OK MSc thesis: chapters 1-3 Body

8/14/2019 OK MSc thesis: chapters 1-3 Body

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Figure 1-1. Equivalent circuit of an ideal solar cell.

This ideal model states that a solar cell is merely a p-n junction. As in any semiconductor, a

photon of energy above the band gap may excite an electron-hole pair. If the e-h pair

reaches the depletion layer before it recombines then the depletion layers electric field

separates the pair. This procedure is represented in the equivalent circuit by the current

source.

In addition to the photo-generated current the ideal model also accounts for parasitic

current named dark current ( I D) which corresponds to the junction voltage. The dark current

I D of an ideal p-n junction (with no recombination within the depletion layer) is given by

the Shockley equation [3]:

( 10 = kT qV D e I I , (1.1)

where I 0 is the reverse saturation current of the p-n junction, q is the magnitude of the

charge of electron, T is the absolute temperature, k is the Boltzmanns constant, and V is the

voltage.

The net current is simply the voltage independent I p minus the voltage dependnent I D .

( )10 = kT qV p e I I I , (1.2)

2

R L

I p

ID

I

V


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After a convention adjustment of the current axis direction, the I-V curve is a vertical shift

by I p of I D-V.

Figure 1-2. Current-voltage (I-V) characteristics of ideal diode in light and dark.

Three parameters usually characterize solar cell outputs. One is the short circuit current, I sc.

(I at V=0). Ideally, it is equal to the light generated current, I p. The second parameter is the

open-circuit voltage; V oc. Setting I to zero in Eq. (1.2) gives the ideal value:

+= 1ln

0 I

I

qkT

V p

oc . (1.3)

The third parameter, the fill factor, FF , is defined as

oc sc

mpmp

oc sc V I

V I

V I P

FF == max , (1.4)

where P max is the maximum power output and (I mp, V mp) is the point on the I-V curve at

which P max occurres. FF is a measure of how square the output characteristics are.

The energy-conversion efficiency, , is given by

3

Vmp

Isc

Imp

V

I

Dark

Illuminated

Voc

IP


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, FF GV P

FF V I P

V I oc

in

oc sc

in

mpmp = (1.5)

where the gain factor in sc P I G = and P in is the total power incident on the cell. For an

ideal solar cell the value of G is constant, FF is almost constant, while V oc increases

logarithmically with P in, hence, the efficiency also increases logarithmically with P in, which

is an important fringe benefit of concentration.

1.2.2 The standard real solar cell model

The standard real solar cell model, also known as lumped series resistance model, is one

evolutionary step beyond the ideal model. Real solar cells generally have parasitic series

and shunt resistances associated with them. The standard real model treats them as lumped

resistances as indicated in the solar cell equivalent circuit of Figure 1-3. [3]. Both types of

parasitic resistance act to reduce the fill factor. Very high R s and very low R sh may also

reduce I sc and V oc.

Figure 1-3. Equivalent circuit of a solar cell with parasitic resistances.

The above circuit diagram along with Eq. (1.2) suggests the general solar cell equation, is

given by

4

R sh

R sI

p

ID

I

V


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( ) sh

s s p R

IRV nkT

IRV q I I I

+

+= 1exp0 , (1.6)

where n is called diode quality factor of the solar cell. This factor enables accounting for

mechanisms such as recombination inside the depletion layer while still using only a single

exponent expression. The single exponent expression enables expressing V in (1.7)

explicitly as a function of I . n=1 per junction is attributed to an absence of recombination in

the p-n junction depletion region while n=2 per junction indicate considerable

recombination in the p-n junction depletion region.

