for ε1 < E < ε2 the allowed states are only in the lowest subband, whilefor ε1 < E < ε2 there are allowed states in the subbands labelled by n = 1, n = 2; we alsonote that for the same E value, the energy partitioning is di�erent according to subbands: in thehigher subband the energy ε for the motion along the z axis is higher, so it is lower the transversekinetic energy. As energies increase there are more and more subbands from which to choose,and electrons with the same total energy have di�erent transverse wave vector k.

Finally, we remeber that the density of states of a 2D free electron gas is the step functionmπ~2 θ(E), so each subbands contributes a step of equal height, starting at εn; the total density

of states is a staircase-like function, with jumps at the energies εn: g2D(E) = m∗

π~2∑nθ (E − εn).

The density per unit area of electrons trapped in a conduction band QW is found as usual byintegrating, over all energies, the product of the density of states g2D(E) and the Fermi-Diracoccupation function fFD(E):

n2D =m∗

π~2

∑n

ˆ ∞εn

fFD(E)dE =m∗

π~2· kT ·

∑n

ln

[1 + exp

(EF − εnkT

)](1.21)

An analogue reasoning is applicable to determine the density per unit area of holes trapped in avalence band QW.


Figure 1.16:The electron wave functions (a), the dispersion relation (b) and the density of states (c) in the case of

in�nitely deep square well of width 10nm

1.3.3.2 DLTS probe of a Schottky barrier with a single quantum well

DLTS is also applicable in studying the electron emission from a quantum well. The localizedcon�ned state in the quantum well, corresponding to the energy level ε1, may be regarded asa �giant trap�, the thermal activation energy of the emission process interpreted as the energyseparation Eb − ε1 between the con�ned state and the top of the barrier.

Let's suppose to use the DLTS technique upon a Schottky barrier with a single quantum wellof width Lz, located at xw below the surface such that it lies within the reverse bias depletionregion (xw < xn), but outside the zero bias one (xo < xw).

When the bias is reduced to zero the carriers into the vicinity are entrapped by the quantumwell. The electron capturing bends the nearby energy bands because of the localized charge,untill the �lling process is interrupted since electrons are repelled. The carriers density nearthe well is so reduced, consequently the capture rate starts falling, reaching a steady state valuebecause of the compensating thermal emission process of electrons out of the well.

Figure 1.17:Conduction band diagram immediately after reducing to zero the bias applied, and when the equilibrium

is reached; d is the width of the depletion regions around the QW


By applying a reverse bias, the depletion region is emptied by the free carriers; the electronsearly entrapped into the well are emitted by thermal process and then swept out, producing acapacitance transient.

Figure 1.18:Charges entrapped by the well, thermally emitted at reverse bias applied

We note that the electrons �rst emitted when the reverse bias is just applied are those whosequantized energies are greater than ε1, which lay under the Fermi energy level at zero bias;because of the energy levels are di�erent, the emission rates are also di�erent and the initialdecay is not exponential, but it is very fast and probably not detectable experimentally. Whatwe observe is the emission of those electrons whose energy is the well ground state ε1, which allhave similar emission rates, and so the slower decay is now exponential.

By application of the principle of detailed balance, the electron capture process, from thebarrier states (corresponding to energies above the band edge Eb) to the well states (whoseenergies stay below Eb), must be compensated by the electron emission process, from the wellstates to the barrier states. The fraction of occupied and unoccupied states in the well andbarrier is determined by the Fermi energy level. The total capture rate of electrons into the wellper unit volume is

Rcapture ≡ cwpw = σw < vn > nbpw

set that nb is the electron density in the barriers , pw the well unoccupied states density. Weassume that only the lower subband is occupied, so pw ∼ mw

π~2Lz

´ Ebε1

(1− fFD(E)) dE ∼E−EF�kT

mwπ~2Lz · (Eb − E1), the factor 1

Lzarising from the fact that the density is per unit volume, set

that mw is the carrier e�ective mass; since Eb − EF � kT , to the electron density in thebarriers does not apply any more the Fermi-Dirac statistics, but the Boltzmann one, so nb ∼Nb exp

(−Eb−EF

kT

), set that Nb = 2

(2πmbkT

h2

)3/2is the states density per unit volume at the

band edge Eb. The total emission rate of electrons from the well per unit volume is

Remission ≡ ewnw

set that nw is the well occupied states density; by remembering equation (4.21), and holding the

approximation ln (1 + x) ∼x�1

x because of ε1 − EF � kT , then nw ∼ mwπ~2Lz · kT · exp

(EF−ε1kT

).

From Rcapture = Remission, we determine

ew = σw < vn > 2 ·(

2mbπkT

h2

)3/2

·(Eb − ε1kT

)exp

(−Eb − ε1

kT

)(1.22)

and assuming σw constant (remembering that < vn >∝ T 1/2) we derive that a plot of ln ewT 2

versus 1T should have slope Eb−ε1

k .


Finally, by assuming that Lz � xw, so that the volume in which the traps are observable issuch that x1 − x2 = Lz and x1 + x2 ∼ 2xw, then we derive the capacitance transient equation

∆C ′(t)

C ′(∞)=xwLzx2n

· nw0

Nd· exp (−ewt) (1.23)

set that nw0 is the initial carriers density into the well. The initial steady state carrier densityis de�ned by the charge neutrality at large distance requirement, i.e. for distances greaterthan wqw if wqw is the width of the depletion region around the QW. Let it be Nd the donordensity in the barriers, then, by application of the depletion approximation, the charge perunit area inside the QW is equal to the charge per unit area in the depleted regions aroundthe QW⇒ nw0Lz = 2wqwNd, set that we get w2

qw = 2εse2Nd

(Eb − EC) from Equation (3.5), and

Eb−EC = [∆EC − (ε1 − εw)]− (EF − ε1)− (EC − EF ) (εw being the QW energy ground state)follows by observing the steady state part of Figure (4.17).

There are important features which make di�erence between a QW and an usual deep state:

V the initial density of trapped electrons is ruled by the doping density of the material aroundthe QW , and not by the trap density;

V as the bands bend with time, the density of free carriers near the QW reduces;

V the carriers are not emitted by each trap from the trap level Et, instead from a continuumof states; a constant emission rate ewis well de�ned only after as many electrons are emittedthat all the remaining trapped electrons can be considered having energy ε1. It is not clearif variation in the band bending around the QW during emission changes the e�ectivecapture cross section σw, making ew time dependent.

deep levels and dlts

Documents

transient

capture cross

conduction

cn ep

majority carrier

energy band

net doping

minority carrier