The main effect of R sh on the I-V curve of solar cells is near I sc, (because at low voltages the

diode constitutes high resistance). Advanced concentrator solar cells in general and the cell

studied here in particular have very high shunt resistance value. As evidenced in their

horizontal I-V curve near I sc. In this case one can safely neglect the last term of Eq. (1.6)

without introducing a significant error and get,

( )

+= 1exp0nkT

IRV q I I I s p (1.7)

Evaluating equation 1.7 at open-circuit condition ( V = V oc and I = 0 ) yields as in the ideal

model

+= 1ln0 I

I

qnkT

V p

oc . (1.8)

From the above equation we observe that according to the lumped resistance model V oc is

independent of the cells series resistance. Nevertheless, from (1.7) we see that the parasitic

dark current increases exponentially with IR S. Therefore for any given series resistance, at

some flux the efficiency will start to decrease with concentration.

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Figure 1-4 . Schematic diagram showing a typical efficiency variation of a solar cell held at fixedtemperature as the concentration ratio is increasing.

1.3 Novel solar cells

1.3.1 Multi-junction solar cells

At high concentration solar cells comprise a small fraction of the total system cost.

Hence solar cell price becomes less of a concern and the most efficient commercial

solar cells, multi-junction solar cells, are to be used.The principle of multi-junction solar cells is converting different ranges of the spectra

(colors) in different semiconductors (with different band gaps). The incident light first

strikes a semiconductor with a large band gap which exploits the high energy of the short

wavelength photons. Light with energy less than the band gap of the first semiconductor

cant be absorbed, so it passes to the next layer which has a smaller band gap and so on.

Figure 1-5 shows the structure of a lattice matched GaInP 2/GaAs/Ge triple junction tandem

solar cell of the same architecture that was used in the experimental part of this thesis.

Multi-junction solar cells have set the record for achieving high conversion efficiency of

over 40 % [4].

6

Concentration Ratio

Eff icie ncy

Increase due to V ocincrease

Decrease due to seriesresistance

T fixed


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Figure 1-5 Schematic cross-section of the cell under study.

1.3.2 Ultrasmall cells

Principally small cells are ideal for high concentration for three reasons: Series resistance,

heat rejection and compactness. Series resistance is lower since the current has to travel a

shorter way. This is reflected in the potential efficiency and the concentration which the

cell peaks as calculated by Sharp in figure 1-6. [5]

Figure 1-6 Efficiency [%] Vs. Concentration [Suns] for several cell sizes at room temperature

Heat rejection is better since at a given concentration less power is incident upon a smaller

cell. Finally, for a given concentrator aspect ratio, smaller cells dictate shallower

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concentrator, which is especially important for glass filled concentrators, as used in the

latest SolFocus product (figure 1-7) [6]. In conclusion, for high concentration, for a given

collection area, there are advantages for using many small cells rather than fewer bigger

cells of the same concentration.

Figure 1-7 SolFocus (Palo Alto, California) Generation 2 moduleAssembles Spectrolabs 1 mm 2 triple-junction solar cells

1.4 Research Objectives

The most advanced commercial solar energy modules (figure 1-7), yet to be mass

produced, incorporate tiny solar cells under high concentration [6].

The first goal of this thesis is characterizing these tiny cells under real operational

conditions, their: peak efficiency and its concentration; spatial and illumination area

dependency; and architectural cones and pros.

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The second objective of this thesis is verifying the standard model presented above (1.2.2)

in light of the results, or developing a new simple model which account for the spatial

dependency.

The third objective is to offer practical improvements suggestions. The complete

knowledge on the exposed surface rather than the unrevealed bulk focuses these

improvement suggestions in the cells metal grid. Basic questions about its architecture as

well as general innovative possibilities for completely eliminating its shading will be

addressed. Pursuing the potential of the innovative possibility is an important part of this

objective.

Chapter 2 - Experimental Protocol

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2.1 Cell module

2.1.1 Solar cell under study

1mm2

GaInP 2/GaAs/Ge triple-junction monolithic solar cell manufactured by Spectrolab

was used [13]. Its metal grid is uniform parallel structure as shown in Figure 2-1 .

Figure 2-1 photograph (left) and schematic drawing (right) of the studied GaInP 2/GaAs/Ge triple- junction solar cells.

The potential drawbacks of small cells might be associated with edge effects such as

recombination and shunt resistance. The specific architecture of these cells might also

incorporate relatively high parasitic dark current as a result of the high percentage of active

area covered by the busbars (see figure 2-1).

2.1.2 Heat sink (passive cooling) and thermocouple

Passive cooling is just a means of transporting heat from where it is generated (in the PV

cells) to where it can be dissipated (the ambient). The cell was bonded to a thin (1.6 mm)

Kovar plate for robustness. The Kovar plate in turn was mounted on copper plate of

30 40 3 mm 3, which was screwed onto an aluminum (black anodized) heat sink. We used

thermal contact paste to attach the cell to the copper plate. A thermocouple was located

inside the copper plate under the center of the cell. Temperatures measured at the cell-heat

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sink boundary between copper and Kovar plates in the center were no more than 5K above

the (indoor) ambient 296 K.

2.1.3 I-V curve tracer

The I-V tracer was designed and built by our dedicated super-technician Vladimir

Melnichack. It is much faster than the previous I-V tracer that was used in our lab due to

upgrade of the load from a variable resistor to a discharging capacitor. This advantage

ensures fix condition during each measurement. In addition this I-V tracer enables dark-

current measurements.

Picoscope

A B

N.O.N.O.

N.O.

N.O.

Relay1

Relay 2

Relay 3

Relay 4

Rs

PV

N.O. N.O.

Cpacitor

Figure 2-2: I-V tracer. The cells voltage is measured with Picoscope slots A & B while the other slot is used for the current.

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2.2 Illumination module

2.2.1 The mini-dish system

To concentrate the solar radiation we used a parabolic mini-dish system. The set-up of this

mini-dish system included: dual axis tracker, collector, a pizza iris to modulate the power

input and an optical fiber which deliver the collected power into the lab.

Figure 2-3 Mini-dish concentration set-up [7]. a) Outdoors b) Indoors c) Collector d) In reality

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)C( )d(

)b(


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This mini-dish system was originally developed by Feuermann et. al. [7] for maximum-flux

solar surgery applications [8-11]. Though not designed for photovoltaics purposes, taking

advantage of the fact the spectrum shift through the system is insignificant (see figure 2-4),

the mini-dish is adopted as a legitimate solar concentrator for lab purposes, more powerful

than any commercial solar concentrator.

Figure 2-4. Comparison between measured solar spectrum outdoors and through the mini-dishsystem shows excellent agreement. All curves are normalized to one sun (1 mW/mm 2) [12].

2.2.2 Illumination modes

Kaleidoscope - A kaleidoscope works as a flux redistributor. To uniformly distribute the

radiation coming out of the 1mm diameter fiber tip on the square 1 mm by 1 mm active

area within the busbars of the triple-junction solar cell we used a 1mm 2-cross-section

kaleidoscope.

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2000

0.0 0.5 0.1 1 .5 0.2 2 .5

htgnelevaW

)(m

]

t c e p S l a r

r e w o p t i s n e d

/ W ] y m ) 2 - [ ( m

m r o N d e z i l a

n a o t a e r a

1 f o m / W m

m 2 noissimsnart hsid iniM

murtceps ralos roodtuO

0061

0021

008

004

0


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The optical loss within the kaleidoscope is negligible because (a) ray containment is based

on total internal reflection and (b) the optical path length is of the order of centimeter,

which, when combined with low attenuation coefficients, yields insignificant absorption.

LIP In a Local illumination Probe (LIP) the fiber nearly touches the cell (~5

microns gap) therefore the illumination profile is the area delimitated by the fiber.

Either 1.0 or 0.6 mm fibers were used. Both imitate realistic irradiation conditions.

(The latter is occasionally used to provide tolerances for tracking errors, so even

when the illuminated area is decentralized, it is still within the solar cells active

area).

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Figure 2-5: Areas of the 0.6 and 1.0 mm LIP.

2.2.3 Input power

All I-V curve measurements were taken during clear-sky periods, two hours about solar

noon, over the course of one year in Sde Boqer, Israel (latitude 30.9N) where the clear-sky

global and direct-beam solar spectra at mid-day are nearly invariant and close to the AM1.5

spectrum commonly used in PV testing [14-15].

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We measured P in with a spectrum blind pyrometer of 5% accuracy, calibrated both

calorimetrically and against a precision solar pyranometer. P in was measured before and

after each LIP illumination. In the scarce case of disagreement between the two P in

measurements, the I-V curve was disqualified.

Power input was modulated by a pizza-slice iris that preserves the angular distribution of

delivered sun light was mounted on the window of the mini-dish.

Figure 2-6 Experiment setup in lab

S o la rC e ll

X Y ZTranslator

Power meter 5% Accuracy

SpectrumblindPyrometer

Fiber

Heatsink

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3. Model for realistic operational conditions

3.1. Quantifying temperature effectsAn important effect in realistic operational conditions, such as those simulated in the

experimental part of this thesis, is temperature elevation at high power input. Though the

thermocouple showed only a small increase in temperature, the open circuit voltage

measurements suggest that the cells temperature was higher, as will be discussed in the

results chapter.

3.1.1. Voltage dependence

Since the current is less sensitive to temperature variation [16], let us try to obtain T V

or

express it in terms of the well documented and easily measured T V OC / . Success in this

exercise enables predicting temperature correction for V(I) curves.

Extracting the voltage from (1.7) leads to

S I I I I

qn kT R I V P =

+)ln (

0

0 (3.1)

In order to differentiate the voltage with respect to temperature, let us find each

components temperature dependence. The band gap has fairly linear-weak dependency at

the relevant temperature range:

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Figure 3-1: Bandgap temperature dependence [3]. The right hand is a zoom in for Si.

Therefore I 0 has exponential dependence:

)e x p ()e x p ('0 kT E

n k T E G OG aa I == (3.2)

Where a s temperature dependence)2/3(~ +T is not important compared with the

exponential term [3], q and k are temperature independent. R S and n are assumed to be

independent of temperature as well. As mentioned above, I has a weak temperature

dependence; therefore the dependence of Ln(I P-I+I 0) Ln(I P-I) on temperature is evenweaker.

Now (3.1) is ready to be differentiated to yield:

(3.3)

)]ln ()[ln ()ln ()[ln ( a I I Ra I I P qn k T I S kT E

kT E

P qn k

T V G OG O +=

There is no need to estimate the constant a in (3.3) in order to express T V / in terms of

T V OC / . Substituting in T V OC / : IP IP-I yields the correct expression for

T V / at I P, I.

))0(,''()','( =====

I I I I I I I I P P T

V P P T

V O C (3.4)

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Since T V / is almost independent of temperature (3.3), V(T) can be easily obtained:

)()''()',',(

)()','()',',(

)',','(

re f P P T V

P P re f

re f P P T V

P P re f

P P

T T I I I I I I I T T V

T T I I I I I I I I T T V

I I I I T T V

OC =+====

==+====

====

(3.5)

Assuming T=T-T ref is proportional to the power input, (3.5) becomes:

IN P P T V

re f P P P R I I I T T V I I I I T T V O C =+====

)''()()',','( (3.6)

Where R is the heat resistance.

3.1.2. Heat resistance

Let us try estimating / predicting what the proportionality constant in (3.6), R, should be.

The thickest cell layer is by far the Germanium substrate. Its heat conductance coefficient is

60 W/m = 60 mW/mm. Namely, one degree difference on the terminals of a 0.15 mm

thick Germanium of a 1 mm 2 cross section will induce a heat flow of 601/0.15=400 mW.

Since the active region of the cell is an order of magnitude shallower than the rest of the

substrate, but of nearly the same conductance coefficient, a one degree difference on the

terminals of the active region will induce a 4 Watt heat flow. Though this approximation is

very crude, the bottom line is that even for the maximum irradiation the cell was exposed,

the depletion layer should have heated up by the order of 1 degree. Nevertheless, when it

comes to practical considerations, such a thin cell can not be mounted directly on the heat

sink. It must first be mounted on an intermediate layer. The reasons are the manufacturer

cannot deliver a product as thin as 160 microns (it is too fragile) and the cell should be

mounted on a layer with thermal expansion similar to Germanium. Ultimately, this layer

should be constituted as a good heat conductor. Apparently in the architecture of these cells

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this point was not the main concern. Eventually, a 1.6 mm gold-coated Kovar whose heat

conductance is only 16.6 mW/mmK was used as the intermediate layer. It is not obvious

whether there is a better solution because complicated technological processes having to do

with the cell encapsulation exclude usage of many materials. The Kovar layer results in an

increase in the order of magnitude of temperature difference. The detailed temperature

profile and its detailed effect are difficult to calculate or measure directly. However, fitting

the experimental results to the model for V(T=T REFF ) described in the following section,

supported by the corrections for temperature effects as described in this section (3.6),

agrees with the order of magnitude of the heat resistance coefficient R.

kovar

Gold

Ge

Silver appoxy

GaIP

GaAs

Gold

Alumina ceramic 96%

Figure 3-2: Solar cell layers: 1.6 mm thick Kovar layer is the main heat flow resistor. Germanium is

160 microns thick and the rest of the layers are an order of magnitude thinner.

3.2. New semi-analytic model for non-uniform illumination

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Figure 3.3: Different locations within the cell will correspond to different building blocks of theequivalent circuit used by the distributed series resistance model. [17]

The main disadvantage of the distributed series resistance model is it can not be solved

analytically; hence, it is not physically transparent, and it is difficult to deduce insights.

A far simpler model constituted from only two unit cells, illuminated and shaded, was used

in our group to account for the voltage drop effect for the LIP-tested cells [18]. As an

exercise I formulated a similar model which is a bit more complicated to derive but easier

to use since it yields explicit V( I ) dependency.

3.2.2. Model assumptions

General assumptions:

1) Superposition assumption: between photogenerated current I p and dark current I d:

I=I p-Id.

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2) Concentrator photovoltaics (CPV) assumption: I p= CI SC@1sun, where C stands for

concentration in suns. This relation expresses the observation that at 1 sun, I SC=I p since

the series resistance effect is negligible in small currents in high quality cells such as

CPV.

3) Dark currents voltage dependence: = A

T V

y x J V

d dxdy J I ]1[exp),(

0

where J 0 is the saturation current density and V J is the junction voltage.

The key difference between all the models mentioned in the previous section is the

evaluation of the above integral. The lumped series resistance assumes V J = V load + IR S

(location independent). The other two models discretisize the integral, thus changing it to

summation. The value of each V J in the sum is found iteratively by solving conjugated

implicit equations.

In this model the above integral will be estimated in a more intuitive way. For clarity, a

specific case of highly localized irradiation profile (HLIP) will be considered first. f

[illuminated area] / [S.C. area including busbars]


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) )1/1(()( m i n +++ f I I r I I r V P sa te d il l ud a r k

s c s(3.7)

where r s (or R S) is the cells resistance under uniform illumination.

The simplest topography which obeys the two constraints imposed for V J is a pyramid. Its

base is V L high and its peak is V L + V + high. The realistic potential drop is expected to be

linear along the metal fingers, as is a pyramid. Yet, across the metal fingers the realistic

potential drop should have a positive second derivative (positive curvature). To account for

this fact, the width of the pyramid is reduced (or equivalently, the effective area of the

busbars is increased), resulting in a pyramid base area somewhat smaller than the cells

active area.

3.2.3. Derivation

3.2.3.1 HLIP (Highly localized irradiation profile)

Under the pyramid assumption, I ds value can be obtained using the identity:

)()ln()1()(0

x Ei xe x Ein t dt x

t +=

where

x

dt t

t x Ei

)exp()( , 0.58 is Euler gamma constant.

}1)1)]/ln()/([/()/{exp( 220 += +++ T T T T Ld V V V V EiV V V V I I (3.8)

where:

2 pyramid base area / Cells Gross area (including busbars). f< 2


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The busbars shading effect is buried in 2 as well.

VT=nkT/q , ))1/1(()( ++ f I I r V V MaxV P s L J

4) Rearrangement and taking advantage of assumption 1 (I=I p-Id) yields:

)ln( )1)]/ln()/([/( 2200

++

+++=

T T T

P

L V V V V EiV V I

I I I T

hotspot V V (3.9)

3.2.3.2 Finite disc irradiation

The HLIP irradiation pattern model can be expanded to a finite disc. It will be further

assumed that a disc irradiation is equivalent to irradiation of a square of the same size of the

disc. Again, the pyramid simplification will be applied, but this time a truncated pyramid

will be used:

Figure 3-4: Truncated pyramid simulates the lateral voltage profile of finite area illumination

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y

V J

VJ

=VL

VJ VL +V+

V+

x


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The dark current can be expressed as the sum of three terms 321 4 I I I I d ++= where I 1,2,3

are the dark currents described in the figure (3-5).

Figure 3-5: Dark current parts following the truncated pyramid profile

Each of the dark currents can be easily derived under the assumptions made in the HLIP

model:

)1(exp1 +

= +T

Lo V

V V f I I where again: ))1/1(( ++ f I I r V P s

I1 I

2

I2

I2

I2

I3/4 I

3/4

I3/4 I

3/4

L1

L2

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}1)1)1( e x p({ e x p

)1( e x p)1(

)1)(

( e x p

2

2

0

2/12

}

{2

+=

+

++=

+

+

+

T

T

T

L f f o

T

L

L

T

Lo

V V

V V

V V

I

V V

L

d xV L xV V

L J I

Following the HLIP assumption of linear voltage drop along each of the two axes, I 3 is an

integral over a pyramid. The solution of this integral is given by (3.8), with a minor

modification: the area of integration is 2)1( f of the cells area:

}1)1)]/ln()/([/()/{exp()1( 22203 += +++ T T T T L V V V V EiV V V V f I I

VL can be obtained explicitly by rearranging I I I I I I P d

=++=321

4 :

}

{

) }/e x p ()1)1)/( e x p (/()(2

)1) ]/l n ()/([/()1l n { ()l n ( 2220

0

T T T

T T T p

V V V V V V

V V V V E iV V V V

f f f

f I I I I

T L

+++

+++

+

+++

++=

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(3.10)

3.3 Analyzing the model

There is one qualitative counter intuitive prediction of this model worth mentioning. For

f


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increased more than I P, the local dark current at the periphery will decrease, making the cell

operate at a lower voltage working point in the periphery; hence, V OC, which is equal to the

voltage of the dark current working point, is decreased.

Since this result is counter intuitive let us again check the assumptions it stems from: (a)

There is a voltage gradient across the cell which peaks at the illuminated area . At V OC

there is a kind of current circulation which produces this voltage gradient. The local dark

current is proportional to the local voltage. This is a mature assumption in solid state

physics. (b) The local voltage across the cell is influenced mainly by the currents directly

beneath it . This assumption is justified by the shallow location of the depletion layers,only a few microns below the cell surface. (c) V + is proportional to P in. This is the least

rigorous assumption. Nevertheless, it makes sense that the potential difference between the

depletion layer and the cell surface is proportional to the vertical current density (between

the depletion layer and the cell surface), which is nearly proportional to P in at the

illuminated area. (The fact that the cells under study are triple rather than single junction is

irrelevant for this models sake). The potential difference is not necessarily identical to V +

since the depletion layer is not an equipotential surface. However, the local dark current is

exponentially dependent on the potential difference, so it is more of a semantic issue. In

conclusion, if the potential difference is indeed proportional to I P and the other two

assumptions are correct, it is possible to have dV OC/dP IN


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